The 3D Coordinates System
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Equations of Lines
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Equations of Planes
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Quadratic Surfaces
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Functions of Several Variables
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Vector Functions in 3D Coordinates System
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Vector Calculus in 3D Coordinates System
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Tangent, Normal and Binormal Vectors
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Arc Length with Vector Functions
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Curvatures in 2D and 3D Space
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Velocity and Acceleration in Space
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Cylindrical and Spherical Coordinate Systems
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Integrals of Vector Functions
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[{"Name":"The 3D Coordinates System","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System","Duration":"19m 22s","ChapterTopicVideoID":9879,"CourseChapterTopicPlaylistID":57474,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9879.jpeg","UploadDate":"2021-01-26T03:46:07.5670000","DurationForVideoObject":"PT19M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.740","Text":"We\u0027re starting a new topic."},{"Start":"00:01.740 ","End":"00:05.310","Text":"The 3D or 3-dimensional coordinate system."},{"Start":"00:05.310 ","End":"00:08.040","Text":"Up till now, we\u0027ve been dealing in the 2-dimensional,"},{"Start":"00:08.040 ","End":"00:11.475","Text":"which is the plane and X and Y."},{"Start":"00:11.475 ","End":"00:14.850","Text":"Now we\u0027re going to talk about 3 dimensional space,"},{"Start":"00:14.850 ","End":"00:16.410","Text":"and we need an extra coordinate,"},{"Start":"00:16.410 ","End":"00:18.825","Text":"and this is going to be Z."},{"Start":"00:18.825 ","End":"00:22.905","Text":"Let\u0027s start by taking a typical point in space."},{"Start":"00:22.905 ","End":"00:28.830","Text":"I\u0027ll give it a name P. To give you an impression of space,"},{"Start":"00:28.830 ","End":"00:34.215","Text":"I\u0027ve dropped perpendiculars to the 3-coordinate planes."},{"Start":"00:34.215 ","End":"00:41.670","Text":"This here, the plane containing these 2 axes is called the xy-plane,"},{"Start":"00:41.670 ","End":"00:49.560","Text":"and the plane containing the Z and the y-axis is called the"},{"Start":"00:49.560 ","End":"01:01.065","Text":"yz-plane and the plane containing the Z and the x-axis would be called the xz-plane."},{"Start":"01:01.065 ","End":"01:08.130","Text":"Also, I\u0027ve only shown the positive directions of the axes."},{"Start":"01:08.130 ","End":"01:10.699","Text":"At the moment, we don\u0027t need the negative,"},{"Start":"01:10.699 ","End":"01:12.740","Text":"it just complicates the picture."},{"Start":"01:12.740 ","End":"01:17.695","Text":"But yeah, these axes can continue in the other direction too."},{"Start":"01:17.695 ","End":"01:20.930","Text":"How do I find the coordinates of this point?"},{"Start":"01:20.930 ","End":"01:22.760","Text":"Well, there\u0027s going to be 3 coordinates."},{"Start":"01:22.760 ","End":"01:27.325","Text":"One way to do it is to take this point,"},{"Start":"01:27.325 ","End":"01:33.305","Text":"and I\u0027ll call it Q, and just drop perpendiculars in the xy-plane."},{"Start":"01:33.305 ","End":"01:36.920","Text":"Let\u0027s say this is the point where it hits"},{"Start":"01:36.920 ","End":"01:42.390","Text":"that will be called A the distance from here to here,"},{"Start":"01:42.390 ","End":"01:45.315","Text":"and from here to here will be B,"},{"Start":"01:45.315 ","End":"01:48.440","Text":"and the height of this would be C or"},{"Start":"01:48.440 ","End":"01:51.725","Text":"you could just take something with the same height on here."},{"Start":"01:51.725 ","End":"01:55.735","Text":"Hard to say exactly where, probably here."},{"Start":"01:55.735 ","End":"02:05.210","Text":"This distance is also C. Then this point becomes the point a,"},{"Start":"02:05.210 ","End":"02:12.965","Text":"b, c. Actually, I want to give the coordinates of the other points."},{"Start":"02:12.965 ","End":"02:18.839","Text":"I\u0027ll call this one R and this one S,"},{"Start":"02:18.839 ","End":"02:27.900","Text":"after P and Q come R and S. This would be the point where on the zy-plane,"},{"Start":"02:27.900 ","End":"02:29.640","Text":"the X is 0."},{"Start":"02:29.640 ","End":"02:33.705","Text":"A here is 0, so this is the point 0, b,"},{"Start":"02:33.705 ","End":"02:42.680","Text":"c. Similarly, the point S is a,"},{"Start":"02:42.680 ","End":"02:47.915","Text":"0, c. Here\u0027s where the Y is 0 on the xz-plane."},{"Start":"02:47.915 ","End":"02:51.650","Text":"On the xy-plane, the Z is 0,"},{"Start":"02:51.650 ","End":"02:57.295","Text":"so this would be the point a, b, 0."},{"Start":"02:57.295 ","End":"03:01.805","Text":"Another term I want to mention is the term projection."},{"Start":"03:01.805 ","End":"03:10.505","Text":"In this case, we would have that Q is the projection of P onto the xy-plane,"},{"Start":"03:10.505 ","End":"03:18.055","Text":"and R is the projection of P onto the zy-plane,"},{"Start":"03:18.055 ","End":"03:26.970","Text":"and S is the projection of P onto the xz-plane."},{"Start":"03:26.970 ","End":"03:31.210","Text":"Another point of notation, in general,"},{"Start":"03:31.210 ","End":"03:36.050","Text":"and we want to talk about 3D or any number of dimensions,"},{"Start":"03:37.940 ","End":"03:45.920","Text":"Gothic R on the line here represents the real numbers or that\u0027s 1 dimension."},{"Start":"03:47.330 ","End":"03:52.440","Text":"When we write R with a 2 or R-squared,"},{"Start":"03:52.440 ","End":"03:55.785","Text":"that\u0027s 2-dimensions or the plane."},{"Start":"03:55.785 ","End":"03:58.545","Text":"If I write R^3,"},{"Start":"03:58.545 ","End":"04:03.180","Text":"that would be our 3D space,"},{"Start":"04:03.180 ","End":"04:06.510","Text":"like 3 real numbers, a, b,"},{"Start":"04:06.510 ","End":"04:08.955","Text":"and c. In general,"},{"Start":"04:08.955 ","End":"04:12.320","Text":"we could have any number of dimensions."},{"Start":"04:12.320 ","End":"04:18.270","Text":"R^n would be n-dimensional space,"},{"Start":"04:18.430 ","End":"04:23.255","Text":"where n could be any positive whole number."},{"Start":"04:23.255 ","End":"04:26.510","Text":"This is 1D, which is just a line."},{"Start":"04:26.510 ","End":"04:28.750","Text":"R^2 is 2D."},{"Start":"04:28.750 ","End":"04:30.540","Text":"This is like a line,"},{"Start":"04:30.540 ","End":"04:32.490","Text":"this would be like a plane,"},{"Start":"04:32.490 ","End":"04:34.800","Text":"this would be like space,"},{"Start":"04:34.800 ","End":"04:36.990","Text":"and R^n is more abstract,"},{"Start":"04:36.990 ","End":"04:41.280","Text":"n-dimensional space, to say."},{"Start":"04:41.280 ","End":"04:44.280","Text":"Some terminology."},{"Start":"04:44.280 ","End":"04:47.930","Text":"Many of the formulas that we learned in 2D,"},{"Start":"04:47.930 ","End":"04:51.030","Text":"they can be generalized, not always,"},{"Start":"04:51.030 ","End":"04:54.150","Text":"but they often can be generalized to 3D."},{"Start":"04:54.150 ","End":"04:56.165","Text":"I want to give you the first example."},{"Start":"04:56.165 ","End":"04:59.870","Text":"Suppose I want to know the formula for the distance between 2 points."},{"Start":"04:59.870 ","End":"05:04.340","Text":"Let\u0027s say I have point p_1, which is say,"},{"Start":"05:04.340 ","End":"05:09.600","Text":"a_1, b_1, c_1, or x_1, y_1, z_1,"},{"Start":"05:09.600 ","End":"05:12.285","Text":"whatever, and we have another point,"},{"Start":"05:12.285 ","End":"05:16.875","Text":"p_2, which is a_2, b_2,"},{"Start":"05:16.875 ","End":"05:25.565","Text":"c_2, and I want to know the distance between the points p_1 and p_2."},{"Start":"05:25.565 ","End":"05:27.380","Text":"I don\u0027t know what that is,"},{"Start":"05:27.380 ","End":"05:32.900","Text":"but I can make a good guess if I take the 2-dimensional case."},{"Start":"05:32.900 ","End":"05:39.350","Text":"I\u0027ll use a different color so we know that this is the 2D and this is the 3D."},{"Start":"05:39.350 ","End":"05:42.800","Text":"In 2D, a point would just have 2 coordinates,"},{"Start":"05:42.800 ","End":"05:46.765","Text":"say a_1 and b_1."},{"Start":"05:46.765 ","End":"05:50.610","Text":"Let say another point which was a_2,"},{"Start":"05:50.610 ","End":"05:56.280","Text":"b_2, then the distance between the 2 points, p_1,"},{"Start":"05:56.280 ","End":"06:04.155","Text":"p_2 in 2-dimensions would be the square root of the difference between the a\u0027s,"},{"Start":"06:04.155 ","End":"06:12.180","Text":"let\u0027s say a_2 minus a_1 squared plus the difference between the b\u0027s,"},{"Start":"06:12.180 ","End":"06:16.780","Text":"b_2 minus b_1 squared."},{"Start":"06:17.060 ","End":"06:23.535","Text":"Sorry, the 2 is outside the brackets, of course."},{"Start":"06:23.535 ","End":"06:31.250","Text":"What I would do is I would generalize this by"},{"Start":"06:31.250 ","End":"06:39.890","Text":"adding c_2 minus c_1 squared and just extending the square root sign."},{"Start":"06:39.890 ","End":"06:45.320","Text":"This gives me a formula for the distance in 3 dimensions, very similar."},{"Start":"06:45.320 ","End":"06:49.535","Text":"The proof for the 2D case was based on Pythagoras\u0027 theorem."},{"Start":"06:49.535 ","End":"06:52.730","Text":"This is based on the generalized Pythagoras theorem,"},{"Start":"06:52.730 ","End":"06:57.215","Text":"which we actually mentioned when we talked about the magnitude of vectors."},{"Start":"06:57.215 ","End":"07:00.120","Text":"Anyway, you don\u0027t have to know the reason,"},{"Start":"07:00.120 ","End":"07:02.570","Text":"you can just accept it as a formula."},{"Start":"07:02.570 ","End":"07:06.725","Text":"I\u0027ll give another example of generalizing from 2D to 3D."},{"Start":"07:06.725 ","End":"07:10.980","Text":"Let\u0027s take the equation of a circle in 2D."},{"Start":"07:12.430 ","End":"07:21.360","Text":"A circle is given by a center and say that the center, say,"},{"Start":"07:21.360 ","End":"07:28.530","Text":"is h, k, and the radius,"},{"Start":"07:28.530 ","End":"07:33.160","Text":"we need to know so let\u0027s call that r."},{"Start":"07:33.750 ","End":"07:40.090","Text":"The equation for this circle is x minus h"},{"Start":"07:40.090 ","End":"07:48.430","Text":"squared plus y minus k squared equals r squared."},{"Start":"07:48.430 ","End":"07:50.350","Text":"You may have seen this with different letters,"},{"Start":"07:50.350 ","End":"07:52.255","Text":"but I\u0027m sure you\u0027ve come across it."},{"Start":"07:52.255 ","End":"07:57.715","Text":"Now, the generalization to 3D is not a circle,"},{"Start":"07:57.715 ","End":"08:00.805","Text":"it\u0027s actually a sphere."},{"Start":"08:00.805 ","End":"08:03.205","Text":"Because if you think about it,"},{"Start":"08:03.205 ","End":"08:10.840","Text":"the points with equal distance r to another point in 2D give us a circle."},{"Start":"08:10.840 ","End":"08:17.455","Text":"But in 3D, the points equidistant from a given point actually form a sphere."},{"Start":"08:17.455 ","End":"08:23.140","Text":"In 3D, let\u0027s say we want to know the equation of a sphere and let\u0027s"},{"Start":"08:23.140 ","End":"08:28.870","Text":"say we know its center is lets say h,"},{"Start":"08:28.870 ","End":"08:37.810","Text":"k, l, and we\u0027ll also have a radius of r. But this time,"},{"Start":"08:37.810 ","End":"08:42.160","Text":"the equation it\u0027s generalized from this,"},{"Start":"08:42.160 ","End":"08:47.680","Text":"it\u0027s x minus h squared plus y minus k squared."},{"Start":"08:47.680 ","End":"08:56.020","Text":"You just do the natural thing of adding z minus l squared also equals r squared."},{"Start":"08:56.020 ","End":"08:58.780","Text":"That\u0027s the equation of a sphere."},{"Start":"08:58.780 ","End":"09:02.170","Text":"It basically comes from the distance formula,"},{"Start":"09:02.170 ","End":"09:06.430","Text":"that the distance of x, y,"},{"Start":"09:06.430 ","End":"09:07.960","Text":"z from the point h, k,"},{"Start":"09:07.960 ","End":"09:09.940","Text":"l is r only,"},{"Start":"09:09.940 ","End":"09:12.144","Text":"it\u0027s like this formula but squared."},{"Start":"09:12.144 ","End":"09:18.310","Text":"Anyway, we now have the equation of a sphere in 3D."},{"Start":"09:18.310 ","End":"09:22.930","Text":"Let\u0027s take another example which actually"},{"Start":"09:22.930 ","End":"09:27.445","Text":"uses all of these and shows you some of the differences."},{"Start":"09:27.445 ","End":"09:37.480","Text":"I\u0027d like to take an example of an equation and the equation will be x equals 3."},{"Start":"09:37.480 ","End":"09:40.374","Text":"But I want to interpret it in 1D,"},{"Start":"09:40.374 ","End":"09:44.050","Text":"in 2D, and in 3D because we have an x in all of them."},{"Start":"09:44.050 ","End":"09:45.130","Text":"Here we have x, here we have x,"},{"Start":"09:45.130 ","End":"09:47.050","Text":"y and we have x, y, z."},{"Start":"09:47.050 ","End":"09:53.395","Text":"In 1 dimension, x equals 3 on the number line is just a point."},{"Start":"09:53.395 ","End":"09:56.390","Text":"Here\u0027s what it looks like in 1D."},{"Start":"09:56.910 ","End":"10:01.015","Text":"It\u0027s just a point on the number line, this."},{"Start":"10:01.015 ","End":"10:05.005","Text":"Let\u0027s now look at it in 2D in the x, y plane."},{"Start":"10:05.005 ","End":"10:08.080","Text":"If x equals 3, y is not restricted,"},{"Start":"10:08.080 ","End":"10:11.455","Text":"so we should get a vertical line through x equals 3."},{"Start":"10:11.455 ","End":"10:14.365","Text":"Here\u0027s what it looks like in 2D."},{"Start":"10:14.365 ","End":"10:20.529","Text":"The same equation in 3D means that the x coordinate is 3,"},{"Start":"10:20.529 ","End":"10:23.905","Text":"the first coordinate, but the other 2 could be anything."},{"Start":"10:23.905 ","End":"10:27.085","Text":"Actually what we get is a plane."},{"Start":"10:27.085 ","End":"10:29.800","Text":"It\u0027s a bit difficult to sketch,"},{"Start":"10:29.800 ","End":"10:36.265","Text":"but this is the x-axis and this would be the point where x equals 3 on the x axis."},{"Start":"10:36.265 ","End":"10:44.709","Text":"What you do is you just take a plane which is orthogonal perpendicular to this x axis,"},{"Start":"10:44.709 ","End":"10:47.620","Text":"and this is what it looks like."},{"Start":"10:47.620 ","End":"10:50.395","Text":"It\u0027s a plane through x equals 3,"},{"Start":"10:50.395 ","End":"10:53.125","Text":"and y and z can be anything."},{"Start":"10:53.125 ","End":"10:58.705","Text":"Now, this actually brings me to a point,"},{"Start":"10:58.705 ","End":"11:06.310","Text":"no pun intended, that we can now describe these 3 coordinate planes."},{"Start":"11:06.310 ","End":"11:10.330","Text":"The x, y plane, for example."},{"Start":"11:10.330 ","End":"11:15.595","Text":"Just write that the x, y plane,"},{"Start":"11:15.595 ","End":"11:17.920","Text":"which is just this here,"},{"Start":"11:17.920 ","End":"11:22.030","Text":"is given by the equation z equals 0."},{"Start":"11:22.030 ","End":"11:25.540","Text":"Because everywhere on this plane we have something of the form any x,"},{"Start":"11:25.540 ","End":"11:27.475","Text":"any y but no z."},{"Start":"11:27.475 ","End":"11:30.160","Text":"Similarly, if I take the x,"},{"Start":"11:30.160 ","End":"11:36.160","Text":"z plane, then its equation actually, it\u0027s the missing 1."},{"Start":"11:36.160 ","End":"11:37.585","Text":"Or you can do it, look what\u0027s missing."},{"Start":"11:37.585 ","End":"11:45.895","Text":"What\u0027s missing is y. y is restricted to being 0."},{"Start":"11:45.895 ","End":"11:49.285","Text":"When y is 0, x and z can be anything,"},{"Start":"11:49.285 ","End":"11:50.710","Text":"but this is where y is 0."},{"Start":"11:50.710 ","End":"11:53.200","Text":"So it\u0027s this plane here. I won\u0027t shade it."},{"Start":"11:53.200 ","End":"11:56.530","Text":"It might look messy. But it looks something like this,"},{"Start":"11:56.530 ","End":"12:01.475","Text":"that\u0027s through this line and this line and the last 1, which is the y,"},{"Start":"12:01.475 ","End":"12:05.760","Text":"z plane, which would be the back wall here,"},{"Start":"12:05.760 ","End":"12:07.575","Text":"including the z-axis and the y-axis,"},{"Start":"12:07.575 ","End":"12:11.745","Text":"this plane is where simply x is equal to 0."},{"Start":"12:11.745 ","End":"12:16.150","Text":"We have 3 equations of the coordinate plane and I\u0027ve shown you how"},{"Start":"12:16.150 ","End":"12:21.460","Text":"an equation of a plane could be by just restricting 1 of the variables to be a constant."},{"Start":"12:21.460 ","End":"12:26.830","Text":"Here\u0027s an example of the similarity or differences between the different dimensions."},{"Start":"12:26.830 ","End":"12:31.585","Text":"Now I want to take another example which just applies to 2 and 3 dimensions."},{"Start":"12:31.585 ","End":"12:34.970","Text":"I need some more space here."},{"Start":"12:34.970 ","End":"12:41.050","Text":"The question will be to graph"},{"Start":"12:41.050 ","End":"12:48.295","Text":"the equation y equals 2x minus 3 but in 2 scenarios."},{"Start":"12:48.295 ","End":"12:49.960","Text":"In 2D, in other words,"},{"Start":"12:49.960 ","End":"12:57.025","Text":"in the case of R^2 and in 3D in other words, in R^3."},{"Start":"12:57.025 ","End":"12:59.245","Text":"Let\u0027s see how they differ."},{"Start":"12:59.245 ","End":"13:03.010","Text":"Here\u0027s the graph in 2D and you would have known how to do"},{"Start":"13:03.010 ","End":"13:06.189","Text":"this you could choose either define the intercepts."},{"Start":"13:06.189 ","End":"13:08.590","Text":"For example, when x is 0,"},{"Start":"13:08.590 ","End":"13:12.730","Text":"then y is minus 3 and when y is 0,"},{"Start":"13:12.730 ","End":"13:15.820","Text":"x is 3 over 2, 1 and half."},{"Start":"13:15.820 ","End":"13:18.625","Text":"You can get this point, draw a line through it"},{"Start":"13:18.625 ","End":"13:22.285","Text":"or you could use the slope and intercept method."},{"Start":"13:22.285 ","End":"13:26.620","Text":"Again, you\u0027d need the minus 3 and then you draw a line with slope 2 by going"},{"Start":"13:26.620 ","End":"13:31.840","Text":"some number of units across and then twice that number of units up."},{"Start":"13:31.840 ","End":"13:34.465","Text":"Note that these are different scales on x and y."},{"Start":"13:34.465 ","End":"13:37.600","Text":"Anyway, that\u0027s y equals 2x minus 3 in 2D."},{"Start":"13:37.600 ","End":"13:44.140","Text":"Now in 3D, this would be how it would look from above, a cross-section."},{"Start":"13:44.140 ","End":"13:48.625","Text":"What we would do is take this line and extend it vertically,"},{"Start":"13:48.625 ","End":"13:51.460","Text":"infinitely up and down."},{"Start":"13:51.460 ","End":"13:56.230","Text":"It would look something like this here."},{"Start":"13:56.230 ","End":"14:04.270","Text":"Well, we\u0027ve twisted it so that the y-axis is here and the x-axis is here."},{"Start":"14:04.270 ","End":"14:06.985","Text":"Then we also added another dimension."},{"Start":"14:06.985 ","End":"14:11.005","Text":"This would be this line and you need a vertical plane through it."},{"Start":"14:11.005 ","End":"14:12.745","Text":"That\u0027s how it looks like."},{"Start":"14:12.745 ","End":"14:16.840","Text":"In general, a linear equation,"},{"Start":"14:16.840 ","End":"14:19.420","Text":"it turns out, will always be a plane in 3D,"},{"Start":"14:19.420 ","End":"14:21.595","Text":"but in 2D it\u0027s a line."},{"Start":"14:21.595 ","End":"14:23.845","Text":"That\u0027s another difference."},{"Start":"14:23.845 ","End":"14:30.730","Text":"Now this concept of drawing a graph in 2D without z and then adding the z"},{"Start":"14:30.730 ","End":"14:34.060","Text":"in will always have the same result of taking whatever graph it is"},{"Start":"14:34.060 ","End":"14:38.290","Text":"here and extending it infinitely upwards and downwards."},{"Start":"14:38.290 ","End":"14:40.900","Text":"I\u0027ll give another example of that,"},{"Start":"14:40.900 ","End":"14:43.060","Text":"instead the line, we\u0027ll take circle."},{"Start":"14:43.060 ","End":"14:45.940","Text":"For this we want a different equation."},{"Start":"14:45.940 ","End":"14:50.005","Text":"What we want is the graph of a circle."},{"Start":"14:50.005 ","End":"14:56.980","Text":"Let\u0027s take x squared plus y squared equals 4."},{"Start":"14:56.980 ","End":"14:59.785","Text":"We\u0027ve seen the equation of a circle."},{"Start":"14:59.785 ","End":"15:02.590","Text":"The center of the circle like this is 0,"},{"Start":"15:02.590 ","End":"15:07.100","Text":"0 because it\u0027s x minus 0 squared plus y minus 0 squared."},{"Start":"15:07.410 ","End":"15:10.915","Text":"The center is 0, 0."},{"Start":"15:10.915 ","End":"15:13.540","Text":"Well, 4 is 2 squared,"},{"Start":"15:13.540 ","End":"15:19.070","Text":"so radius is 2."},{"Start":"15:20.070 ","End":"15:22.970","Text":"Here we are in 2D."},{"Start":"15:22.970 ","End":"15:28.940","Text":"Now in 3D, what we would do if we extend this vertically up and down infinitely,"},{"Start":"15:28.940 ","End":"15:32.355","Text":"which like putting vertical lines through every point,"},{"Start":"15:32.355 ","End":"15:34.765","Text":"clearly we\u0027ll get a cylinder and in fact,"},{"Start":"15:34.765 ","End":"15:36.640","Text":"here\u0027s what it looks like."},{"Start":"15:36.640 ","End":"15:40.010","Text":"It\u0027s a bit hard to sketch the 3D stuff."},{"Start":"15:40.010 ","End":"15:43.160","Text":"But for example, this point here where x is 2,"},{"Start":"15:43.160 ","End":"15:46.820","Text":"that would correspond to the point on the x-axis,"},{"Start":"15:46.820 ","End":"15:54.610","Text":"which is 2 here and this point where y is 2 would be over here somewhere."},{"Start":"15:54.610 ","End":"15:57.400","Text":"Then we just extend it upwards."},{"Start":"15:57.400 ","End":"16:06.440","Text":"In fact, this whole part here would be the part that\u0027s in the x,"},{"Start":"16:06.440 ","End":"16:08.270","Text":"y plane, that would be the original circle,"},{"Start":"16:08.270 ","End":"16:12.810","Text":"but we get a lot more because that is unrestricted z, sorry."},{"Start":"16:13.480 ","End":"16:18.860","Text":"Now so far, we\u0027ve typically seen that in 3D we get a surface,"},{"Start":"16:18.860 ","End":"16:22.970","Text":"we had some planes and now we have a cylinder."},{"Start":"16:22.970 ","End":"16:25.535","Text":"Usually this is the case."},{"Start":"16:25.535 ","End":"16:29.695","Text":"Typically as an equation will give a surface rather than a curve."},{"Start":"16:29.695 ","End":"16:31.820","Text":"A line is a curve, a circle is a curve,"},{"Start":"16:31.820 ","End":"16:39.315","Text":"but a plane and a cylinder are surfaces but there are ways of drawing curves in 3D."},{"Start":"16:39.315 ","End":"16:41.150","Text":"We\u0027ll see more of that later."},{"Start":"16:41.150 ","End":"16:43.445","Text":"But just for now, I could give you an example."},{"Start":"16:43.445 ","End":"16:49.880","Text":"Suppose I wanted the circle that went through z equals 4."},{"Start":"16:49.880 ","End":"16:51.560","Text":"Let me see where is 4, 0,"},{"Start":"16:51.560 ","End":"16:55.870","Text":"1, 2, 3, I don\u0027t know."},{"Start":"16:55.870 ","End":"16:58.300","Text":"I think this is 4 here."},{"Start":"16:58.300 ","End":"17:02.415","Text":"Let\u0027s say this was z equals 4."},{"Start":"17:02.415 ","End":"17:05.330","Text":"Let me try and get this circled."},{"Start":"17:05.330 ","End":"17:08.210","Text":"It\u0027s not a great job,"},{"Start":"17:08.210 ","End":"17:09.260","Text":"but you get the idea."},{"Start":"17:09.260 ","End":"17:12.430","Text":"Suppose this height was at z equals 4"},{"Start":"17:12.430 ","End":"17:17.475","Text":"and I wanted the equation of just this circle in 3D."},{"Start":"17:17.475 ","End":"17:21.055","Text":"1 way to do it would be to say, okay,"},{"Start":"17:21.055 ","End":"17:30.270","Text":"x squared plus y squared equals 4 and z equals 4."},{"Start":"17:30.270 ","End":"17:36.820","Text":"That would do it. I can put a curly brackets and say it\u0027s 2 equations."},{"Start":"17:36.820 ","End":"17:40.690","Text":"But if you don\u0027t like the end and you think this is cheating,"},{"Start":"17:40.690 ","End":"17:44.410","Text":"well, I\u0027ve got a trick that bits that cheating."},{"Start":"17:44.410 ","End":"17:49.690","Text":"You can always take 2 equations and make them into 1."},{"Start":"17:49.690 ","End":"17:55.795","Text":"I could always say x squared plus y squared minus 4."},{"Start":"17:55.795 ","End":"17:58.825","Text":"I\u0027ll write it and I\u0027ll explain what my logic is,"},{"Start":"17:58.825 ","End":"18:06.150","Text":"plus z minus 4 squared equals 0."},{"Start":"18:06.150 ","End":"18:07.775","Text":"Now it\u0027s 1 equation."},{"Start":"18:07.775 ","End":"18:09.550","Text":"Now what did I just do here?"},{"Start":"18:09.550 ","End":"18:11.350","Text":"It\u0027s an old trick."},{"Start":"18:11.350 ","End":"18:16.450","Text":"If I want to say that a equals 0 and b equals 0,"},{"Start":"18:16.450 ","End":"18:20.830","Text":"but I want to not use and just have 1 equation,"},{"Start":"18:20.830 ","End":"18:24.970","Text":"I could write a squared plus b squared equals 0."},{"Start":"18:24.970 ","End":"18:29.470","Text":"You see a and b if they\u0027re real numbers,"},{"Start":"18:29.470 ","End":"18:31.090","Text":"a squared is non-negative,"},{"Start":"18:31.090 ","End":"18:33.865","Text":"it\u0027s 0 or positive and this is 0 or positive."},{"Start":"18:33.865 ","End":"18:40.165","Text":"The only way the sum of 0 or positive could be 0 if they\u0027re both 0."},{"Start":"18:40.165 ","End":"18:42.850","Text":"This is actually both ways."},{"Start":"18:42.850 ","End":"18:45.805","Text":"2 equations could become 1 equation."},{"Start":"18:45.805 ","End":"18:49.010","Text":"I just use that trick here by just putting the 4 over to"},{"Start":"18:49.010 ","End":"18:52.580","Text":"the other side and making it 0 and also here."},{"Start":"18:52.580 ","End":"18:53.885","Text":"If I write it this way,"},{"Start":"18:53.885 ","End":"18:56.375","Text":"it\u0027s 1 equation and it\u0027s a circle,"},{"Start":"18:56.375 ","End":"18:59.710","Text":"and it\u0027s parallel to the x,"},{"Start":"18:59.710 ","End":"19:03.785","Text":"y plane, and it\u0027s at height z equals 4."},{"Start":"19:03.785 ","End":"19:05.030","Text":"It\u0027s this circle here,"},{"Start":"19:05.030 ","End":"19:07.740","Text":"which I drew badly."},{"Start":"19:08.070 ","End":"19:10.625","Text":"They are straightened it out a bit."},{"Start":"19:10.625 ","End":"19:15.590","Text":"Anyway, later on we\u0027ll see other techniques of describing curves in 3D,"},{"Start":"19:15.590 ","End":"19:18.035","Text":"for example, parametric equations."},{"Start":"19:18.035 ","End":"19:23.250","Text":"But meanwhile, we\u0027re done for this introduction."}],"ID":9748},{"Watched":false,"Name":"Exercises 1","Duration":"1m 32s","ChapterTopicVideoID":9794,"CourseChapterTopicPlaylistID":57474,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9794.jpeg","UploadDate":"2017-08-01T05:00:12.8700000","DurationForVideoObject":"PT1M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this exercise, we\u0027re given a point"},{"Start":"00:02.340 ","End":"00:04.800","Text":"in 3D space and we want"},{"Start":"00:04.800 ","End":"00:10.395","Text":"its projection onto each of the 3 coordinate planes."},{"Start":"00:10.395 ","End":"00:13.755","Text":"A diagram might help."},{"Start":"00:13.755 ","End":"00:17.595","Text":"Note that if I project onto the xy-plane,"},{"Start":"00:17.595 ","End":"00:19.785","Text":"that\u0027s where z is 0."},{"Start":"00:19.785 ","End":"00:23.100","Text":"In general, if I have a point x, y, z,"},{"Start":"00:23.100 ","End":"00:26.670","Text":"all I have to do is set the z component to be 0"},{"Start":"00:26.670 ","End":"00:28.965","Text":"and I\u0027ve got it onto the xy-plane."},{"Start":"00:28.965 ","End":"00:34.080","Text":"In our case, the projection onto"},{"Start":"00:34.080 ","End":"00:40.780","Text":"the xy-plane will just be 4,7,0,"},{"Start":"00:41.030 ","End":"00:44.445","Text":"just setting z as 0."},{"Start":"00:44.445 ","End":"00:50.720","Text":"Similarly, the projection onto the xz-plane"},{"Start":"00:50.720 ","End":"00:54.780","Text":"is obtained by letting y equal 0."},{"Start":"00:54.780 ","End":"01:01.535","Text":"In this case, we\u0027ll have our original point 4 here minus 5,"},{"Start":"01:01.535 ","End":"01:03.215","Text":"except that in the middle,"},{"Start":"01:03.215 ","End":"01:06.215","Text":"for the y, we put 0."},{"Start":"01:06.215 ","End":"01:10.890","Text":"Finally, if we wanted on to the,"},{"Start":"01:11.730 ","End":"01:14.620","Text":"let\u0027s see what\u0027s missing,"},{"Start":"01:14.620 ","End":"01:20.175","Text":"the yz-plane, that\u0027s this 1 here."},{"Start":"01:20.175 ","End":"01:22.860","Text":"That\u0027s where we let the x equals 0."},{"Start":"01:22.860 ","End":"01:28.680","Text":"We\u0027ve got 0 and then 7 minus 5."},{"Start":"01:28.680 ","End":"01:32.230","Text":"That\u0027s all, so we\u0027re done here."}],"ID":9749},{"Watched":false,"Name":"Exercises 2","Duration":"3m 40s","ChapterTopicVideoID":9795,"CourseChapterTopicPlaylistID":57474,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9795.jpeg","UploadDate":"2017-08-01T05:00:25.7670000","DurationForVideoObject":"PT3M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.475","Text":"Here\u0027s an exercise in 2 parts."},{"Start":"00:02.475 ","End":"00:07.350","Text":"Let\u0027s just do part a first and then b will make more sense."},{"Start":"00:07.350 ","End":"00:12.570","Text":"In part a, we want to know the distance of this point from the xy-plane."},{"Start":"00:12.570 ","End":"00:16.170","Text":"Now, the distance you can get just"},{"Start":"00:16.170 ","End":"00:22.350","Text":"by taking the projection of this point onto this plane that will give"},{"Start":"00:22.350 ","End":"00:26.610","Text":"the closest point on the xy-plane to this because"},{"Start":"00:26.610 ","End":"00:36.240","Text":"it\u0027s going to be perpendicular, the line from the point to the projection."},{"Start":"00:36.240 ","End":"00:39.030","Text":"I brought the picture from the previous clip,"},{"Start":"00:39.030 ","End":"00:45.290","Text":"it might help that here\u0027s the point P. If I want to know the distance to the xy- plane,"},{"Start":"00:45.290 ","End":"00:50.300","Text":"I just take the distance from here to"},{"Start":"00:50.300 ","End":"00:53.540","Text":"here because the projection has the property that"},{"Start":"00:53.540 ","End":"00:57.200","Text":"its 90 degrees and that\u0027s what the distance means."},{"Start":"00:57.200 ","End":"01:00.215","Text":"The distance is the perpendicular distance."},{"Start":"01:00.215 ","End":"01:09.465","Text":"We need the distance from this original point,"},{"Start":"01:09.465 ","End":"01:15.420","Text":"4, 7 minus 5 to the projection."},{"Start":"01:15.420 ","End":"01:23.630","Text":"The projection we get just by setting the z equals 4, 7, 0."},{"Start":"01:23.630 ","End":"01:24.920","Text":"That\u0027s this point here."},{"Start":"01:24.920 ","End":"01:29.720","Text":"It\u0027s labeled Q in this diagram."},{"Start":"01:29.720 ","End":"01:36.140","Text":"There\u0027s no point in using the distance formula."},{"Start":"01:36.140 ","End":"01:37.550","Text":"I mean, you could use it."},{"Start":"01:37.550 ","End":"01:40.880","Text":"You could say it\u0027s 4 minus 4 squared plus 7"},{"Start":"01:40.880 ","End":"01:44.570","Text":"minus 7 squared plus this minus this squared square root."},{"Start":"01:44.570 ","End":"01:47.855","Text":"But because the first 2 coordinates are the same,"},{"Start":"01:47.855 ","End":"01:52.010","Text":"all we need is the distance from minus 5-0."},{"Start":"01:52.010 ","End":"01:55.490","Text":"We subtract and take the absolute value,"},{"Start":"01:55.490 ","End":"01:59.875","Text":"so the answer is 5."},{"Start":"01:59.875 ","End":"02:03.430","Text":"That\u0027s the answer to a."},{"Start":"02:03.500 ","End":"02:13.020","Text":"Now, in b, the closer point means the one with the lesser distance."},{"Start":"02:14.500 ","End":"02:19.190","Text":"This one is the same point as this point."},{"Start":"02:19.190 ","End":"02:23.760","Text":"We know the distance from 4,"},{"Start":"02:23.760 ","End":"02:29.190","Text":"7 negative 5 to the xy-plane. Maybe I\u0027ll write that."},{"Start":"02:29.190 ","End":"02:31.365","Text":"I could copy-paste here."},{"Start":"02:31.365 ","End":"02:38.340","Text":"The distance to the xy-plane is 5."},{"Start":"02:38.340 ","End":"02:48.045","Text":"Then similarly, to get the distance of the other point is 5 minus 6, 7."},{"Start":"02:48.045 ","End":"02:50.700","Text":"It\u0027s not the same Q as this, I don\u0027t know,"},{"Start":"02:50.700 ","End":"02:51.960","Text":"I will call this some other letter,"},{"Start":"02:51.960 ","End":"02:55.575","Text":"I will call it T or something."},{"Start":"02:55.575 ","End":"03:00.920","Text":"The distance of this point is where we use the same trick."},{"Start":"03:00.920 ","End":"03:03.230","Text":"You first of all, find the projection."},{"Start":"03:03.230 ","End":"03:06.860","Text":"The projection of this onto the xy-plane is"},{"Start":"03:06.860 ","End":"03:14.810","Text":"5 minus 6,0."},{"Start":"03:14.810 ","End":"03:21.340","Text":"Because these 2 are the same as the distance from 7-0 is 7."},{"Start":"03:21.340 ","End":"03:27.075","Text":"Now, because 5 is less than 7,"},{"Start":"03:27.075 ","End":"03:32.475","Text":"the answer is this point,"},{"Start":"03:32.475 ","End":"03:39.220","Text":"P is the closer one. That\u0027s it."}],"ID":9750},{"Watched":false,"Name":"Exercises 3","Duration":"4m 26s","ChapterTopicVideoID":9792,"CourseChapterTopicPlaylistID":57474,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9792.jpeg","UploadDate":"2017-08-01T04:59:54.3730000","DurationForVideoObject":"PT4M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In part a of this exercise,"},{"Start":"00:02.490 ","End":"00:09.090","Text":"we want to find the distance of a point from the z-axis."},{"Start":"00:09.090 ","End":"00:17.830","Text":"Previously we\u0027ve had the distance of a point from the x,y plane, from its projection."},{"Start":"00:18.020 ","End":"00:24.179","Text":"The distance of a point to an axis is similar but different."},{"Start":"00:24.179 ","End":"00:31.450","Text":"We need to drop a perpendicular from this point to this axis."},{"Start":"00:32.260 ","End":"00:37.415","Text":"It doesn\u0027t look perpendicular because of the perspective,"},{"Start":"00:37.415 ","End":"00:41.225","Text":"but this is supposed to be at 90 degrees."},{"Start":"00:41.225 ","End":"00:44.120","Text":"You can see it better if I complete the rectangle,"},{"Start":"00:44.120 ","End":"00:46.010","Text":"just join these 2."},{"Start":"00:46.010 ","End":"00:50.930","Text":"This is in the x,y plane, so obviously perpendicular."},{"Start":"00:50.930 ","End":"00:55.010","Text":"What I\u0027m saying is this height is the same as this height."},{"Start":"00:55.010 ","End":"00:59.210","Text":"Leave enough room here. That\u0027s better."},{"Start":"00:59.210 ","End":"01:04.435","Text":"Let\u0027s call this point p bar."},{"Start":"01:04.435 ","End":"01:10.125","Text":"It\u0027s associated with P. It\u0027s a projection of P on to the z-axis."},{"Start":"01:10.125 ","End":"01:14.125","Text":"What I\u0027m doing is this height is the same as this height."},{"Start":"01:14.125 ","End":"01:19.130","Text":"This is the same as this and perhaps this is"},{"Start":"01:19.130 ","End":"01:24.890","Text":"a long-winded approach to stating the obvious is that this point,"},{"Start":"01:24.890 ","End":"01:34.790","Text":"its coordinates will be 0,0,z."},{"Start":"01:34.790 ","End":"01:39.575","Text":"In our case, in our particular p,"},{"Start":"01:39.575 ","End":"01:43.145","Text":"that p bar we\u0027ll call it,"},{"Start":"01:43.145 ","End":"01:46.265","Text":"would be, I mean,"},{"Start":"01:46.265 ","End":"01:50.820","Text":"the same z as this but the x and the y are 0."},{"Start":"01:52.670 ","End":"01:58.470","Text":"The distance between these 2,"},{"Start":"01:58.470 ","End":"02:08.645","Text":"let\u0027s call it d from p to p bar is the distance formula is the square root of 4"},{"Start":"02:08.645 ","End":"02:14.599","Text":"minus 0 squared plus 7 minus 0"},{"Start":"02:14.599 ","End":"02:21.420","Text":"squared plus negative 5"},{"Start":"02:21.420 ","End":"02:25.600","Text":"minus negative 5 squared."},{"Start":"02:26.540 ","End":"02:34.020","Text":"This part is 0 so I just get the square root of 4"},{"Start":"02:34.020 ","End":"02:41.600","Text":"squared plus 7 squared and this is the square root of,"},{"Start":"02:41.600 ","End":"02:48.950","Text":"let\u0027s see, 49 and 16, is 65."},{"Start":"02:48.950 ","End":"02:53.735","Text":"Now I\u0027m not going to do the whole thing from scratch with Q."},{"Start":"02:53.735 ","End":"02:58.340","Text":"There\u0027s a Q and then there\u0027s the projection just"},{"Start":"02:58.340 ","End":"03:03.110","Text":"like P was projected onto the z-axis here,"},{"Start":"03:03.110 ","End":"03:08.450","Text":"the Q would have its corresponding Q bar on"},{"Start":"03:08.450 ","End":"03:14.980","Text":"the z-axis and it would be 0,0,7."},{"Start":"03:14.980 ","End":"03:22.924","Text":"Now if I did the distance between Q and Q bar,"},{"Start":"03:22.924 ","End":"03:32.550","Text":"what we would get basically is just this squared because its this minus 0 squared,"},{"Start":"03:32.550 ","End":"03:38.865","Text":"and this minus this is negative 6 squared."},{"Start":"03:38.865 ","End":"03:40.940","Text":"The last one\u0027s going to be 0 squared,"},{"Start":"03:40.940 ","End":"03:42.815","Text":"just like it was here,"},{"Start":"03:42.815 ","End":"03:44.570","Text":"so we don\u0027t need that."},{"Start":"03:44.570 ","End":"03:54.335","Text":"This comes out to be 25 and 36 is 61."},{"Start":"03:54.335 ","End":"03:58.480","Text":"This is the square root of 61."},{"Start":"03:58.480 ","End":"04:13.020","Text":"Now, obviously this is less than this."},{"Start":"04:13.020 ","End":"04:15.820","Text":"The closer 1 is the Q."},{"Start":"04:16.190 ","End":"04:26.880","Text":"This Q is the closer 1 to the z-axis because this is smaller than this. That\u0027s it."}],"ID":9751},{"Watched":false,"Name":"Exercises 4","Duration":"4m 40s","ChapterTopicVideoID":9793,"CourseChapterTopicPlaylistID":57474,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9793.jpeg","UploadDate":"2017-08-01T05:00:07.8430000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"The purpose of this exercise is just to show you that the same equation in"},{"Start":"00:05.310 ","End":"00:11.520","Text":"x and y could be considered as an equation in 2 variables and in 2D,"},{"Start":"00:11.520 ","End":"00:13.620","Text":"what we call R-squared,"},{"Start":"00:13.620 ","End":"00:15.825","Text":"the real number squared."},{"Start":"00:15.825 ","End":"00:18.915","Text":"Or it could be in R3."},{"Start":"00:18.915 ","End":"00:22.560","Text":"In each case, it would represent a different shape,"},{"Start":"00:22.560 ","End":"00:27.540","Text":"get an extra dimension in each of part B."},{"Start":"00:27.540 ","End":"00:29.520","Text":"So each case they\u0027re giving you an equation,"},{"Start":"00:29.520 ","End":"00:34.050","Text":"in x and y, and we want to interpret it in 2D and in 3D."},{"Start":"00:34.050 ","End":"00:36.210","Text":"I don\u0027t want to get too technical,"},{"Start":"00:36.210 ","End":"00:39.580","Text":"just wanted you to see if you can identify the shape."},{"Start":"00:39.580 ","End":"00:45.440","Text":"Well, in 2D we\u0027ve learned about equations like this, linear equations."},{"Start":"00:45.440 ","End":"00:46.550","Text":"This would be a line,"},{"Start":"00:46.550 ","End":"00:53.600","Text":"I could even sketch it if I let x equal 0."},{"Start":"00:53.600 ","End":"00:57.089","Text":"If I let x is 0,"},{"Start":"00:57.089 ","End":"01:00.000","Text":"then I\u0027ll get that 3_y is 6,"},{"Start":"01:00.000 ","End":"01:04.830","Text":"so y is 2 and the other way of y is 0, x is 3."},{"Start":"01:05.380 ","End":"01:09.470","Text":"If I just do something rough like this,"},{"Start":"01:09.470 ","End":"01:11.809","Text":"we said that when x was 0,"},{"Start":"01:11.809 ","End":"01:14.590","Text":"then y was 2,"},{"Start":"01:14.590 ","End":"01:17.285","Text":"and when y was 0, x was 3."},{"Start":"01:17.285 ","End":"01:20.375","Text":"So we get some straight line here."},{"Start":"01:20.375 ","End":"01:24.170","Text":"This is y, this is x,"},{"Start":"01:24.170 ","End":"01:26.164","Text":"and that\u0027s our line."},{"Start":"01:26.164 ","End":"01:27.950","Text":"Here\u0027s 2, here\u0027s 3."},{"Start":"01:27.950 ","End":"01:33.890","Text":"Now, if I was looking at this as an equation in 3 dimensions, x, y,"},{"Start":"01:33.890 ","End":"01:37.085","Text":"z, and z just happens to be missing,"},{"Start":"01:37.085 ","End":"01:40.400","Text":"what we do is we just add a third dimension."},{"Start":"01:40.400 ","End":"01:42.170","Text":"I can\u0027t really sketch it here."},{"Start":"01:42.170 ","End":"01:45.050","Text":"But through each of these points on the line,"},{"Start":"01:45.050 ","End":"01:50.450","Text":"we\u0027ll take a vertical line perpendicular to the plane that we\u0027re seeing."},{"Start":"01:50.450 ","End":"01:54.020","Text":"What we actually get is a plane."},{"Start":"01:54.020 ","End":"02:00.395","Text":"Here I just wanted you to identify this is the equation of a line."},{"Start":"02:00.395 ","End":"02:05.450","Text":"But in 3-dimensions, it\u0027s the equation of a plane."},{"Start":"02:05.450 ","End":"02:07.490","Text":"It\u0027s actually a vertical plane."},{"Start":"02:07.490 ","End":"02:12.320","Text":"It\u0027s parallel to the z-axis,"},{"Start":"02:12.320 ","End":"02:19.910","Text":"which is a vertical line through the origin."},{"Start":"02:19.910 ","End":"02:24.900","Text":"That\u0027s all. In Part 2,"},{"Start":"02:26.200 ","End":"02:32.280","Text":"if you studied a little bit of implicit equations and circles,"},{"Start":"02:32.280 ","End":"02:36.315","Text":"you should recognize that this is the equation of a circle."},{"Start":"02:36.315 ","End":"02:38.380","Text":"Not that it\u0027s important,"},{"Start":"02:38.380 ","End":"02:40.300","Text":"but as a matter of fact,"},{"Start":"02:40.300 ","End":"02:41.800","Text":"we even know the center,"},{"Start":"02:41.800 ","End":"02:44.065","Text":"it\u0027s 1 comma 0."},{"Start":"02:44.065 ","End":"02:46.150","Text":"This here might be 1,"},{"Start":"02:46.150 ","End":"02:48.460","Text":"and this would be 0,"},{"Start":"02:48.460 ","End":"02:51.585","Text":"y, and then x."},{"Start":"02:51.585 ","End":"02:54.060","Text":"Then 9 is 3 squared,"},{"Start":"02:54.060 ","End":"02:57.540","Text":"so the radius would be 3."},{"Start":"02:57.540 ","End":"03:00.730","Text":"So if I go 3 in this direction,"},{"Start":"03:00.730 ","End":"03:02.680","Text":"that will take me to 4,"},{"Start":"03:02.680 ","End":"03:04.720","Text":"and if I take 3 in this direction,"},{"Start":"03:04.720 ","End":"03:07.360","Text":"it\u0027ll take me to minus 2,"},{"Start":"03:07.360 ","End":"03:16.195","Text":"and then we get some kind of a circle which it really doesn\u0027t matter."},{"Start":"03:16.195 ","End":"03:19.640","Text":"Don\u0027t have to be exact."},{"Start":"03:20.010 ","End":"03:24.670","Text":"Tidy it up a bit and change color and the line."},{"Start":"03:24.670 ","End":"03:33.180","Text":"Anyway, the idea is just to identify that this is a circle in 2D,"},{"Start":"03:33.180 ","End":"03:37.960","Text":"but if I add the Z dimension through each of these points,"},{"Start":"03:37.960 ","End":"03:43.780","Text":"I can draw a straight line extending infinitely in both directions."},{"Start":"03:43.780 ","End":"03:50.150","Text":"What we\u0027ll get, and we\u0027ve seen this before is a cylinder."},{"Start":"03:53.880 ","End":"03:56.140","Text":"As I said, the whole idea"},{"Start":"03:56.140 ","End":"04:01.600","Text":"here is just to emphasize that if you see an equation in x and y,"},{"Start":"04:01.600 ","End":"04:06.565","Text":"you don\u0027t know if it\u0027s in 2D or in 3D,"},{"Start":"04:06.565 ","End":"04:09.020","Text":"and you get a different shape,"},{"Start":"04:09.020 ","End":"04:10.190","Text":"you get an extra dimension."},{"Start":"04:10.190 ","End":"04:13.700","Text":"The line becomes a plane if you extend it infinitely upward and"},{"Start":"04:13.700 ","End":"04:18.330","Text":"downwards and a circle becomes a cylinder."},{"Start":"04:18.470 ","End":"04:22.010","Text":"These are both equations in x, y, and z,"},{"Start":"04:22.010 ","End":"04:24.215","Text":"possibly in part B,"},{"Start":"04:24.215 ","End":"04:30.080","Text":"but you just don\u0027t see z because not every variable has to explicitly appear."},{"Start":"04:30.080 ","End":"04:36.840","Text":"If you want to, you could always write plus 0 z and then force z to be in here or here."},{"Start":"04:36.840 ","End":"04:39.560","Text":"That\u0027s all I wanted to say."}],"ID":9752}],"Thumbnail":null,"ID":57474},{"Name":"Equations of Lines","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Equations of Lines","Duration":"7m 23s","ChapterTopicVideoID":9880,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9880.jpeg","UploadDate":"2020-01-16T07:11:55.8130000","DurationForVideoObject":"PT7M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"Continuing with the 3D coordinate system,"},{"Start":"00:02.280 ","End":"00:07.395","Text":"the subtopic is going to be; Equations of Lines."},{"Start":"00:07.395 ","End":"00:15.075","Text":"I\u0027m going to assume that you\u0027ve already studied parametric equations at least in 2D,"},{"Start":"00:15.075 ","End":"00:17.235","Text":"which is what we\u0027re going to start off with."},{"Start":"00:17.235 ","End":"00:20.269","Text":"In 2D we often gave 2 functions,"},{"Start":"00:20.269 ","End":"00:24.900","Text":"we would say that x is some function of t or just say x is a function of t,"},{"Start":"00:24.900 ","End":"00:27.765","Text":"and we give another function y in terms of"},{"Start":"00:27.765 ","End":"00:32.295","Text":"t. Then we also maybe restricted t to a certain range,"},{"Start":"00:32.295 ","End":"00:35.895","Text":"maybe t was between 0 and 2Pi or whatever,"},{"Start":"00:35.895 ","End":"00:42.250","Text":"t might have been between something and something or less than or whatever."},{"Start":"00:43.040 ","End":"00:46.690","Text":"I\u0027m not going to repeat all that,"},{"Start":"00:46.970 ","End":"00:52.210","Text":"perhaps I\u0027ll just give an example."},{"Start":"00:52.220 ","End":"00:57.100","Text":"I brought an example of an ellipse even though the subject is going to be lines,"},{"Start":"00:57.100 ","End":"00:59.530","Text":"but it\u0027s a good example."},{"Start":"00:59.530 ","End":"01:03.935","Text":"In this case what we get is not this general,"},{"Start":"01:03.935 ","End":"01:11.140","Text":"we have specifically x equals 6 cosine t,"},{"Start":"01:11.140 ","End":"01:14.245","Text":"the 6 is actually related to the 6 here,"},{"Start":"01:14.245 ","End":"01:23.190","Text":"and y is 3 sine t and that\u0027s related to the 3 here and often we put curly braces."},{"Start":"01:23.190 ","End":"01:33.135","Text":"Then t and it happens here goes between 0 and 2Pi or 360 degrees,"},{"Start":"01:33.135 ","End":"01:35.745","Text":"but I\u0027m taking it in radians."},{"Start":"01:35.745 ","End":"01:41.960","Text":"Now, this is all a prelude to something called vector functions,"},{"Start":"01:41.960 ","End":"01:45.200","Text":"and I want to do this in the vector notation."},{"Start":"01:45.200 ","End":"01:52.680","Text":"I also hope you remember the concept of position vector,"},{"Start":"01:52.680 ","End":"01:58.535","Text":"and if you don\u0027t remember it perhaps you might want to go back to the chapter on vectors."},{"Start":"01:58.535 ","End":"02:08.715","Text":"In any event, the vector that connects the point (0,0) the origin to"},{"Start":"02:08.715 ","End":"02:12.120","Text":"a general point (x,y)"},{"Start":"02:12.120 ","End":"02:20.849","Text":"the position vector is what we get when we subtract the x\u0027s and we get x,"},{"Start":"02:20.849 ","End":"02:22.290","Text":"and subtract the y\u0027s,"},{"Start":"02:22.290 ","End":"02:23.850","Text":"y minus 0 is y."},{"Start":"02:23.850 ","End":"02:26.920","Text":"The angular brackets means this is a vector notation,"},{"Start":"02:26.920 ","End":"02:32.535","Text":"and this is the position vector from the origin to the point \u003cx,y\u003e."},{"Start":"02:32.535 ","End":"02:35.710","Text":"Often we call this the letter"},{"Start":"02:35.710 ","End":"02:43.095","Text":"r. Instead of a pair of numbers \u003cx,y\u003e,"},{"Start":"02:43.095 ","End":"02:44.549","Text":"we have a vector."},{"Start":"02:44.549 ","End":"02:49.980","Text":"In fact, we can also use r in a function notation,"},{"Start":"02:49.980 ","End":"03:00.825","Text":"and what I wrote above I could write r as a function of t is 6 cosine t,"},{"Start":"03:00.825 ","End":"03:10.500","Text":"3 sine t, and here t is restricted between 0 and 2Pi."},{"Start":"03:10.500 ","End":"03:11.960","Text":"Actually you don\u0027t have to restrict it,"},{"Start":"03:11.960 ","End":"03:15.845","Text":"but then it just goes round forever and ever and so on."},{"Start":"03:15.845 ","End":"03:21.140","Text":"But often t is not restricted as we\u0027ll see later when we talk about straight lines,"},{"Start":"03:21.140 ","End":"03:22.540","Text":"which I\u0027m coming to."},{"Start":"03:22.540 ","End":"03:25.995","Text":"I forgot to write the vector there."},{"Start":"03:25.995 ","End":"03:33.260","Text":"What I\u0027m going to talk about now is how to find the parametric equation of a line in 2D,"},{"Start":"03:33.260 ","End":"03:37.770","Text":"especially if we\u0027re given say 2 points on the line."},{"Start":"03:38.180 ","End":"03:46.530","Text":"Here I have a diagram which I\u0027m going to use for the equation of a line in 2D,"},{"Start":"03:46.530 ","End":"03:51.610","Text":"this is the line I\u0027ll just call it the line."},{"Start":"03:51.800 ","End":"03:55.775","Text":"Here I have a specific point on the line P0,"},{"Start":"03:55.775 ","End":"04:02.415","Text":"let\u0027s give it coordinates (x0,y0) and then we have another point,"},{"Start":"04:02.415 ","End":"04:05.495","Text":"this will be a more general point (x,y)."},{"Start":"04:05.495 ","End":"04:08.840","Text":"We have position vectors 2P and 2P0,"},{"Start":"04:08.840 ","End":"04:13.210","Text":"we call them r vector and r0 vector."},{"Start":"04:13.210 ","End":"04:18.000","Text":"Included i and j, it came with the picture of the standard basis vectors,"},{"Start":"04:18.000 ","End":"04:27.840","Text":"and I want to know how to write the equation of this line in vector parametric form."},{"Start":"04:27.910 ","End":"04:32.270","Text":"What I can say is, and this vector,"},{"Start":"04:32.270 ","End":"04:35.830","Text":"this red 1 here I\u0027ll call it v,"},{"Start":"04:35.830 ","End":"04:40.040","Text":"this is just any direction vector for the line."},{"Start":"04:40.040 ","End":"04:43.160","Text":"Of course a vector would be drawn usually through the origin,"},{"Start":"04:43.160 ","End":"04:44.585","Text":"it doesn\u0027t matter where you place it,"},{"Start":"04:44.585 ","End":"04:50.270","Text":"but it\u0027s got to be a vector non-0 and parallel to the line any direction vector."},{"Start":"04:50.270 ","End":"04:56.485","Text":"What I want to say is that I\u0027m going to develop the formula by saying that I can get to"},{"Start":"04:56.485 ","End":"05:05.180","Text":"a general point (x,y) on the line by going first to this specific point on the line here,"},{"Start":"05:05.180 ","End":"05:08.960","Text":"and then adding to it the vector from P0 to P."},{"Start":"05:08.960 ","End":"05:13.410","Text":"Remember the triangle rule for addition of vectors."},{"Start":"05:13.410 ","End":"05:21.200","Text":"What I\u0027m saying is that the vector r is equal"},{"Start":"05:21.200 ","End":"05:30.195","Text":"to the vector r0 plus the position vector from P0 to P,"},{"Start":"05:30.195 ","End":"05:40.260","Text":"I\u0027ll just call it for the moment P0P."},{"Start":"05:40.260 ","End":"05:45.405","Text":"On the other hand, the vector P0P,"},{"Start":"05:45.405 ","End":"05:51.840","Text":"wherever P is on the line is going to be a multiple of this position vector,"},{"Start":"05:51.840 ","End":"05:53.330","Text":"and conversely any multiple of"},{"Start":"05:53.330 ","End":"05:58.025","Text":"this position vector if I place it from P0 will be on the line."},{"Start":"05:58.025 ","End":"05:59.885","Text":"This is some constant,"},{"Start":"05:59.885 ","End":"06:09.720","Text":"let\u0027s use the parameter t times the position vector v. After I put that in here I"},{"Start":"06:09.720 ","End":"06:17.715","Text":"get that r is equal to r0 plus t"},{"Start":"06:17.715 ","End":"06:20.975","Text":"times v. If I have"},{"Start":"06:20.975 ","End":"06:26.555","Text":"a point on the line in vector form and a direction vector for the line,"},{"Start":"06:26.555 ","End":"06:28.880","Text":"then this is the equation of the line."},{"Start":"06:28.880 ","End":"06:38.480","Text":"What we do is we say that because r is \u003cx,y\u003e I can write this equivalently as"},{"Start":"06:38.480 ","End":"06:44.910","Text":"\u003cx,y\u003e is equal to \u003cx0,y0\u003e"},{"Start":"06:44.910 ","End":"06:54.780","Text":"plus t. Let\u0027s say this vector v is let\u0027s say \u003ca,b\u003e,"},{"Start":"06:54.780 ","End":"07:01.170","Text":"so here I can write \u003ca,b\u003e."},{"Start":"07:01.170 ","End":"07:06.075","Text":"Either of these, either this form or"},{"Start":"07:06.075 ","End":"07:13.925","Text":"this form will give the equation of a line in parametric form in 2D."},{"Start":"07:13.925 ","End":"07:18.305","Text":"Every value of t gives me a different point on the line,"},{"Start":"07:18.305 ","End":"07:19.800","Text":"and t is actually not restricted,"},{"Start":"07:19.800 ","End":"07:24.390","Text":"it can go from minus infinity to infinity and gives us the whole line."}],"ID":9753},{"Watched":false,"Name":"The 3D Coordinate System - Equations of Lines (continued)","Duration":"15m 19s","ChapterTopicVideoID":9881,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9881.jpeg","UploadDate":"2020-01-16T07:12:36.2170000","DurationForVideoObject":"PT15M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Actually, I wasn\u0027t being quite precise with the terminology,"},{"Start":"00:03.330 ","End":"00:04.500","Text":"I was saying parametric,"},{"Start":"00:04.500 ","End":"00:10.090","Text":"but actually, this is called the vector form."},{"Start":"00:10.330 ","End":"00:15.630","Text":"There are actually 3 forms used more in 3D,"},{"Start":"00:15.630 ","End":"00:19.170","Text":"but I want to stay with 2D just because it\u0027s easier and then we\u0027ll generalize."},{"Start":"00:19.170 ","End":"00:22.170","Text":"There is something called the parametric form also."},{"Start":"00:22.170 ","End":"00:25.120","Text":"It looks like what we had above."},{"Start":"00:25.160 ","End":"00:27.330","Text":"Something is bothering me."},{"Start":"00:27.330 ","End":"00:31.500","Text":"I think I\u0027d rather put a variable t here just to"},{"Start":"00:31.500 ","End":"00:36.060","Text":"emphasize that the position vector is a function of a parameter t,"},{"Start":"00:36.060 ","End":"00:37.470","Text":"although this is called the vector form."},{"Start":"00:37.470 ","End":"00:40.135","Text":"At least both of these are called the vector form."},{"Start":"00:40.135 ","End":"00:42.810","Text":"Now, the parametric form is,"},{"Start":"00:42.810 ","End":"00:45.695","Text":"if I just write this one component-wise,"},{"Start":"00:45.695 ","End":"00:51.115","Text":"then I get that x equals x-naught plus ta,"},{"Start":"00:51.115 ","End":"00:56.545","Text":"and y equals y-naught plus tb."},{"Start":"00:56.545 ","End":"01:02.860","Text":"This is called the parametric form."},{"Start":"01:05.990 ","End":"01:10.210","Text":"There is a 3rd form which is not usually used in 2D,"},{"Start":"01:10.210 ","End":"01:14.605","Text":"but I\u0027m going to give it anyway because it will just be practice for 3D."},{"Start":"01:14.605 ","End":"01:19.540","Text":"If I isolate t from each of the equations here,"},{"Start":"01:19.540 ","End":"01:25.740","Text":"I can say that t is equal to x"},{"Start":"01:25.740 ","End":"01:32.625","Text":"minus x-naught over a."},{"Start":"01:32.625 ","End":"01:42.645","Text":"From here, I get the t equals y minus y-naught over b."},{"Start":"01:42.645 ","End":"01:47.510","Text":"Then I can throw out the t. This form,"},{"Start":"01:47.510 ","End":"01:48.875","Text":"I\u0027m not going to give it a name."},{"Start":"01:48.875 ","End":"01:50.570","Text":"I\u0027ll give it a name when we get to 3D,"},{"Start":"01:50.570 ","End":"01:52.265","Text":"it begins with the word symmetric,"},{"Start":"01:52.265 ","End":"01:54.815","Text":"but it\u0027s not obvious in 2D."},{"Start":"01:54.815 ","End":"02:02.280","Text":"Let me now generalize this to 3D and I\u0027ll get a bit more space."},{"Start":"02:02.530 ","End":"02:05.570","Text":"I\u0027m not going to develop everything from scratch,"},{"Start":"02:05.570 ","End":"02:07.730","Text":"but I have included a 3D picture"},{"Start":"02:07.730 ","End":"02:13.340","Text":"where this time this direction vector is written down here,"},{"Start":"02:13.340 ","End":"02:16.775","Text":"but it\u0027s the same v only this time,"},{"Start":"02:16.775 ","End":"02:18.650","Text":"it is not a, b,"},{"Start":"02:18.650 ","End":"02:22.505","Text":"it\u0027s a, b, c,"},{"Start":"02:22.505 ","End":"02:25.620","Text":"and the r-naught is,"},{"Start":"02:25.620 ","End":"02:29.985","Text":"instead of being x-naught, y-naught is x-naught,"},{"Start":"02:29.985 ","End":"02:35.325","Text":"y-naught, z-naught, and r is x,"},{"Start":"02:35.325 ","End":"02:38.940","Text":"y, z, and so on."},{"Start":"02:38.940 ","End":"02:47.660","Text":"I\u0027ll just write the analogs of these 2D formulas in 3D."},{"Start":"02:47.660 ","End":"02:49.555","Text":"Let\u0027s look at them."},{"Start":"02:49.555 ","End":"02:52.215","Text":"I\u0027ll start with the vector form."},{"Start":"02:52.215 ","End":"02:54.705","Text":"I\u0027ll use different color for 3D."},{"Start":"02:54.705 ","End":"03:04.220","Text":"We get that r as a function of t is r-naught,"},{"Start":"03:04.220 ","End":"03:10.085","Text":"which is a position vector of any point on the line,"},{"Start":"03:10.085 ","End":"03:16.040","Text":"plus a parameter t times a direction vector of the line,"},{"Start":"03:16.040 ","End":"03:18.950","Text":"which in this case I also call v. Actually,"},{"Start":"03:18.950 ","End":"03:22.760","Text":"this part\u0027s identical, but when I write it out in components,"},{"Start":"03:22.760 ","End":"03:28.175","Text":"the same thing, then I get the analog which is x, y,"},{"Start":"03:28.175 ","End":"03:32.570","Text":"z equals x-naught, y-naught,"},{"Start":"03:32.570 ","End":"03:36.830","Text":"z-naught plus t, and not a, b, but a,"},{"Start":"03:36.830 ","End":"03:44.345","Text":"b, c. I label these vector form,"},{"Start":"03:44.345 ","End":"03:46.535","Text":"either or both of these."},{"Start":"03:46.535 ","End":"03:49.580","Text":"Now let\u0027s get to the parametric form."},{"Start":"03:49.580 ","End":"03:53.780","Text":"Also, we just break this up component-wise."},{"Start":"03:53.780 ","End":"03:56.390","Text":"Maybe put some curly braces."},{"Start":"03:56.390 ","End":"04:01.100","Text":"Only this time we\u0027re going to have x, y,"},{"Start":"04:01.100 ","End":"04:08.420","Text":"and z. I\u0027ll just say this is x_0 plus ta,"},{"Start":"04:08.420 ","End":"04:15.850","Text":"y_0 plus tb,"},{"Start":"04:15.850 ","End":"04:20.790","Text":"z_0 plus tc."},{"Start":"04:20.790 ","End":"04:24.630","Text":"The last form which is equivalent to this,"},{"Start":"04:26.470 ","End":"04:33.720","Text":"I label this parametric form and I need to scroll down."},{"Start":"04:33.720 ","End":"04:38.990","Text":"We\u0027re heading towards something called the symmetric equations of the line."},{"Start":"04:38.990 ","End":"04:40.375","Text":"That\u0027s the 3rd form."},{"Start":"04:40.375 ","End":"04:44.840","Text":"I\u0027m just going to extend that and I will write,"},{"Start":"04:44.840 ","End":"04:46.880","Text":"and this is what we get when we isolate t,"},{"Start":"04:46.880 ","End":"04:49.210","Text":"where t equals to each of these 3 things,"},{"Start":"04:49.210 ","End":"04:57.050","Text":"we get that x minus x-naught over a equals y minus"},{"Start":"04:57.050 ","End":"05:01.484","Text":"y-naught over b equals z"},{"Start":"05:01.484 ","End":"05:07.640","Text":"minus z-naught over c. This is not really an equation."},{"Start":"05:07.640 ","End":"05:10.000","Text":"Here, it\u0027s 2 equations,"},{"Start":"05:10.000 ","End":"05:13.290","Text":"so it\u0027s a set of equations."},{"Start":"05:13.290 ","End":"05:19.605","Text":"There is a small technical snag with this form because 1 of these could be 0,"},{"Start":"05:19.605 ","End":"05:21.700","Text":"they can\u0027t all be 0,"},{"Start":"05:21.700 ","End":"05:24.280","Text":"but 1 or even 2 of them could be 0."},{"Start":"05:24.280 ","End":"05:25.975","Text":"What we do is that,"},{"Start":"05:25.975 ","End":"05:27.799","Text":"if say, b is 0,"},{"Start":"05:27.799 ","End":"05:32.270","Text":"we understand this to mean that y minus y-naught is 0."},{"Start":"05:32.270 ","End":"05:35.060","Text":"If denominator is 0, we force the numerator to be 0."},{"Start":"05:35.060 ","End":"05:37.700","Text":"That means that y is a constant y-naught,"},{"Start":"05:37.700 ","End":"05:39.080","Text":"just like with the parameters."},{"Start":"05:39.080 ","End":"05:42.109","Text":"If b is 0, then y would be y-naught,"},{"Start":"05:42.109 ","End":"05:44.000","Text":"so we take it with that understanding."},{"Start":"05:44.000 ","End":"05:51.530","Text":"That turns out to be a line that\u0027s parallel to the x, z plane."},{"Start":"05:51.530 ","End":"05:53.300","Text":"Actually, if 2 of them are 0,"},{"Start":"05:53.300 ","End":"05:55.070","Text":"say, a and b are 0,"},{"Start":"05:55.070 ","End":"05:58.660","Text":"then x is x-naught and y is y-naught,"},{"Start":"05:58.660 ","End":"06:00.840","Text":"but then there\u0027s no equation."},{"Start":"06:00.840 ","End":"06:07.970","Text":"It just means that it\u0027s a line parallel to the z-axis through x-naught, y-naught."},{"Start":"06:07.970 ","End":"06:10.820","Text":"But I don\u0027t want to get into all these peculiar cases."},{"Start":"06:10.820 ","End":"06:11.900","Text":"If you do get them,"},{"Start":"06:11.900 ","End":"06:14.580","Text":"you can always use 1 of the other forms."},{"Start":"06:15.730 ","End":"06:19.830","Text":"Best now to do an example."},{"Start":"06:19.830 ","End":"06:23.219","Text":"For the example, I\u0027m going to give 2 points."},{"Start":"06:23.219 ","End":"06:25.530","Text":"I\u0027m going to give a point P,"},{"Start":"06:25.530 ","End":"06:30.165","Text":"which is 2, minus 1, 3,"},{"Start":"06:30.165 ","End":"06:32.015","Text":"and another point, I don\u0027t know,"},{"Start":"06:32.015 ","End":"06:39.270","Text":"Q will equal 1, 4, minus 3."},{"Start":"06:39.270 ","End":"06:42.835","Text":"My question is to find"},{"Start":"06:42.835 ","End":"06:49.010","Text":"the equation or equations of the line passing through these 2 points."},{"Start":"06:49.010 ","End":"06:51.305","Text":"I want all 3 forms, the vector form,"},{"Start":"06:51.305 ","End":"06:55.800","Text":"parametric form, and the symmetric equations."},{"Start":"06:56.060 ","End":"07:02.275","Text":"What I can do is for r-naught,"},{"Start":"07:02.275 ","End":"07:05.660","Text":"I could take just the position vector of P,"},{"Start":"07:05.660 ","End":"07:07.340","Text":"this could be P-naught."},{"Start":"07:07.340 ","End":"07:13.890","Text":"I\u0027m going to say that r-naught is a point on the line or the position vector,"},{"Start":"07:13.890 ","End":"07:16.395","Text":"so 2, minus 1,"},{"Start":"07:16.395 ","End":"07:21.310","Text":"3, and now I need also a direction vector."},{"Start":"07:21.310 ","End":"07:25.245","Text":"Need a position vector of a point on the line and a direction vector."},{"Start":"07:25.245 ","End":"07:26.640","Text":"For the direction vector,"},{"Start":"07:26.640 ","End":"07:32.835","Text":"the most obvious thing to do is to take the vector that joins P to Q."},{"Start":"07:32.835 ","End":"07:35.555","Text":"The tail is P and the head is Q."},{"Start":"07:35.555 ","End":"07:41.000","Text":"We subtract Q minus P coordinate-wise or component-wise,"},{"Start":"07:41.000 ","End":"07:44.310","Text":"1 minus 2 is minus 1,"},{"Start":"07:44.310 ","End":"07:48.620","Text":"4 minus minus 1 is 5,"},{"Start":"07:48.620 ","End":"07:53.825","Text":"and minus 3 minus 3 is minus 6."},{"Start":"07:53.825 ","End":"08:01.010","Text":"Now I have r naught and v. I can now say that the vector form is,"},{"Start":"08:01.010 ","End":"08:04.075","Text":"I\u0027ll use this form,"},{"Start":"08:04.075 ","End":"08:07.365","Text":"both of these are vector forms."},{"Start":"08:07.365 ","End":"08:10.470","Text":"I\u0027ll use that x,"},{"Start":"08:10.470 ","End":"08:15.615","Text":"y, z is equal to 2,"},{"Start":"08:15.615 ","End":"08:19.950","Text":"minus 1, 3; plus t,"},{"Start":"08:19.950 ","End":"08:26.740","Text":"a parameter; times minus 1, 5, minus 6."},{"Start":"08:27.200 ","End":"08:30.555","Text":"That\u0027s the vector form."},{"Start":"08:30.555 ","End":"08:40.360","Text":"The parametric form is just take some curly braces and say, x equals,"},{"Start":"08:40.360 ","End":"08:43.855","Text":"y equals, z equals,"},{"Start":"08:43.855 ","End":"08:50.390","Text":"and then it comes out to 2 minus t,"},{"Start":"08:51.050 ","End":"08:59.140","Text":"minus 1 plus 5t,"},{"Start":"08:59.140 ","End":"09:04.270","Text":"and z is going to be 3 minus 6t."},{"Start":"09:05.580 ","End":"09:10.010","Text":"I think I can squeeze in the other form as well."},{"Start":"09:10.010 ","End":"09:16.140","Text":"What we get is x minus the x naught,"},{"Start":"09:16.140 ","End":"09:21.185","Text":"x minus the 2 over the minus"},{"Start":"09:21.185 ","End":"09:27.610","Text":"1 is equal to y minus this,"},{"Start":"09:27.610 ","End":"09:35.305","Text":"so it\u0027s y plus 1 over the 5,"},{"Start":"09:35.305 ","End":"09:45.260","Text":"and then z minus 3 over minus 6."},{"Start":"09:45.840 ","End":"09:48.964","Text":"That answers the questions,"},{"Start":"09:48.964 ","End":"09:54.530","Text":"vector form, parametric form, symmetric equations."},{"Start":"09:55.400 ","End":"09:57.640","Text":"Before I finish this chapter,"},{"Start":"09:57.640 ","End":"10:01.485","Text":"I think we\u0027ll do 1 more example so I\u0027ll erase this 1."},{"Start":"10:01.485 ","End":"10:06.430","Text":"In this question, we\u0027re given a line and I\u0027m going to give it"},{"Start":"10:06.430 ","End":"10:10.870","Text":"in parametric form but I won\u0027t use the braces."},{"Start":"10:10.870 ","End":"10:16.985","Text":"I\u0027ll just write x equals 10 plus 3t,"},{"Start":"10:16.985 ","End":"10:20.585","Text":"and y equals 12t,"},{"Start":"10:20.585 ","End":"10:28.415","Text":"and z equals 3 minus t, that\u0027s 1 line."},{"Start":"10:28.415 ","End":"10:34.130","Text":"There\u0027s another line, and this is the 1 we\u0027re going to look for."},{"Start":"10:34.130 ","End":"10:41.780","Text":"What we know about this 1 is that it\u0027s parallel to the above line,"},{"Start":"10:41.780 ","End":"10:50.675","Text":"to this line, and it passes through a given point"},{"Start":"10:50.675 ","End":"11:00.364","Text":"and that point is 0, minus 3, 8."},{"Start":"11:00.364 ","End":"11:02.735","Text":"I haven\u0027t given you a question yet."},{"Start":"11:02.735 ","End":"11:08.260","Text":"The question is, does it cross the x-z plane?"},{"Start":"11:08.260 ","End":"11:10.490","Text":"Pass-through it?"},{"Start":"11:10.910 ","End":"11:14.280","Text":"If so, then where?"},{"Start":"11:14.280 ","End":"11:16.085","Text":"Give the coordinates of the point."},{"Start":"11:16.085 ","End":"11:20.555","Text":"I know that the middle coordinate will be 0 because y is 0, well, we\u0027ll see."},{"Start":"11:20.555 ","End":"11:23.540","Text":"What we do here,"},{"Start":"11:24.810 ","End":"11:29.815","Text":"the first thing to wonder is what does parallel mean?"},{"Start":"11:29.815 ","End":"11:32.585","Text":"Well, parallel means they go in the same direction,"},{"Start":"11:32.585 ","End":"11:35.040","Text":"they have the same direction vector."},{"Start":"11:35.040 ","End":"11:37.755","Text":"I know that the direction vector"},{"Start":"11:37.755 ","End":"11:41.745","Text":"of this line will be the same as the direction vector of this line."},{"Start":"11:41.745 ","End":"11:43.450","Text":"But I know the direction vector,"},{"Start":"11:43.450 ","End":"11:45.270","Text":"let\u0027s call it v, of this line."},{"Start":"11:45.270 ","End":"11:48.680","Text":"I just take the coefficients of the t,"},{"Start":"11:48.680 ","End":"11:52.310","Text":"the a, b, and c here are the direction vectors."},{"Start":"11:52.310 ","End":"12:01.180","Text":"In our case, we take 3,"},{"Start":"12:01.180 ","End":"12:04.965","Text":"12, minus 1 will be a direction vector."},{"Start":"12:04.965 ","End":"12:09.750","Text":"I also know a position vector because I have a point on the line,"},{"Start":"12:09.750 ","End":"12:20.595","Text":"so I can take my r naught to be 0, minus 3, 8."},{"Start":"12:20.595 ","End":"12:28.600","Text":"I already have the parametric form of the line where r or x,"},{"Start":"12:28.600 ","End":"12:30.310","Text":"y, z, let\u0027s write it as x, y,"},{"Start":"12:30.310 ","End":"12:32.910","Text":"z. I know that x,"},{"Start":"12:32.910 ","End":"12:41.970","Text":"y, z is equal to 0,"},{"Start":"12:41.970 ","End":"12:44.749","Text":"minus 3, 8,"},{"Start":"12:46.970 ","End":"12:53.825","Text":"plus t times the same direction vector,"},{"Start":"12:53.825 ","End":"12:58.970","Text":"3, 12, minus 1."},{"Start":"12:59.690 ","End":"13:06.365","Text":"Actually, it suits me to write it in parametric form,"},{"Start":"13:06.365 ","End":"13:09.885","Text":"but like before, I\u0027ll just write it all in 1 row."},{"Start":"13:09.885 ","End":"13:19.170","Text":"x equals 0 plus 3t,"},{"Start":"13:19.170 ","End":"13:23.860","Text":"y equals minus 3 plus 12t,"},{"Start":"13:25.590 ","End":"13:34.650","Text":"and z equals 8 minus 1t."},{"Start":"13:34.650 ","End":"13:38.850","Text":"Now I have the parametric equation and the vector equation of this line."},{"Start":"13:38.850 ","End":"13:40.474","Text":"Now what about the question,"},{"Start":"13:40.474 ","End":"13:42.695","Text":"does it cross the x-z plane?"},{"Start":"13:42.695 ","End":"13:45.170","Text":"What\u0027s special about the x-z plane?"},{"Start":"13:45.170 ","End":"13:46.600","Text":"In the x-z plane,"},{"Start":"13:46.600 ","End":"13:50.370","Text":"we know that y equals 0."},{"Start":"13:50.370 ","End":"13:53.880","Text":"That\u0027s the equation of the x-z plane, we\u0027ve seen it before."},{"Start":"13:53.880 ","End":"13:57.545","Text":"The other letter y that\u0027s missing is 0 there."},{"Start":"13:57.545 ","End":"14:03.815","Text":"All I have to do is compare this to this,"},{"Start":"14:03.815 ","End":"14:13.940","Text":"and I get an equation that minus 3 plus 12t equals 0."},{"Start":"14:13.940 ","End":"14:15.335","Text":"If I solve that,"},{"Start":"14:15.335 ","End":"14:17.805","Text":"I get 12t equals 3,"},{"Start":"14:17.805 ","End":"14:20.840","Text":"so t equals 3 over 12,"},{"Start":"14:20.840 ","End":"14:23.415","Text":"3 over 12 is 1/4."},{"Start":"14:23.415 ","End":"14:28.520","Text":"That\u0027s just the t. That answers the question,"},{"Start":"14:28.520 ","End":"14:30.050","Text":"the first 1, yes,"},{"Start":"14:30.050 ","End":"14:31.355","Text":"there is a solution,"},{"Start":"14:31.355 ","End":"14:36.650","Text":"but the \"where\" we have to put t equals 1/4 into here,"},{"Start":"14:36.650 ","End":"14:43.750","Text":"and so the where would be this plus 1/4,"},{"Start":"14:43.750 ","End":"14:46.325","Text":"well, I can do it here, put a 1/4 in here."},{"Start":"14:46.325 ","End":"14:52.430","Text":"x is going to be equal to 3/4 if t is 1/4;"},{"Start":"14:52.430 ","End":"14:55.650","Text":"y will equal 0 of course,"},{"Start":"14:55.650 ","End":"14:59.345","Text":"I mean that\u0027s the whole point of passing through the x-z plane;"},{"Start":"14:59.345 ","End":"15:05.555","Text":"and z will equal 8 minus 1/4,"},{"Start":"15:05.555 ","End":"15:10.490","Text":"so that would be 7 and 3/4."},{"Start":"15:10.490 ","End":"15:13.045","Text":"That\u0027s the answer to the where."},{"Start":"15:13.045 ","End":"15:14.960","Text":"This exercise is solved,"},{"Start":"15:14.960 ","End":"15:20.170","Text":"and we\u0027re done with lines in 3D."}],"ID":9754},{"Watched":false,"Name":"Exercises 5","Duration":"4m 50s","ChapterTopicVideoID":9799,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9799.jpeg","UploadDate":"2017-08-01T05:01:20.7930000","DurationForVideoObject":"PT4M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.370","Text":"In this exercise, we have to find the line which passes through 2 given points."},{"Start":"00:08.370 ","End":"00:12.750","Text":"We want all 3 forms of the line that we\u0027ve learned: the vector form,"},{"Start":"00:12.750 ","End":"00:16.240","Text":"parametric form, and symmetric form."},{"Start":"00:16.250 ","End":"00:19.405","Text":"Let\u0027s start with the vector form."},{"Start":"00:19.405 ","End":"00:26.270","Text":"In general, the vector form of a line would be that the general point r on"},{"Start":"00:26.270 ","End":"00:36.530","Text":"the line is equal to some position vector of a point on the line plus parameter t,"},{"Start":"00:36.530 ","End":"00:43.160","Text":"times the direction vector of the line meaning a vector which is parallel to the line."},{"Start":"00:43.160 ","End":"00:49.445","Text":"Now we can easily get both r naught and v because I could take"},{"Start":"00:49.445 ","End":"00:53.840","Text":"r naught to be the position vector of"},{"Start":"00:53.840 ","End":"00:58.625","Text":"either one of these and this one\u0027s the first one so I\u0027ll take this one."},{"Start":"00:58.625 ","End":"01:03.759","Text":"R naught I can take as minus 10, 4, 0."},{"Start":"01:03.759 ","End":"01:06.300","Text":"We\u0027re using bracket notation for vectors,"},{"Start":"01:06.300 ","End":"01:08.955","Text":"not the i, j, k, here."},{"Start":"01:08.955 ","End":"01:11.325","Text":"As the vector v,"},{"Start":"01:11.325 ","End":"01:14.480","Text":"there\u0027s any number of vectors parallel to the line,"},{"Start":"01:14.480 ","End":"01:19.980","Text":"but the easiest is to take the vector from here to here."},{"Start":"01:19.980 ","End":"01:22.745","Text":"It\u0027s the displacement vector."},{"Start":"01:22.745 ","End":"01:24.890","Text":"It would be a direction vector,"},{"Start":"01:24.890 ","End":"01:28.620","Text":"because v I could take that,"},{"Start":"01:28.620 ","End":"01:31.810","Text":"subtract the second minus the first."},{"Start":"01:33.380 ","End":"01:36.870","Text":"I\u0027ll just write it that way as 1 minus,"},{"Start":"01:36.870 ","End":"01:40.035","Text":"minus 10, and then afterwards we\u0027ll do the computation."},{"Start":"01:40.035 ","End":"01:43.575","Text":"Minus 4, minus 4,"},{"Start":"01:43.575 ","End":"01:47.895","Text":"and 2 minus 0,"},{"Start":"01:47.895 ","End":"01:52.410","Text":"and this comes out to be 1 minus,"},{"Start":"01:52.410 ","End":"01:55.905","Text":"minus 10 is 11."},{"Start":"01:55.905 ","End":"02:00.980","Text":"This comes out minus 8 and this is 2."},{"Start":"02:00.980 ","End":"02:03.425","Text":"Now that I have r naught and v,"},{"Start":"02:03.425 ","End":"02:13.940","Text":"I can say that the equation is that r is equal to minus 10,"},{"Start":"02:13.940 ","End":"02:18.740","Text":"4, 0, plus t"},{"Start":"02:18.740 ","End":"02:26.645","Text":"times 11, minus 8, 2."},{"Start":"02:26.645 ","End":"02:32.925","Text":"That\u0027s the vector form of the line."},{"Start":"02:32.925 ","End":"02:36.060","Text":"From the vector, it\u0027s easy to get to the parametric,"},{"Start":"02:36.060 ","End":"02:41.445","Text":"because basically you just say that r is x, y, z."},{"Start":"02:41.445 ","End":"02:47.255","Text":"The parametric, we can get from here component-wise by saying x equals,"},{"Start":"02:47.255 ","End":"02:50.145","Text":"y equals, z equals."},{"Start":"02:50.145 ","End":"02:58.710","Text":"Then we take first component which would be minus 10 plus 11t."},{"Start":"02:58.710 ","End":"03:01.450","Text":"In fact, you know what?"},{"Start":"03:05.180 ","End":"03:09.160","Text":"Let me go back and take this one further."},{"Start":"03:09.160 ","End":"03:15.795","Text":"I can do it component-wise and say this is minus 10 plus 11t."},{"Start":"03:15.795 ","End":"03:17.265","Text":"That\u0027s the first component."},{"Start":"03:17.265 ","End":"03:20.710","Text":"Then 4 minus 8t."},{"Start":"03:21.110 ","End":"03:27.225","Text":"Then 0 plus 2t which is just 2t."},{"Start":"03:27.225 ","End":"03:29.280","Text":"That\u0027s the vector form."},{"Start":"03:29.280 ","End":"03:30.730","Text":"Now I don\u0027t have to work hard,"},{"Start":"03:30.730 ","End":"03:36.445","Text":"I can just copy minus 10 plus 11t,"},{"Start":"03:36.445 ","End":"03:44.150","Text":"4 minus 8t, z equals 2t."},{"Start":"03:44.150 ","End":"03:51.045","Text":"This is the parametric form of the line."},{"Start":"03:51.045 ","End":"03:53.325","Text":"T is the parameter."},{"Start":"03:53.325 ","End":"03:56.260","Text":"For the symmetric form of the line,"},{"Start":"03:56.260 ","End":"04:01.100","Text":"we just isolate what t is."},{"Start":"04:01.520 ","End":"04:12.735","Text":"I can say from the first one that t is x plus 10 over 11."},{"Start":"04:12.735 ","End":"04:15.460","Text":"If I isolate t from the second one,"},{"Start":"04:15.460 ","End":"04:23.950","Text":"I\u0027ll get y minus 4 over minus 8."},{"Start":"04:24.050 ","End":"04:32.715","Text":"T from the last one is just z over 2."},{"Start":"04:32.715 ","End":"04:36.520","Text":"Then I don\u0027t need this."},{"Start":"04:36.800 ","End":"04:42.200","Text":"This would be the symmetric equation of the line."},{"Start":"04:42.200 ","End":"04:49.430","Text":"We\u0027re done."}],"ID":9755},{"Watched":false,"Name":"Exercises 6","Duration":"4m 13s","ChapterTopicVideoID":9800,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9800.jpeg","UploadDate":"2017-08-01T05:01:32.6400000","DurationForVideoObject":"PT4M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.140","Text":"In this exercise, we have a line or given a point on the line,"},{"Start":"00:07.140 ","End":"00:11.940","Text":"and another line that our line is parallel to."},{"Start":"00:11.940 ","End":"00:15.420","Text":"We want to find the line in 3 different forms,"},{"Start":"00:15.420 ","End":"00:18.610","Text":"vector, parametric, and symmetric."},{"Start":"00:18.710 ","End":"00:22.860","Text":"Easiest to start with the vector form of the line."},{"Start":"00:22.860 ","End":"00:28.700","Text":"For our line, what we need is a point on the line and a direction vector because then"},{"Start":"00:28.700 ","End":"00:34.340","Text":"we can say that the general point r on the line,"},{"Start":"00:34.340 ","End":"00:38.525","Text":"is the specific given point on the line,"},{"Start":"00:38.525 ","End":"00:44.080","Text":"plus parameter t, times a direction vector of the line."},{"Start":"00:44.080 ","End":"00:48.725","Text":"For r_0, we\u0027re okay,"},{"Start":"00:48.725 ","End":"00:53.960","Text":"because we can take that as the position vector of this point,"},{"Start":"00:53.960 ","End":"00:55.945","Text":"which just means this."},{"Start":"00:55.945 ","End":"01:02.665","Text":"Same numbers but different kinds of brackets to indicate that it\u0027s a vector, not a point."},{"Start":"01:02.665 ","End":"01:07.960","Text":"Now what we need is a direction vector for the line."},{"Start":"01:07.960 ","End":"01:12.095","Text":"Direction vector means a vector parallel to the line."},{"Start":"01:12.095 ","End":"01:15.310","Text":"If our line is parallel to this line,"},{"Start":"01:15.310 ","End":"01:18.460","Text":"and if I can find a vector parallel to this line,"},{"Start":"01:18.460 ","End":"01:23.275","Text":"it\u0027s also going to be parallel to our line because parallel to parallel is parallel."},{"Start":"01:23.275 ","End":"01:25.660","Text":"If a is parallel to b and b is parallel to c,"},{"Start":"01:25.660 ","End":"01:28.960","Text":"then a is parallel to c. We know that when"},{"Start":"01:28.960 ","End":"01:32.410","Text":"we have an equation like this in the parametric form,"},{"Start":"01:32.410 ","End":"01:37.060","Text":"that the coefficients of the t are a direction vector of the line."},{"Start":"01:37.060 ","End":"01:41.215","Text":"I\u0027m just looking here, here, and here."},{"Start":"01:41.215 ","End":"01:49.980","Text":"I\u0027ve got my v to be 4, 3, minus 5."},{"Start":"01:49.980 ","End":"01:51.770","Text":"If I plug-in here,"},{"Start":"01:51.770 ","End":"01:57.260","Text":"I\u0027ve got my vector equation of the line that r is equal to,"},{"Start":"01:57.260 ","End":"02:00.949","Text":"copying from here, minus 10, 4,"},{"Start":"02:00.949 ","End":"02:04.190","Text":"0 plus t times,"},{"Start":"02:04.190 ","End":"02:06.500","Text":"the direction vector of this line,"},{"Start":"02:06.500 ","End":"02:08.045","Text":"hence of our line,"},{"Start":"02:08.045 ","End":"02:12.750","Text":"which is 4, 3, minus 5."},{"Start":"02:14.240 ","End":"02:17.835","Text":"Let\u0027s just expand it component-wise."},{"Start":"02:17.835 ","End":"02:21.370","Text":"First component minus 10 plus 4t,"},{"Start":"02:21.770 ","End":"02:26.260","Text":"and the next component 4 plus 3t,"},{"Start":"02:27.320 ","End":"02:29.880","Text":"and what\u0027s the next one?"},{"Start":"02:29.880 ","End":"02:34.150","Text":"0 minus 5t, so just minus 5t."},{"Start":"02:34.790 ","End":"02:39.390","Text":"That\u0027s the vector form."},{"Start":"02:39.390 ","End":"02:43.880","Text":"For the parametric, we just take it component-wise."},{"Start":"02:43.880 ","End":"02:45.470","Text":"Sometimes I do it horizontally,"},{"Start":"02:45.470 ","End":"02:47.180","Text":"sometimes when have room,"},{"Start":"02:47.180 ","End":"02:50.370","Text":"I like to do it with a curly brace."},{"Start":"02:50.560 ","End":"02:52.680","Text":"Can do x, y, z,"},{"Start":"02:52.680 ","End":"02:54.125","Text":"one on top of the other."},{"Start":"02:54.125 ","End":"02:55.790","Text":"But just copying from here,"},{"Start":"02:55.790 ","End":"02:59.005","Text":"minus 10 plus 4t,"},{"Start":"02:59.005 ","End":"03:03.930","Text":"4 plus 3t, minus 5t."},{"Start":"03:03.930 ","End":"03:07.035","Text":"That\u0027s the parametric."},{"Start":"03:07.035 ","End":"03:09.055","Text":"For the symmetric,"},{"Start":"03:09.055 ","End":"03:14.340","Text":"what we do is we isolate t from each of these."},{"Start":"03:14.340 ","End":"03:24.440","Text":"From here, we would have t equals x plus 10 over 4, from this one,"},{"Start":"03:24.440 ","End":"03:31.420","Text":"we would get that t equals y minus 4/3,"},{"Start":"03:31.420 ","End":"03:39.615","Text":"and from the last one we would get that t equals z over minus 5."},{"Start":"03:39.615 ","End":"03:41.665","Text":"If you compare the ts,"},{"Start":"03:41.665 ","End":"03:43.850","Text":"you get the symmetric form,"},{"Start":"03:43.850 ","End":"03:51.655","Text":"which is x plus 10 over 4."},{"Start":"03:51.655 ","End":"03:55.430","Text":"Then that\u0027s going to equal the t from here,"},{"Start":"03:55.430 ","End":"04:00.170","Text":"which is y, minus 4 over 3."},{"Start":"04:00.170 ","End":"04:07.335","Text":"This is going to equal z over minus 5."},{"Start":"04:07.335 ","End":"04:12.970","Text":"That gives us the symmetric and that\u0027s it."}],"ID":9756},{"Watched":false,"Name":"Exercises 7","Duration":"6m 49s","ChapterTopicVideoID":9801,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9801.jpeg","UploadDate":"2017-08-01T05:01:52.8670000","DurationForVideoObject":"PT6M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"In this exercise, we have 2 lines, l_1 and l_1."},{"Start":"00:04.725 ","End":"00:09.000","Text":"L_1 is given by 2 points that it goes through,"},{"Start":"00:09.000 ","End":"00:14.805","Text":"and l_2 is given in the form of the vector equation of the line."},{"Start":"00:14.805 ","End":"00:18.720","Text":"We want to know whether these 2 lines are parallel,"},{"Start":"00:18.720 ","End":"00:22.240","Text":"or perpendicular, or neither."},{"Start":"00:22.730 ","End":"00:27.480","Text":"The most useful thing we can do for either the parallel or the"},{"Start":"00:27.480 ","End":"00:32.340","Text":"perpendicular is to find direction vectors for each of the lines."},{"Start":"00:32.340 ","End":"00:36.520","Text":"For l_1, let\u0027s say that the direction vector is v_1."},{"Start":"00:36.520 ","End":"00:39.265","Text":"Actually, I shouldn\u0027t say the direction vector,"},{"Start":"00:39.265 ","End":"00:44.030","Text":"a direction vector because is infinitely number."},{"Start":"00:44.030 ","End":"00:48.980","Text":"I mean, anything times a scalar will still be a direction vector, I\u0027m just mentioning."},{"Start":"00:48.980 ","End":"00:54.245","Text":"But the easiest direction vector to find is"},{"Start":"00:54.245 ","End":"00:59.510","Text":"the displacement vector that would take me from this point to this point."},{"Start":"00:59.510 ","End":"01:05.105","Text":"What I\u0027ll do is say that direction vector for line 1, call it v_1."},{"Start":"01:05.105 ","End":"01:08.525","Text":"One possibility is to subtract,"},{"Start":"01:08.525 ","End":"01:13.400","Text":"say 2 minus 4, 0 minus 1,"},{"Start":"01:13.400 ","End":"01:17.280","Text":"9 minus minus 5,"},{"Start":"01:17.280 ","End":"01:21.720","Text":"which is minus 2,"},{"Start":"01:21.720 ","End":"01:26.830","Text":"minus 1, 9 plus 5 is 14."},{"Start":"01:26.960 ","End":"01:32.954","Text":"Now, a direction vector for line 2,"},{"Start":"01:32.954 ","End":"01:37.340","Text":"call it v_2 can easily be seen from"},{"Start":"01:37.340 ","End":"01:43.205","Text":"the vector form by taking the coefficients of the t in each of the components."},{"Start":"01:43.205 ","End":"01:45.130","Text":"This would be."},{"Start":"01:45.130 ","End":"01:47.100","Text":"We don\u0027t have a t here,"},{"Start":"01:47.100 ","End":"01:49.110","Text":"so the coefficient, it\u0027s like 0,"},{"Start":"01:49.110 ","End":"01:51.280","Text":"like 5 plus 0t."},{"Start":"01:51.280 ","End":"01:54.875","Text":"So 0 here minus 9,"},{"Start":"01:54.875 ","End":"01:58.240","Text":"and here minus 4."},{"Start":"01:58.240 ","End":"02:01.840","Text":"No need to simplify, that\u0027s that."},{"Start":"02:01.840 ","End":"02:11.850","Text":"Now, this direction vector is parallel to l_1 and this is parallel to l_2."},{"Start":"02:11.850 ","End":"02:16.700","Text":"In order for l_1 and l_2 to be parallel or not,"},{"Start":"02:16.700 ","End":"02:21.945","Text":"it\u0027s the same thing as seeing if v_1 and v_2 are parallel or not."},{"Start":"02:21.945 ","End":"02:26.015","Text":"How can we check if these 2 vectors are parallel?"},{"Start":"02:26.015 ","End":"02:30.545","Text":"How do I know if this one or this one are parallel?"},{"Start":"02:30.545 ","End":"02:35.770","Text":"Well, one of them has to be a scalar multiple of the other."},{"Start":"02:35.770 ","End":"02:37.905","Text":"Now if the vectors are not 0,"},{"Start":"02:37.905 ","End":"02:41.690","Text":"0 is one of those debatable vectors if it\u0027s parallel or not."},{"Start":"02:41.690 ","End":"02:43.325","Text":"Anyway, neither of them is 0."},{"Start":"02:43.325 ","End":"02:45.935","Text":"One of them has to be some constant,"},{"Start":"02:45.935 ","End":"02:53.520","Text":"a non-zero constant, say v_2 has got to be some non-zero constant times v_1."},{"Start":"02:54.070 ","End":"02:58.465","Text":"The question is whether there is such a k,"},{"Start":"02:58.465 ","End":"03:03.030","Text":"and I claim there\u0027s no such k. Whichever side you put the k on,"},{"Start":"03:03.030 ","End":"03:06.195","Text":"if you went for v_1 is k times v_2."},{"Start":"03:06.195 ","End":"03:11.445","Text":"The reason is, because look I have a 0 here."},{"Start":"03:11.445 ","End":"03:21.910","Text":"So the only way v_2 could be k times v_1 is if 0 is k times minus 2."},{"Start":"03:22.520 ","End":"03:26.330","Text":"Maybe I\u0027ll write this down. Looking at the first component,"},{"Start":"03:26.330 ","End":"03:29.930","Text":"we get 0 is k times minus 2,"},{"Start":"03:29.930 ","End":"03:34.445","Text":"and that would mean that k is 0, and if k is 0,"},{"Start":"03:34.445 ","End":"03:40.205","Text":"then v_2 would be the 0 vector, which it isn\u0027t."},{"Start":"03:40.205 ","End":"03:42.440","Text":"So the answer is no,"},{"Start":"03:42.440 ","End":"03:46.295","Text":"there is no such k, doesn\u0027t exist."},{"Start":"03:46.295 ","End":"03:50.730","Text":"They are not parallel."},{"Start":"03:52.160 ","End":"03:55.175","Text":"If you tried it the other way around,"},{"Start":"03:55.175 ","End":"03:57.905","Text":"v_1 is k times v_2,"},{"Start":"03:57.905 ","End":"04:01.010","Text":"then you\u0027d have minus 2 is k times 0,"},{"Start":"04:01.010 ","End":"04:03.310","Text":"not going to work either."},{"Start":"04:03.310 ","End":"04:07.285","Text":"Either way, it\u0027s no good, so not parallel."},{"Start":"04:07.285 ","End":"04:10.995","Text":"Next, let\u0027s go for perpendicular."},{"Start":"04:10.995 ","End":"04:17.930","Text":"Now l_1 and l_2 for them to be perpendicular,"},{"Start":"04:17.930 ","End":"04:21.000","Text":"it\u0027s actually 2 conditions."},{"Start":"04:22.190 ","End":"04:25.765","Text":"That means they intersect."},{"Start":"04:25.765 ","End":"04:28.205","Text":"They could be skewed to each other."},{"Start":"04:28.205 ","End":"04:35.090","Text":"They have to be in the intersect and they have to intersect at 90 degrees."},{"Start":"04:35.090 ","End":"04:38.555","Text":"For working in radians is Pi over 2."},{"Start":"04:38.555 ","End":"04:43.070","Text":"90 degrees to each other means the dot product is 0."},{"Start":"04:43.070 ","End":"04:46.775","Text":"Let\u0027s see. Let\u0027s try the second condition."},{"Start":"04:46.775 ","End":"04:50.000","Text":"I mean both of these have to happen, they have to intersect,"},{"Start":"04:50.000 ","End":"04:52.969","Text":"and the angle between them has to be 90 degrees."},{"Start":"04:52.969 ","End":"04:55.069","Text":"Now if the angle is 90 degrees,"},{"Start":"04:55.069 ","End":"05:04.350","Text":"that means that they\u0027re"},{"Start":"05:04.350 ","End":"05:06.170","Text":"perpendicular to each other."},{"Start":"05:06.170 ","End":"05:08.240","Text":"They\u0027re also going to be perpendicular."},{"Start":"05:08.240 ","End":"05:12.890","Text":"But with vectors, we just have to have that"},{"Start":"05:12.890 ","End":"05:18.350","Text":"they are 90 degrees to each other because vectors are not tied to a specific place."},{"Start":"05:18.350 ","End":"05:26.285","Text":"In short, we have to check that v_1.v_2, is it equal to 0?"},{"Start":"05:26.285 ","End":"05:31.355","Text":"Because that\u0027s the condition for perpendicularity with vectors."},{"Start":"05:31.355 ","End":"05:34.145","Text":"These are parallel to these,"},{"Start":"05:34.145 ","End":"05:37.295","Text":"and these are 90 degrees and these are at 90 degrees."},{"Start":"05:37.295 ","End":"05:39.995","Text":"Let\u0027s see what this is."},{"Start":"05:39.995 ","End":"05:45.365","Text":"Here\u0027s v_1 minus 2 minus 1,"},{"Start":"05:45.365 ","End":"05:54.090","Text":"14.product with 0 minus 9 minus 4,"},{"Start":"05:54.090 ","End":"05:55.860","Text":"and this is equal to."},{"Start":"05:55.860 ","End":"05:57.700","Text":"This times this is 0,"},{"Start":"05:57.700 ","End":"06:09.160","Text":"this times this is 9, 9 minus 56,"},{"Start":"06:09.160 ","End":"06:11.530","Text":"which was not 0 in any rate."},{"Start":"06:11.530 ","End":"06:13.120","Text":"What is it equal to?"},{"Start":"06:13.120 ","End":"06:17.445","Text":"Minus 47, I believe,"},{"Start":"06:17.445 ","End":"06:20.340","Text":"anyway, it\u0027s not 0."},{"Start":"06:20.340 ","End":"06:25.255","Text":"Not perpendicular, so I don\u0027t even have to check if they intersect or not."},{"Start":"06:25.255 ","End":"06:27.190","Text":"But if I did get 0,"},{"Start":"06:27.190 ","End":"06:30.970","Text":"then it wouldn\u0027t be enough for lines to be perpendicular."},{"Start":"06:30.970 ","End":"06:33.520","Text":"They also have to intersect in any event,"},{"Start":"06:33.520 ","End":"06:34.990","Text":"so they\u0027re not 90 degrees,"},{"Start":"06:34.990 ","End":"06:41.180","Text":"so we\u0027re done, not parallel, not perpendicular."},{"Start":"06:41.180 ","End":"06:43.945","Text":"Abbreviate that to PERP."},{"Start":"06:43.945 ","End":"06:49.470","Text":"The answer is neither and we\u0027re done."}],"ID":9757},{"Watched":false,"Name":"Exercises 8","Duration":"7m 9s","ChapterTopicVideoID":9796,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9796.jpeg","UploadDate":"2017-08-01T05:00:33.1470000","DurationForVideoObject":"PT7M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this exercise, we\u0027re given 2 lines, slightly different forms."},{"Start":"00:05.070 ","End":"00:12.120","Text":"This l1 is given in parametric form and l2 is in vector form. Very similar."},{"Start":"00:12.120 ","End":"00:17.790","Text":"I\u0027ll just write them both in parametric form just for uniformity."},{"Start":"00:18.620 ","End":"00:24.720","Text":"L1 could be written as x equals y equals,"},{"Start":"00:24.720 ","End":"00:27.565","Text":"z equals, first thing in a moment."},{"Start":"00:27.565 ","End":"00:31.980","Text":"L2 we can also write x equals,"},{"Start":"00:31.980 ","End":"00:34.875","Text":"y equals, z equals."},{"Start":"00:34.875 ","End":"00:40.095","Text":"In the first 1, just copying minus 7 plus 12t,"},{"Start":"00:40.095 ","End":"00:44.010","Text":"from here, 3 minus t,"},{"Start":"00:44.010 ","End":"00:47.160","Text":"and then 14 plus 8t."},{"Start":"00:47.160 ","End":"00:50.390","Text":"From the second 1, just component-wise,"},{"Start":"00:50.390 ","End":"00:57.480","Text":"8 plus t, 5 plus 6t, 4 minus 2t."},{"Start":"00:57.480 ","End":"01:01.270","Text":"Now, for these to intersect,"},{"Start":"01:01.270 ","End":"01:05.380","Text":"you\u0027ve got to have some value of t here that will give us a point x, y,"},{"Start":"01:05.380 ","End":"01:10.050","Text":"z, and a different value of t possibly here,"},{"Start":"01:10.050 ","End":"01:12.720","Text":"that will give us the same x, y, z."},{"Start":"01:12.720 ","End":"01:15.820","Text":"The point is that it doesn\u0027t have to be the same t and it,"},{"Start":"01:15.820 ","End":"01:17.745","Text":"usually, will not be."},{"Start":"01:17.745 ","End":"01:21.300","Text":"Let\u0027s say, I\u0027ll use some couple of other letters,"},{"Start":"01:21.300 ","End":"01:23.295","Text":"let\u0027s say when I plug in here,"},{"Start":"01:23.295 ","End":"01:25.910","Text":"t equals, what\u0027s the next letter?"},{"Start":"01:25.910 ","End":"01:29.900","Text":"U. I plug in here t equals v,"},{"Start":"01:29.900 ","End":"01:32.860","Text":"I should get the same x, y, z."},{"Start":"01:32.860 ","End":"01:37.155","Text":"That will give me 3 equations."},{"Start":"01:37.155 ","End":"01:45.510","Text":"Here I\u0027ll get minus 7 plus 12u."},{"Start":"01:45.510 ","End":"01:48.750","Text":"I\u0027ll just write these first,"},{"Start":"01:48.750 ","End":"01:49.950","Text":"plug in u here."},{"Start":"01:49.950 ","End":"01:52.065","Text":"I\u0027ve got 3 minus u,"},{"Start":"01:52.065 ","End":"01:56.895","Text":"I\u0027m plugging u here, 14 plus 8u."},{"Start":"01:56.895 ","End":"01:58.780","Text":"Now here I\u0027ll put it in another value,"},{"Start":"01:58.780 ","End":"02:02.020","Text":"v. So 8 plus v,"},{"Start":"02:02.020 ","End":"02:08.720","Text":"5 plus 6v, 4 minus 2v."},{"Start":"02:09.140 ","End":"02:16.960","Text":"Any letters, this is just to emphasize that it\u0027s important that we can\u0027t assume it\u0027s"},{"Start":"02:16.960 ","End":"02:21.140","Text":"the same letter t."},{"Start":"02:22.130 ","End":"02:27.675","Text":"What we have here is 3 equations and 2 unknowns."},{"Start":"02:27.675 ","End":"02:32.000","Text":"In general, when you have more equations than unknowns,"},{"Start":"02:32.000 ","End":"02:37.020","Text":"typically there won\u0027t be a solution, but there could be."},{"Start":"02:37.070 ","End":"02:39.675","Text":"That\u0027s what we\u0027re going to do here."},{"Start":"02:39.675 ","End":"02:43.205","Text":"What we are going to do,"},{"Start":"02:43.205 ","End":"02:46.040","Text":"what I suggest is to choose 2 of these 3,"},{"Start":"02:46.040 ","End":"02:47.830","Text":"it doesn\u0027t really matter."},{"Start":"02:47.830 ","End":"02:50.610","Text":"Let\u0027s say I\u0027ll choose the first 2."},{"Start":"02:50.610 ","End":"02:53.745","Text":"Then I\u0027ll have 2 equations and 2 unknowns,"},{"Start":"02:53.745 ","End":"02:56.370","Text":"solve for u and v m,"},{"Start":"02:56.370 ","End":"03:01.260","Text":"and see if the u and v and that I found fit the third equation also."},{"Start":"03:01.260 ","End":"03:05.890","Text":"Then we\u0027ll say, \u0027\u0027Yes, we have a solution,\u0027\u0027 or, \u0027\u0027No we don\u0027t.\u0027\u0027"},{"Start":"03:06.980 ","End":"03:10.695","Text":"I\u0027ll just copy the first 2 equations,"},{"Start":"03:10.695 ","End":"03:12.555","Text":"these 2 over here,"},{"Start":"03:12.555 ","End":"03:18.180","Text":"minus 7 plus 12u equals 8 plus v,"},{"Start":"03:18.180 ","End":"03:25.520","Text":"3 minus u equals 5 plus 6v."},{"Start":"03:25.520 ","End":"03:27.275","Text":"There\u0027s many ways to solve this."},{"Start":"03:27.275 ","End":"03:30.200","Text":"Let me just see if there\u0027s anything easy to do."},{"Start":"03:30.200 ","End":"03:35.930","Text":"I think I\u0027ll isolate v from the first equation and get that v equals,"},{"Start":"03:35.930 ","End":"03:37.985","Text":"bring the 8 to the other side,"},{"Start":"03:37.985 ","End":"03:39.850","Text":"so I have 12u,"},{"Start":"03:39.850 ","End":"03:43.955","Text":"minus 7 minus 8 is minus 15."},{"Start":"03:43.955 ","End":"03:49.855","Text":"Then I can put this into the v here."},{"Start":"03:49.855 ","End":"04:00.790","Text":"I will get 3 minus u equals 5 plus 6 times 12u minus 15."},{"Start":"04:00.790 ","End":"04:03.630","Text":"Then, let\u0027s expand the brackets."},{"Start":"04:03.630 ","End":"04:07.290","Text":"3 minus u equals 5,"},{"Start":"04:07.290 ","End":"04:14.910","Text":"plus 6 times 12 is 72u, minus 90."},{"Start":"04:14.910 ","End":"04:18.870","Text":"I\u0027ll put the us on the right,"},{"Start":"04:18.870 ","End":"04:20.894","Text":"but then I\u0027ll switch sides."},{"Start":"04:20.894 ","End":"04:25.380","Text":"I\u0027ll get here, 73u,"},{"Start":"04:25.380 ","End":"04:30.315","Text":"and the numbers on the left which will then go on the right,"},{"Start":"04:30.315 ","End":"04:40.485","Text":"3 plus 90 is 93 minus 5 is 88."},{"Start":"04:40.485 ","End":"04:51.480","Text":"We get that u is equal to 88 over 73."},{"Start":"04:51.480 ","End":"04:54.900","Text":"Numbers often come out messy,"},{"Start":"04:54.900 ","End":"05:01.260","Text":"I cooked it up so it will come out nice, 88 over 73."},{"Start":"05:01.260 ","End":"05:04.885","Text":"Now let\u0027s continue. We have u,"},{"Start":"05:04.885 ","End":"05:11.719","Text":"so we can plug u in this value into here,"},{"Start":"05:11.719 ","End":"05:16.630","Text":"and then we\u0027ll get that v equals"},{"Start":"05:16.630 ","End":"05:26.265","Text":"12 times 88 over 73 minus 15."},{"Start":"05:26.265 ","End":"05:31.310","Text":"This comes out minus 39 over 73."},{"Start":"05:31.310 ","End":"05:36.150","Text":"I won\u0027t show you all the work, exercise in fractions."},{"Start":"05:36.150 ","End":"05:46.080","Text":"I have now u and I have v. What we have to do is just to highlight the important."},{"Start":"05:46.080 ","End":"05:51.750","Text":"That\u0027s u and v is this."},{"Start":"05:51.750 ","End":"05:59.670","Text":"Then I\u0027ll plug in v here and u here."},{"Start":"05:59.670 ","End":"06:02.520","Text":"We\u0027ll do it over here."},{"Start":"06:02.520 ","End":"06:12.840","Text":"The question is 14 plus 8 times u, 88 over 73."},{"Start":"06:12.840 ","End":"06:15.589","Text":"Now we\u0027re verifying, we\u0027re checking,"},{"Start":"06:15.589 ","End":"06:17.645","Text":"I\u0027m putting a question mark here."},{"Start":"06:17.645 ","End":"06:23.390","Text":"Does it equal 4 minus twice v,"},{"Start":"06:23.390 ","End":"06:29.090","Text":"which is minus 39 over 73?"},{"Start":"06:29.090 ","End":"06:37.010","Text":"This comes out 1726 over 73,"},{"Start":"06:37.010 ","End":"06:44.865","Text":"and this comes out to be 370 over 73."},{"Start":"06:44.865 ","End":"06:48.880","Text":"These are definitely not equal."},{"Start":"06:50.090 ","End":"06:57.680","Text":"That means that there are no u and v which satisfy all 3 of these."},{"Start":"06:57.680 ","End":"07:00.245","Text":"Do l1 and l2 intersect?"},{"Start":"07:00.245 ","End":"07:02.850","Text":"The answer is no."},{"Start":"07:02.850 ","End":"07:05.959","Text":"Then there\u0027s no point in continuing to find the intersection"},{"Start":"07:05.959 ","End":"07:09.960","Text":"if they don\u0027t. That\u0027s the answer."}],"ID":9758},{"Watched":false,"Name":"Exercises 9","Duration":"8m 38s","ChapterTopicVideoID":9797,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9797.jpeg","UploadDate":"2017-08-01T05:00:56.8900000","DurationForVideoObject":"PT8M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have 2 lines, l_1 and l_2."},{"Start":"00:04.260 ","End":"00:10.725","Text":"L_1 is defined by 2 points on the line,"},{"Start":"00:10.725 ","End":"00:13.920","Text":"which is less convenient than l_2,"},{"Start":"00:13.920 ","End":"00:18.030","Text":"which is given to us as a vector equation with a parameter"},{"Start":"00:18.030 ","End":"00:23.355","Text":"t. I\u0027d prefer to have l_1 in this form also."},{"Start":"00:23.355 ","End":"00:24.870","Text":"Then we can answer the question,"},{"Start":"00:24.870 ","End":"00:28.020","Text":"if these 2 lines intersect and if they do,"},{"Start":"00:28.020 ","End":"00:30.105","Text":"let\u0027s find the intersection point."},{"Start":"00:30.105 ","End":"00:32.955","Text":"What I can do for l_1,"},{"Start":"00:32.955 ","End":"00:36.810","Text":"would be to write it as the usual formula."},{"Start":"00:36.810 ","End":"00:39.225","Text":"We take one of the points,"},{"Start":"00:39.225 ","End":"00:40.410","Text":"say the first one,"},{"Start":"00:40.410 ","End":"00:44.190","Text":"its position vector is minus 5, 0,"},{"Start":"00:44.190 ","End":"00:48.590","Text":"2 plus t times the direction vector,"},{"Start":"00:48.590 ","End":"00:51.140","Text":"which I can get by subtracting 1 from the other set,"},{"Start":"00:51.140 ","End":"00:56.180","Text":"a second minus the first,13 less minus 5 is 18,"},{"Start":"00:56.180 ","End":"01:02.450","Text":"minus 2, minus 1."},{"Start":"01:02.450 ","End":"01:06.660","Text":"If I put it in parametric form,"},{"Start":"01:06.660 ","End":"01:08.100","Text":"it\u0027s even more convenient."},{"Start":"01:08.100 ","End":"01:09.779","Text":"I can say x equals,"},{"Start":"01:09.779 ","End":"01:12.510","Text":"y equals, z equals,"},{"Start":"01:12.510 ","End":"01:18.540","Text":"component-wise minus 5 plus 18t."},{"Start":"01:18.540 ","End":"01:22.710","Text":"Then 0 minus 2t and,"},{"Start":"01:22.710 ","End":"01:27.390","Text":"2 minus t, that\u0027s l_1."},{"Start":"01:27.390 ","End":"01:34.320","Text":"Now l_2 from here,"},{"Start":"01:34.320 ","End":"01:37.260","Text":"I\u0027ll just write it in x, y,"},{"Start":"01:37.260 ","End":"01:44.180","Text":"z form, the vector form is almost the same as the parametric, very close."},{"Start":"01:44.180 ","End":"01:47.720","Text":"Here we have x equals 3,"},{"Start":"01:47.720 ","End":"01:52.580","Text":"y equals minus 1 minus t,"},{"Start":"01:52.580 ","End":"01:56.790","Text":"and z equals 2 plus 4t."},{"Start":"01:57.380 ","End":"02:00.840","Text":"Now, if they intersect,"},{"Start":"02:00.840 ","End":"02:05.795","Text":"the sum value of t here and sum value of t here,"},{"Start":"02:05.795 ","End":"02:08.635","Text":"that will give us the same x, y, z,"},{"Start":"02:08.635 ","End":"02:11.590","Text":"but they\u0027re going to be different values of t."},{"Start":"02:11.590 ","End":"02:15.740","Text":"The t for here and the t for here may not be the same."},{"Start":"02:15.740 ","End":"02:18.380","Text":"Let me use 2 other letters like,"},{"Start":"02:18.380 ","End":"02:22.865","Text":"u and v. If I let t equals u here,"},{"Start":"02:22.865 ","End":"02:25.490","Text":"and t equals v here,"},{"Start":"02:25.490 ","End":"02:29.000","Text":"I hopefully will, what I\u0027m looking for,"},{"Start":"02:29.000 ","End":"02:31.395","Text":"get the same point."},{"Start":"02:31.395 ","End":"02:35.015","Text":"If I do that, I\u0027ll get the following equations."},{"Start":"02:35.015 ","End":"02:37.190","Text":"If I plug in u here,"},{"Start":"02:37.190 ","End":"02:41.480","Text":"I\u0027ve got minus 5 plus 18u."},{"Start":"02:41.480 ","End":"02:43.855","Text":"If I plug in v here,"},{"Start":"02:43.855 ","End":"02:46.590","Text":"I think the plug in with x,"},{"Start":"02:46.590 ","End":"02:48.345","Text":"that will equal 3."},{"Start":"02:48.345 ","End":"02:54.810","Text":"Here I\u0027ll get minus 2 u equals,"},{"Start":"02:54.810 ","End":"03:00.780","Text":"and here minus 1 minus v. The last one for z,"},{"Start":"03:00.780 ","End":"03:04.499","Text":"I\u0027ll get 2 minus u"},{"Start":"03:04.499 ","End":"03:11.100","Text":"equals 2 plus 4v."},{"Start":"03:11.100 ","End":"03:15.600","Text":"The equation for x, for y and for z,"},{"Start":"03:15.600 ","End":"03:20.475","Text":"and I don\u0027t know if it has a solution because very often,"},{"Start":"03:20.475 ","End":"03:26.215","Text":"if you have more equations than unknowns,"},{"Start":"03:26.215 ","End":"03:29.890","Text":"you\u0027re more likely not to have a solution but we don\u0027t know,"},{"Start":"03:29.890 ","End":"03:31.435","Text":"there could be a solution."},{"Start":"03:31.435 ","End":"03:38.445","Text":"What we\u0027re going to do, 1 way is to take just 2 of these 3 equations."},{"Start":"03:38.445 ","End":"03:40.590","Text":"Let\u0027s say take the first 2,"},{"Start":"03:40.590 ","End":"03:43.080","Text":"then I have 2 equations and 2 unknowns."},{"Start":"03:43.080 ","End":"03:45.060","Text":"Solve that for u and v,"},{"Start":"03:45.060 ","End":"03:49.180","Text":"and then check if it satisfies the third one,"},{"Start":"03:49.180 ","End":"03:51.100","Text":"the values of u and v that we get."},{"Start":"03:51.100 ","End":"03:54.020","Text":"I\u0027ll write these 2 over here."},{"Start":"03:54.060 ","End":"04:00.505","Text":"Just copying minus 5 plus 18u, equals 3,"},{"Start":"04:00.505 ","End":"04:08.450","Text":"and minus 2u equals minus 1 minus v. Looking at this,"},{"Start":"04:08.450 ","End":"04:09.710","Text":"what I think we could do,"},{"Start":"04:09.710 ","End":"04:16.940","Text":"would be to extract v from the second one and then substitute in the first."},{"Start":"04:16.940 ","End":"04:21.755","Text":"Actually there is no v in the first."},{"Start":"04:21.755 ","End":"04:27.420","Text":"On second thoughts, we\u0027ll just solve the first one for u because v is missing from it."},{"Start":"04:27.420 ","End":"04:34.325","Text":"What we\u0027ll get is 18u equals 3 plus 5 is 8,"},{"Start":"04:34.325 ","End":"04:39.440","Text":"which gives us that u is 8 over 18,"},{"Start":"04:39.440 ","End":"04:43.085","Text":"which I can cancel to 4/9."},{"Start":"04:43.085 ","End":"04:53.840","Text":"Then I\u0027ll plug the value of u from here into here and so I\u0027ll get minus 2u,"},{"Start":"04:53.840 ","End":"05:00.770","Text":"that\u0027s minus 8/9, equals minus"},{"Start":"05:00.770 ","End":"05:08.985","Text":"1 minus v. Bring the 1 to the other side,"},{"Start":"05:08.985 ","End":"05:12.479","Text":"we got minus 8/9 plus 1,"},{"Start":"05:12.479 ","End":"05:15.100","Text":"it\u0027s going to be 1/9."},{"Start":"05:15.290 ","End":"05:24.280","Text":"That\u0027s minus v, so v will be minus a 1/9."},{"Start":"05:25.940 ","End":"05:31.959","Text":"We found u and v for the first 2 equations,"},{"Start":"05:31.959 ","End":"05:35.120","Text":"and now we need to check with the last equation."},{"Start":"05:35.120 ","End":"05:43.985","Text":"I\u0027ll take this u and put it into here and I\u0027ll take v from here and put it into here."},{"Start":"05:43.985 ","End":"05:46.025","Text":"Let\u0027s see what we get,"},{"Start":"05:46.025 ","End":"05:53.235","Text":"2 minus u is 4/9 equals, but question mark,"},{"Start":"05:53.235 ","End":"05:54.945","Text":"that\u0027s what we\u0027re going to check,"},{"Start":"05:54.945 ","End":"05:57.110","Text":"we know the first 2 are satisfied,"},{"Start":"05:57.110 ","End":"05:58.580","Text":"we don\u0027t know about the third,"},{"Start":"05:58.580 ","End":"06:01.665","Text":"equals 2 plus 4,"},{"Start":"06:01.665 ","End":"06:06.490","Text":"and v is minus 1/9."},{"Start":"06:06.980 ","End":"06:14.800","Text":"Yes, they are equal because here it\u0027s minus 4/9 and here is minus 4/9."},{"Start":"06:14.800 ","End":"06:20.860","Text":"The answer would be 1 and 5/9 for both sides,"},{"Start":"06:20.860 ","End":"06:23.200","Text":"but we can already see that they\u0027re equal."},{"Start":"06:23.200 ","End":"06:26.019","Text":"Yes, it is equal,"},{"Start":"06:26.019 ","End":"06:28.420","Text":"and so they intersect."},{"Start":"06:28.420 ","End":"06:32.240","Text":"Now all we have to do is find the intersection point."},{"Start":"06:32.240 ","End":"06:37.765","Text":"What we do is, take either 1 of these lines,"},{"Start":"06:37.765 ","End":"06:40.150","Text":"whichever is more convenient,"},{"Start":"06:40.150 ","End":"06:45.160","Text":"l_1 or l_2, and plug-in the value."},{"Start":"06:45.160 ","End":"06:50.610","Text":"Let me plug these into l_2,"},{"Start":"06:50.610 ","End":"06:52.840","Text":"I think will be slightly easier."},{"Start":"06:52.840 ","End":"06:54.910","Text":"They should get the same answer for both."},{"Start":"06:54.910 ","End":"06:57.025","Text":"If I plug in to l_2,"},{"Start":"06:57.025 ","End":"07:00.994","Text":"what I need is to let t equal the v,"},{"Start":"07:00.994 ","End":"07:05.510","Text":"which is minus 1/9."},{"Start":"07:06.180 ","End":"07:10.985","Text":"Let just me copy this down here,"},{"Start":"07:10.985 ","End":"07:13.620","Text":"so there\u0027s a copy paste."},{"Start":"07:13.620 ","End":"07:17.920","Text":"Now I just have to plug-in to all these,"},{"Start":"07:19.340 ","End":"07:29.869","Text":"t equals minus 1/9 so we\u0027ll get x equals 3,"},{"Start":"07:29.869 ","End":"07:31.605","Text":"because nothing to substitute."},{"Start":"07:31.605 ","End":"07:35.890","Text":"Y equals minus 1 minus t,"},{"Start":"07:35.890 ","End":"07:41.545","Text":"is minus 1 plus 1/9, is minus 8/9."},{"Start":"07:41.545 ","End":"07:50.740","Text":"Z equals 2 minus 4/9,"},{"Start":"07:50.740 ","End":"08:00.615","Text":"which will be 1 and 5/9."},{"Start":"08:00.615 ","End":"08:07.385","Text":"That will give us the answer that the intersection point is 3 minus"},{"Start":"08:07.385 ","End":"08:14.550","Text":"8/9, 1 and 5/9."},{"Start":"08:14.550 ","End":"08:16.125","Text":"I\u0027ll highlight that."},{"Start":"08:16.125 ","End":"08:18.150","Text":"I\u0027ll just mention that,"},{"Start":"08:18.150 ","End":"08:21.360","Text":"of course, instead of using l_2,"},{"Start":"08:21.360 ","End":"08:23.280","Text":"we could\u0027ve used l_1."},{"Start":"08:23.280 ","End":"08:29.400","Text":"If you substitute t equals 4/9 into here,"},{"Start":"08:29.400 ","End":"08:34.220","Text":"you should get exactly the same 3 values as here."},{"Start":"08:34.220 ","End":"08:39.180","Text":"This is the point of intersection and we\u0027re done."}],"ID":9759},{"Watched":false,"Name":"Exercises 10","Duration":"3m 15s","ChapterTopicVideoID":9798,"CourseChapterTopicPlaylistID":57475,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9798.jpeg","UploadDate":"2017-08-01T05:01:06.4400000","DurationForVideoObject":"PT3M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"Here we have the parametric equation of a line, x, y,"},{"Start":"00:05.880 ","End":"00:12.525","Text":"and z in terms of t. The first question is,"},{"Start":"00:12.525 ","End":"00:15.690","Text":"does this line intersect the xy-plane?"},{"Start":"00:15.690 ","End":"00:19.420","Text":"And if so, find the point of intersection."},{"Start":"00:19.420 ","End":"00:28.769","Text":"Now, the xy-plane is characterized by the equation z equals 0."},{"Start":"00:28.769 ","End":"00:32.370","Text":"So what I\u0027m looking for is the value of t here,"},{"Start":"00:32.370 ","End":"00:35.010","Text":"which will make the z equal 0."},{"Start":"00:35.010 ","End":"00:43.815","Text":"What I will get will be 16 plus 8t equals 0."},{"Start":"00:43.815 ","End":"00:54.905","Text":"Therefore, that will give me that t equals minus 16 over 8 minus 2."},{"Start":"00:54.905 ","End":"00:59.105","Text":"Now the fact that I have a solution for t means that, yes,"},{"Start":"00:59.105 ","End":"01:03.470","Text":"the line will intersect the xy-plane,"},{"Start":"01:03.470 ","End":"01:06.870","Text":"and I even know the z coordinate."},{"Start":"01:08.620 ","End":"01:11.015","Text":"Since we want to know where,"},{"Start":"01:11.015 ","End":"01:14.750","Text":"I just need to complete the x and the y part."},{"Start":"01:14.750 ","End":"01:20.460","Text":"The x of"},{"Start":"01:21.280 ","End":"01:28.895","Text":"the point of intersection will be minus 7 plus 12,"},{"Start":"01:28.895 ","End":"01:32.460","Text":"and then t is minus 2."},{"Start":"01:33.430 ","End":"01:42.060","Text":"Then y will be 3 and z,"},{"Start":"01:42.060 ","End":"01:44.970","Text":"we already know that that\u0027s going to be 0."},{"Start":"01:44.970 ","End":"01:50.160","Text":"We\u0027ve got already 2 out of 3 all we only need is the x now."},{"Start":"01:50.160 ","End":"01:55.860","Text":"Minus 7, minus 24,"},{"Start":"01:55.860 ","End":"02:01.920","Text":"this is minus 31,"},{"Start":"02:01.920 ","End":"02:07.835","Text":"and so the intersection will be minus 31,"},{"Start":"02:07.835 ","End":"02:14.385","Text":"3, 0, and that\u0027s Part A."},{"Start":"02:14.385 ","End":"02:17.115","Text":"Now for Part B,"},{"Start":"02:17.115 ","End":"02:19.380","Text":"same idea."},{"Start":"02:19.380 ","End":"02:21.670","Text":"When we talked about the xy-plane,"},{"Start":"02:21.670 ","End":"02:24.650","Text":"it was characterized by z equals 0,"},{"Start":"02:24.650 ","End":"02:27.239","Text":"now we have the xz-plane,"},{"Start":"02:27.239 ","End":"02:30.755","Text":"which is characterized by y equals 0."},{"Start":"02:30.755 ","End":"02:37.139","Text":"So we have to find the value of t that will make the y here 0,"},{"Start":"02:37.139 ","End":"02:39.240","Text":"but y is 3."},{"Start":"02:39.240 ","End":"02:44.970","Text":"It doesn\u0027t even depend on t. It doesn\u0027t matter what t is, y is 3,"},{"Start":"02:44.970 ","End":"02:52.600","Text":"and 3 will not equal 0 ever regardless of what t is."},{"Start":"02:52.790 ","End":"02:56.625","Text":"So the answer is no,"},{"Start":"02:56.625 ","End":"03:00.705","Text":"the l does not intersect."},{"Start":"03:00.705 ","End":"03:05.100","Text":"There is no intersection,"},{"Start":"03:05.100 ","End":"03:06.780","Text":"and if there is no intersection,"},{"Start":"03:06.780 ","End":"03:10.605","Text":"we can\u0027t find where it is because there isn\u0027t one."},{"Start":"03:10.605 ","End":"03:14.980","Text":"That\u0027s all there is to it. We\u0027re done."}],"ID":9760}],"Thumbnail":null,"ID":57475},{"Name":"Equations of Planes","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Equations of Planes","Duration":"19m 37s","ChapterTopicVideoID":9882,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9882.jpeg","UploadDate":"2020-01-16T07:12:37.2300000","DurationForVideoObject":"PT19M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:04.970","Text":"This is what we just finished in 3D equations of lines."},{"Start":"00:04.970 ","End":"00:07.155","Text":"We\u0027ll go on to the next topic,"},{"Start":"00:07.155 ","End":"00:11.955","Text":"which is going to be the equation of planes."},{"Start":"00:11.955 ","End":"00:15.240","Text":"Basically, what we did with lines,"},{"Start":"00:15.240 ","End":"00:17.910","Text":"the information we needed was 2 things."},{"Start":"00:17.910 ","End":"00:21.460","Text":"We needed a point on the line,"},{"Start":"00:21.710 ","End":"00:29.189","Text":"that was 1, and we needed a direction vector for the line."},{"Start":"00:29.920 ","End":"00:32.705","Text":"Because the line is defined,"},{"Start":"00:32.705 ","End":"00:36.965","Text":"mainly, 1 of the most significant things is that is its direction."},{"Start":"00:36.965 ","End":"00:39.860","Text":"Once we have the direction then everything else will be parallel."},{"Start":"00:39.860 ","End":"00:44.900","Text":"Once we have the point on the line then that fixes it."},{"Start":"00:44.900 ","End":"00:48.710","Text":"Now with planes, it\u0027s similar but different."},{"Start":"00:48.710 ","End":"00:50.810","Text":"What we need also are 2 things."},{"Start":"00:50.810 ","End":"00:56.035","Text":"We need a point on the plane"},{"Start":"00:56.035 ","End":"01:01.340","Text":"but the question is what\u0027s going to take the place of the direction?"},{"Start":"01:01.340 ","End":"01:03.635","Text":"It\u0027s also going to be a vector."},{"Start":"01:03.635 ","End":"01:08.915","Text":"But a plane is actually determined by something called a normal."},{"Start":"01:08.915 ","End":"01:14.965","Text":"I\u0027ll get into that in a moment though we have learned it in the chapter on vectors."},{"Start":"01:14.965 ","End":"01:19.760","Text":"I\u0027ll just emphasize the difference is that here in both cases we have a point,"},{"Start":"01:19.760 ","End":"01:22.145","Text":"but here we have a direction for a line,"},{"Start":"01:22.145 ","End":"01:24.590","Text":"and for planes we need a normal vector."},{"Start":"01:24.590 ","End":"01:27.335","Text":"I\u0027ll best explain it with a diagram all though,"},{"Start":"01:27.335 ","End":"01:29.855","Text":"as I say, you should have studied it with vectors."},{"Start":"01:29.855 ","End":"01:35.100","Text":"Normal also means orthogonal or perpendicular to the plane."},{"Start":"01:35.420 ","End":"01:38.510","Text":"Here we have a diagram."},{"Start":"01:38.510 ","End":"01:41.000","Text":"This is the plane that we want."},{"Start":"01:41.000 ","End":"01:43.790","Text":"We have a point in the plane."},{"Start":"01:43.790 ","End":"01:46.465","Text":"This is what we\u0027re given P naught."},{"Start":"01:46.465 ","End":"01:49.800","Text":"Well, it could be x naught,"},{"Start":"01:49.800 ","End":"01:52.560","Text":"y naught, z naught, or ABC or whatever."},{"Start":"01:52.560 ","End":"01:54.550","Text":"Here\u0027s the normal vector."},{"Start":"01:54.550 ","End":"01:58.045","Text":"Normal vector means it\u0027s perpendicular to the whole plane."},{"Start":"01:58.045 ","End":"02:05.820","Text":"It\u0027s perpendicular to any vector in the plane or any line in the plane if you want."},{"Start":"02:05.820 ","End":"02:08.380","Text":"The fact that this is perpendicular to"},{"Start":"02:08.380 ","End":"02:13.755","Text":"any vector that\u0027s parallel to the plane is what gives us the equation."},{"Start":"02:13.755 ","End":"02:16.815","Text":"If we take any point P,"},{"Start":"02:16.815 ","End":"02:20.385","Text":"preferably not P naught itself,"},{"Start":"02:20.385 ","End":"02:25.600","Text":"then P minus P naught will be a perpendicular vector to n. On the other hand,"},{"Start":"02:25.600 ","End":"02:28.615","Text":"if this is vector r and this is vector r naught,"},{"Start":"02:28.615 ","End":"02:33.110","Text":"then this vector P naught P will be r minus r naught."},{"Start":"02:33.110 ","End":"02:42.175","Text":"In short, the equation of the plane will be that this normal vector."},{"Start":"02:42.175 ","End":"02:46.200","Text":"I\u0027ll call the normal vector here n and I\u0027ll call the"},{"Start":"02:46.200 ","End":"02:50.315","Text":"1 in for the line v. I\u0027ll just want to contrast and compare"},{"Start":"02:50.315 ","End":"02:55.035","Text":"the 2 equations so that we have that n."},{"Start":"02:55.035 ","End":"03:03.390","Text":"product with r minus r naught is equal to 0."},{"Start":"03:03.390 ","End":"03:08.210","Text":"Is this vector r minus r naught is this vector perpendicular means equal 0."},{"Start":"03:08.210 ","End":"03:10.760","Text":"Because it works at P naught itself because"},{"Start":"03:10.760 ","End":"03:15.530","Text":"then this minus this is 0 so of course it\u0027s still 0."},{"Start":"03:15.530 ","End":"03:19.000","Text":"Just compare this."},{"Start":"03:19.130 ","End":"03:27.470","Text":"I\u0027m just for comparison I\u0027ll write the equation of a line is parametric and it\u0027s"},{"Start":"03:27.470 ","End":"03:36.380","Text":"r is equal to r naught plus parameter times direction vector."},{"Start":"03:36.380 ","End":"03:43.305","Text":"Here is normal vector times r minus the r of a particular point equals 0."},{"Start":"03:43.305 ","End":"03:45.850","Text":"There is a slight variation on this."},{"Start":"03:45.850 ","End":"03:48.670","Text":"If we multiply out using the distributive law,"},{"Start":"03:48.670 ","End":"03:53.010","Text":"we get n dot r minus n. r naught."},{"Start":"03:53.010 ","End":"03:57.895","Text":"We can bring it over to the other side so it becomes n."},{"Start":"03:57.895 ","End":"04:05.435","Text":"r equals n.r naught."},{"Start":"04:05.435 ","End":"04:07.950","Text":"Anyway, that\u0027s just a variation of this."},{"Start":"04:07.950 ","End":"04:11.690","Text":"This is the important thing we\u0027re looking for so I\u0027ll highlight it."},{"Start":"04:11.690 ","End":"04:14.920","Text":"Here we go. This is the formula."},{"Start":"04:14.920 ","End":"04:16.510","Text":"Now just like with the line,"},{"Start":"04:16.510 ","End":"04:18.580","Text":"we also had an xyz version."},{"Start":"04:18.580 ","End":"04:21.965","Text":"We\u0027ll have an xyz version for the plane also."},{"Start":"04:21.965 ","End":"04:25.905","Text":"R is the vector x, y,"},{"Start":"04:25.905 ","End":"04:30.000","Text":"z r naught is the vector x naught,"},{"Start":"04:30.000 ","End":"04:32.910","Text":"y naught, z naught."},{"Start":"04:32.910 ","End":"04:38.265","Text":"I need also the n and let\u0027s just assume that that\u0027s a, b,"},{"Start":"04:38.265 ","End":"04:43.560","Text":"c. What we get from this equation is that a,"},{"Start":"04:43.560 ","End":"04:47.680","Text":"b, c which is the n dot with."},{"Start":"04:47.680 ","End":"04:53.240","Text":"Now r minus r naught is going to be x minus x naught,"},{"Start":"04:53.240 ","End":"04:55.910","Text":"y minus y naught,"},{"Start":"04:55.910 ","End":"05:02.800","Text":"z minus z naught is equal to 0."},{"Start":"05:02.900 ","End":"05:08.690","Text":"This gives us the equation a times x minus x naught,"},{"Start":"05:08.690 ","End":"05:10.895","Text":"just by the definition of dot product,"},{"Start":"05:10.895 ","End":"05:15.800","Text":"plus b times y minus y"},{"Start":"05:15.800 ","End":"05:23.200","Text":"naught plus c times z minus z naught equals 0."},{"Start":"05:23.200 ","End":"05:31.145","Text":"This is the equivalent of the formula above, just with xyz."},{"Start":"05:31.145 ","End":"05:34.385","Text":"There is another variation of this."},{"Start":"05:34.385 ","End":"05:37.220","Text":"If we bring all the constants to the other side,"},{"Start":"05:37.220 ","End":"05:39.920","Text":"like ax naught, by naught,"},{"Start":"05:39.920 ","End":"05:42.080","Text":"cz naught there will be the minus,"},{"Start":"05:42.080 ","End":"05:43.835","Text":"on the other side they become a plus."},{"Start":"05:43.835 ","End":"05:46.229","Text":"I put them all together,"},{"Start":"05:46.229 ","End":"05:54.715","Text":"then we get a variant that ax plus by plus cz equals d,"},{"Start":"05:54.715 ","End":"06:04.815","Text":"where as I said, d is equal to ax naught plus by naught plus cz naught."},{"Start":"06:04.815 ","End":"06:10.490","Text":"It happens not by coincidence to also equal n. r naught."},{"Start":"06:10.490 ","End":"06:15.635","Text":"The dot product of these 2 is ax-naught plus by-naught plus cz-naught and"},{"Start":"06:15.635 ","End":"06:20.470","Text":"so really this is just the other variation of this,"},{"Start":"06:20.470 ","End":"06:30.510","Text":"but with this abbreviated to d. That\u0027s a plane in 3D."},{"Start":"06:30.510 ","End":"06:35.760","Text":"In a way it\u0027s analogous to a line in 2D."},{"Start":"06:35.760 ","End":"06:41.090","Text":"We had ax plus by equals c with a symmetric equation of a line."},{"Start":"06:41.090 ","End":"06:43.235","Text":"If we have 3 variables,"},{"Start":"06:43.235 ","End":"06:48.420","Text":"it becomes a plane in 3D."},{"Start":"06:49.370 ","End":"06:51.930","Text":"Just to give them names,"},{"Start":"06:51.930 ","End":"06:58.470","Text":"this 1 here is called the vector equation of the plane."},{"Start":"07:01.130 ","End":"07:07.130","Text":"This 1 here, I suppose you could refer it to this 1"},{"Start":"07:07.130 ","End":"07:13.585","Text":"here is called the scalar equation of the plane."},{"Start":"07:13.585 ","End":"07:16.170","Text":"Just to give them names."},{"Start":"07:16.170 ","End":"07:18.090","Text":"I\u0027m about to do an example,"},{"Start":"07:18.090 ","End":"07:19.460","Text":"just 1 more comment."},{"Start":"07:19.460 ","End":"07:21.770","Text":"If you\u0027re given a plane in this form with something"},{"Start":"07:21.770 ","End":"07:24.260","Text":"x plus something y plus something z equals d,"},{"Start":"07:24.260 ","End":"07:26.110","Text":"then these 3 numbers,"},{"Start":"07:26.110 ","End":"07:28.970","Text":"the a, b, and c,"},{"Start":"07:28.970 ","End":"07:34.685","Text":"will actually be a normal to the plane."},{"Start":"07:34.685 ","End":"07:36.260","Text":"If we\u0027re given an equation in this form,"},{"Start":"07:36.260 ","End":"07:38.135","Text":"we can find the normal."},{"Start":"07:38.135 ","End":"07:43.585","Text":"I\u0027m going to start the examples on a fresh page and I\u0027ll carry these formulas with me."},{"Start":"07:43.585 ","End":"07:46.700","Text":"Now it\u0027s time for a couple of examples."},{"Start":"07:46.700 ","End":"07:52.055","Text":"I just kept whatever I needed from the previous page"},{"Start":"07:52.055 ","End":"07:58.010","Text":"that basically the best way to find the equation of a plane is to find 2 things,"},{"Start":"07:58.010 ","End":"08:02.105","Text":"a point on the plane and the normal vector to the plane, these 2 things."},{"Start":"08:02.105 ","End":"08:05.510","Text":"Then we can either have the vector equation or the scalar equation."},{"Start":"08:05.510 ","End":"08:08.240","Text":"1 of the examples I wanted to"},{"Start":"08:08.240 ","End":"08:13.925","Text":"bring here is how to find the equation of a plane if we\u0027re given 3 points in the plane."},{"Start":"08:13.925 ","End":"08:15.740","Text":"Now it\u0027s well-known that in general,"},{"Start":"08:15.740 ","End":"08:19.250","Text":"3 points on the plane completely determine the plane."},{"Start":"08:19.250 ","End":"08:23.930","Text":"There are exceptions if the 3 points are on the same line or what is called co-linear,"},{"Start":"08:23.930 ","End":"08:25.280","Text":"then it won\u0027t work"},{"Start":"08:25.280 ","End":"08:27.320","Text":"but if they\u0027re not all on the same line,"},{"Start":"08:27.320 ","End":"08:29.555","Text":"then any 3 points determine a plane."},{"Start":"08:29.555 ","End":"08:31.220","Text":"We\u0027ll take an example of this."},{"Start":"08:31.220 ","End":"08:33.144","Text":"I\u0027ll give you 3 points."},{"Start":"08:33.144 ","End":"08:38.445","Text":"I\u0027ll take point P which is 1, 1, 1,"},{"Start":"08:38.445 ","End":"08:44.460","Text":"and I\u0027ll take point Q which is minus 1,"},{"Start":"08:44.460 ","End":"08:51.865","Text":"1, 0, and R which is 2, 0, 3."},{"Start":"08:51.865 ","End":"08:56.750","Text":"We want to find the equation of the plane that goes through these 3 points."},{"Start":"08:56.750 ","End":"09:01.325","Text":"We\u0027ll find out soon enough if these points are co-linear and then we won\u0027t get a plane"},{"Start":"09:01.325 ","End":"09:06.140","Text":"but meanwhile, we have the first part done to find a point we have a choice of 3,"},{"Start":"09:06.140 ","End":"09:07.475","Text":"any on. Of these will do."},{"Start":"09:07.475 ","End":"09:10.775","Text":"I\u0027ll use P. That\u0027s a point on the plane."},{"Start":"09:10.775 ","End":"09:12.835","Text":"Now, what about the normal?"},{"Start":"09:12.835 ","End":"09:16.005","Text":"Here\u0027s the way this thing works."},{"Start":"09:16.005 ","End":"09:18.375","Text":"I\u0027m going to reuse this diagram."},{"Start":"09:18.375 ","End":"09:20.150","Text":"Here I have 2 points on the plane."},{"Start":"09:20.150 ","End":"09:23.035","Text":"Suppose I had a third point."},{"Start":"09:23.035 ","End":"09:28.105","Text":"Here\u0027s a third point in the plane and it doesn\u0027t look like they\u0027re in the same line,"},{"Start":"09:28.105 ","End":"09:31.065","Text":"doesn\u0027t wonder what its name is."},{"Start":"09:31.065 ","End":"09:37.205","Text":"What I do is I join SAP naught to this point."},{"Start":"09:37.205 ","End":"09:39.350","Text":"Here we are."},{"Start":"09:39.350 ","End":"09:43.610","Text":"Now we have another vector that\u0027s in the plane."},{"Start":"09:43.610 ","End":"09:45.800","Text":"A vector doesn\u0027t really have a location."},{"Start":"09:45.800 ","End":"09:47.390","Text":"A vector could have been drawn from here."},{"Start":"09:47.390 ","End":"09:50.420","Text":"A vector just has magnitude and direction,"},{"Start":"09:50.420 ","End":"09:54.285","Text":"but you could sort of think of it as in the plane or it\u0027s parallel to the plane."},{"Start":"09:54.285 ","End":"09:57.500","Text":"Now, the normal, since it\u0027s normal to the plane,"},{"Start":"09:57.500 ","End":"10:00.605","Text":"will be normal to these 2 vectors."},{"Start":"10:00.605 ","End":"10:04.100","Text":"To find a normal, in other words,"},{"Start":"10:04.100 ","End":"10:07.550","Text":"to find something that\u0027s perpendicular or orthogonal to both of these,"},{"Start":"10:07.550 ","End":"10:10.955","Text":"all I have to do is take the cross-product, that\u0027s the key here."},{"Start":"10:10.955 ","End":"10:14.935","Text":"The cross-product of these 2 could be used as a normal vector."},{"Start":"10:14.935 ","End":"10:18.364","Text":"In general, if the cross-product turns out to be 0,"},{"Start":"10:18.364 ","End":"10:22.610","Text":"then it means that these 3 points were co-linear."},{"Start":"10:22.610 ","End":"10:24.680","Text":"We\u0027ll soon enough find out."},{"Start":"10:24.680 ","End":"10:27.265","Text":"Let\u0027s take that here."},{"Start":"10:27.265 ","End":"10:31.890","Text":"Maybe P naught is the first 1 and P is what\u0027s here,"},{"Start":"10:31.890 ","End":"10:34.380","Text":"Q, and this 1 is R, it doesn\u0027t matter."},{"Start":"10:34.380 ","End":"10:38.150","Text":"I take any 2 vectors in the plane that are not parallel."},{"Start":"10:38.150 ","End":"10:40.830","Text":"Let\u0027s say I\u0027ll take PQ."},{"Start":"10:41.870 ","End":"10:45.680","Text":"PQ the vector is,"},{"Start":"10:45.680 ","End":"10:49.760","Text":"I take the coordinates of this minus this so I get minus 1"},{"Start":"10:49.760 ","End":"10:54.990","Text":"minus 1 is minus 2, 0, minus 1."},{"Start":"10:54.990 ","End":"10:57.555","Text":"There\u0027s another 1, let\u0027s take PR."},{"Start":"10:57.555 ","End":"10:59.120","Text":"There\u0027s many possibilities."},{"Start":"10:59.120 ","End":"11:04.530","Text":"I just need 1 possibility for 2 vectors,"},{"Start":"11:04.530 ","End":"11:07.170","Text":"I mean, well, 2 possibilities if you like."},{"Start":"11:07.170 ","End":"11:17.340","Text":"PR is from here to here just subtract 1 from each of these and I\u0027ll get 1, minus 1, 2."},{"Start":"11:17.340 ","End":"11:21.260","Text":"That\u0027s this and this. Now to get a normal you need something orthogonal to both."},{"Start":"11:21.260 ","End":"11:22.835","Text":"I\u0027ll take the cross-product."},{"Start":"11:22.835 ","End":"11:25.910","Text":"I\u0027ll do minus 2, 0,"},{"Start":"11:25.910 ","End":"11:33.320","Text":"minus 1 cross 1, minus 1, 2."},{"Start":"11:33.320 ","End":"11:36.730","Text":"The answer comes out minus 1, 3, 2,"},{"Start":"11:36.730 ","End":"11:39.115","Text":"and I\u0027m not going to go into the computation,"},{"Start":"11:39.115 ","End":"11:43.540","Text":"because you can go to the section on vectors and the cross-product."},{"Start":"11:43.540 ","End":"11:44.860","Text":"Also, there\u0027s many techniques,"},{"Start":"11:44.860 ","End":"11:46.840","Text":"I taught a way using a determinant,"},{"Start":"11:46.840 ","End":"11:48.130","Text":"and the way without it,"},{"Start":"11:48.130 ","End":"11:51.175","Text":"and 1 is easier and you may not have learned the other."},{"Start":"11:51.175 ","End":"11:53.740","Text":"Well, just go refer to the chapter,"},{"Start":"11:53.740 ","End":"11:55.330","Text":"I\u0027ll just quote in the answer."},{"Start":"11:55.330 ","End":"11:59.035","Text":"This I\u0027m some taking as my normal vector."},{"Start":"11:59.035 ","End":"12:05.170","Text":"For the point, I\u0027m going to choose this,"},{"Start":"12:05.170 ","End":"12:09.860","Text":"and for the normal vector, I\u0027ve got this."},{"Start":"12:11.190 ","End":"12:16.450","Text":"Well, let\u0027s get the scalar equation."},{"Start":"12:16.450 ","End":"12:20.470","Text":"Just for reference in the formula,"},{"Start":"12:20.470 ","End":"12:26.965","Text":"this would be the x_naught, y_naught, z_naught,"},{"Start":"12:26.965 ","End":"12:29.455","Text":"and this becomes the a,"},{"Start":"12:29.455 ","End":"12:33.820","Text":"b, c. What we get is a,"},{"Start":"12:33.820 ","End":"12:41.485","Text":"which is minus 1 times x minus 1, plus b,"},{"Start":"12:41.485 ","End":"12:46.510","Text":"times y minus 1"},{"Start":"12:46.510 ","End":"12:53.065","Text":"plus 2 times z minus 1 equals 0."},{"Start":"12:53.065 ","End":"12:55.030","Text":"There is a variation on this."},{"Start":"12:55.030 ","End":"12:57.430","Text":"Like I mentioned, if you just want to take the x, y, and z,"},{"Start":"12:57.430 ","End":"13:04.705","Text":"we got minus x plus 3y plus 2z."},{"Start":"13:04.705 ","End":"13:13.840","Text":"All the constants go on the other side we have 1 minus 3 plus minus 2 is minus 4,"},{"Start":"13:13.840 ","End":"13:16.000","Text":"and on the other side, it\u0027s equal to 4."},{"Start":"13:16.000 ","End":"13:19.840","Text":"This is maybe a nicer way of getting the equation of"},{"Start":"13:19.840 ","End":"13:24.505","Text":"the plane If this was on an exam and you had time,"},{"Start":"13:24.505 ","End":"13:28.480","Text":"I would plug in these 3 points and see that they all work."},{"Start":"13:28.480 ","End":"13:31.585","Text":"Let\u0027s, for example, take the second 1."},{"Start":"13:31.585 ","End":"13:36.714","Text":"If I take x is minus 1 and y is 1 and z is 0,"},{"Start":"13:36.714 ","End":"13:39.850","Text":"this becomes plus 1,"},{"Start":"13:39.850 ","End":"13:44.230","Text":"and this becomes plus 3 plus 0,"},{"Start":"13:44.230 ","End":"13:46.300","Text":"1 plus 3 plus 0 is 4?"},{"Start":"13:46.300 ","End":"13:48.925","Text":"Yes, Q is on the line."},{"Start":"13:48.925 ","End":"13:50.890","Text":"I know that P is on the line I."},{"Start":"13:50.890 ","End":"13:54.040","Text":"If I substituted here x, y, and z are 1,"},{"Start":"13:54.040 ","End":"13:57.760","Text":"I can see 1 minus 1 minus 1 is 0."},{"Start":"13:57.760 ","End":"14:00.745","Text":"You should try the third on your own and make sure that"},{"Start":"14:00.745 ","End":"14:04.930","Text":"this equation really goes through these 3 points."},{"Start":"14:04.930 ","End":"14:09.175","Text":"Okay, 1 more example before we\u0027re done."},{"Start":"14:09.175 ","End":"14:10.840","Text":"In the next example,"},{"Start":"14:10.840 ","End":"14:14.380","Text":"I\u0027m going to give you the equation of a plane,"},{"Start":"14:14.380 ","End":"14:18.020","Text":"and then the equation of a line,"},{"Start":"14:18.090 ","End":"14:21.040","Text":"then I\u0027m going to ask a question."},{"Start":"14:21.040 ","End":"14:27.730","Text":"The plane will be minus x plus 2z equals 10."},{"Start":"14:27.730 ","End":"14:32.950","Text":"Note that the y is missing, but that\u0027s okay."},{"Start":"14:32.950 ","End":"14:37.885","Text":"The line, I\u0027ll write it over here,"},{"Start":"14:37.885 ","End":"14:39.520","Text":"I\u0027ll give it in the form of r,"},{"Start":"14:39.520 ","End":"14:42.850","Text":"which is r of t, but I won\u0027t bother with that, it\u0027s x, y, z,"},{"Start":"14:42.850 ","End":"14:48.055","Text":"r is equal to 5,"},{"Start":"14:48.055 ","End":"14:53.710","Text":"2 minus t, 10 plus 4t."},{"Start":"14:53.710 ","End":"14:57.100","Text":"Now the question, I have the plane and the line,"},{"Start":"14:57.100 ","End":"15:05.095","Text":"and I want to know if relative to each other if they are parallel, that\u0027s 1 possibility."},{"Start":"15:05.095 ","End":"15:08.305","Text":"If they are orthogonal,"},{"Start":"15:08.305 ","End":"15:12.385","Text":"that\u0027s another possibility meaning perpendicular."},{"Start":"15:12.385 ","End":"15:17.515","Text":"Third possibility, neither, none of the above."},{"Start":"15:17.515 ","End":"15:19.525","Text":"We have to decide,"},{"Start":"15:19.525 ","End":"15:23.950","Text":"and it looks like a difficult question."},{"Start":"15:23.950 ","End":"15:25.630","Text":"But when I show you how we do it,"},{"Start":"15:25.630 ","End":"15:27.715","Text":"it\u0027s actually quite easy."},{"Start":"15:27.715 ","End":"15:30.370","Text":"There are 2 vectors that are very important to me,"},{"Start":"15:30.370 ","End":"15:32.290","Text":"and that\u0027s all I need to solve this problem."},{"Start":"15:32.290 ","End":"15:34.765","Text":"I need a normal vector to the plane,"},{"Start":"15:34.765 ","End":"15:37.225","Text":"and a direction vector of the line."},{"Start":"15:37.225 ","End":"15:38.905","Text":"We learned how to do this."},{"Start":"15:38.905 ","End":"15:42.400","Text":"Remember, it\u0027s like we have a plus 0y here."},{"Start":"15:42.400 ","End":"15:47.200","Text":"The coefficients are a normal vector."},{"Start":"15:47.200 ","End":"15:52.540","Text":"I can take a vector n the normal to the plane will be,"},{"Start":"15:52.540 ","End":"16:00.730","Text":"1 possibility is minus1,0 for the missing y, and 2."},{"Start":"16:00.730 ","End":"16:05.095","Text":"The other thing I need is the direction vector of the line."},{"Start":"16:05.095 ","End":"16:11.420","Text":"We already learned that that\u0027s the coefficients of t. Well,"},{"Start":"16:11.420 ","End":"16:14.905","Text":"it\u0027s 0, here, it\u0027s minus 1,"},{"Start":"16:14.905 ","End":"16:17.720","Text":"and here it\u0027s 4."},{"Start":"16:17.880 ","End":"16:25.885","Text":"Now, how can this help me to find out if the plane and line are parallel or orthogonal?"},{"Start":"16:25.885 ","End":"16:28.450","Text":"Well, let\u0027s take the parallel."},{"Start":"16:28.450 ","End":"16:31.570","Text":"If this vector is parallel to the plane,"},{"Start":"16:31.570 ","End":"16:35.170","Text":"it means that I could put it inside the plane."},{"Start":"16:35.170 ","End":"16:38.080","Text":"Any vector inside the plane or parallel to"},{"Start":"16:38.080 ","End":"16:42.035","Text":"the plane has got to be orthogonal to the normal."},{"Start":"16:42.035 ","End":"16:48.255","Text":"The normal is orthogonal to any vector that\u0027s parallel to the plane."},{"Start":"16:48.255 ","End":"16:54.755","Text":"What I have to check is that if these 2 are perpendicular or orthogonal,"},{"Start":"16:54.755 ","End":"16:59.845","Text":"then the line and the plane will be parallel."},{"Start":"16:59.845 ","End":"17:02.050","Text":"Let\u0027s check that."},{"Start":"17:02.050 ","End":"17:09.715","Text":"Remember the test for 2 vectors being orthogonal is if their dot product is 0."},{"Start":"17:09.715 ","End":"17:14.470","Text":"I have to check what is n.v,"},{"Start":"17:14.470 ","End":"17:16.975","Text":"and ask is this equal to 0?"},{"Start":"17:16.975 ","End":"17:20.890","Text":"If so, these 2 vectors are orthogonal,"},{"Start":"17:20.890 ","End":"17:24.940","Text":"but that will mean that the plane on the line are parallel."},{"Start":"17:24.940 ","End":"17:27.880","Text":"If the parallel is orthogonal and later it turns out that"},{"Start":"17:27.880 ","End":"17:31.570","Text":"orthogonal is parallel, actually reverse."},{"Start":"17:31.570 ","End":"17:38.275","Text":"That\u0027s because the normal is perpendicular to the plane, it reverses everything."},{"Start":"17:38.275 ","End":"17:47.365","Text":"Let\u0027s see, n.v is equal to just multiply each 1 with its corresponding 1,"},{"Start":"17:47.365 ","End":"17:49.899","Text":"minus 1 times 0 is 0,"},{"Start":"17:49.899 ","End":"17:53.425","Text":"0 times minus 1 is 0,"},{"Start":"17:53.425 ","End":"17:56.860","Text":"and 2 times 4 is 8."},{"Start":"17:56.860 ","End":"18:01.705","Text":"At any rate, it is not 0."},{"Start":"18:01.705 ","End":"18:06.835","Text":"We\u0027ve ruled out the parallel, no."},{"Start":"18:06.835 ","End":"18:09.400","Text":"Now orthogonal."},{"Start":"18:09.400 ","End":"18:16.300","Text":"If a line is going to be orthogonal to the plane perpendicular,"},{"Start":"18:16.300 ","End":"18:19.870","Text":"it\u0027s got to be parallel to the normal vector."},{"Start":"18:19.870 ","End":"18:24.460","Text":"If think about it, if it\u0027s parallel to a perpendicular to the plane,"},{"Start":"18:24.460 ","End":"18:27.430","Text":"and it\u0027s also perpendicular to the plane or orthogonal."},{"Start":"18:27.430 ","End":"18:35.180","Text":"Now I have to check if they are parallel."},{"Start":"18:35.220 ","End":"18:40.165","Text":"What is the test for these 2 vectors to be parallel?"},{"Start":"18:40.165 ","End":"18:43.225","Text":"Similar, but using the cross-product."},{"Start":"18:43.225 ","End":"18:49.450","Text":"Parallel would mean that n cross v was equal to 0,"},{"Start":"18:49.450 ","End":"18:51.745","Text":"but this time the 0 vector."},{"Start":"18:51.745 ","End":"18:54.820","Text":"Let\u0027s check the cross-product."},{"Start":"18:54.820 ","End":"19:01.915","Text":"Minus 1, 0, 2, cross with 0,"},{"Start":"19:01.915 ","End":"19:08.845","Text":"minus 1, 4, equals 2, 4,1."},{"Start":"19:08.845 ","End":"19:10.390","Text":"I\u0027m not doing the computation,"},{"Start":"19:10.390 ","End":"19:12.505","Text":"if you like I didn\u0027t do in the previous exercise."},{"Start":"19:12.505 ","End":"19:14.410","Text":"Don\u0027t want to waste time with that."},{"Start":"19:14.410 ","End":"19:18.895","Text":"At any rate, this is not the 0 vector."},{"Start":"19:18.895 ","End":"19:24.835","Text":"We\u0027ve also answered that orthogonal, no."},{"Start":"19:24.835 ","End":"19:30.070","Text":"So, yeah, it\u0027s neither parallel nor orthogonal."},{"Start":"19:30.070 ","End":"19:37.010","Text":"I see under this example and we\u0027re done with the 3D equation of planes."}],"ID":9761},{"Watched":false,"Name":"The 3D Coordinate System - Equations of Planes (continued)","Duration":"11m 29s","ChapterTopicVideoID":9883,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9883.jpeg","UploadDate":"2020-01-16T07:13:02.0170000","DurationForVideoObject":"PT11M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"We\u0027re continuing with the 3D coordinate system."},{"Start":"00:03.690 ","End":"00:05.490","Text":"In the previous clip,"},{"Start":"00:05.490 ","End":"00:09.270","Text":"we learned about planes and their equations."},{"Start":"00:09.270 ","End":"00:16.710","Text":"One of the forms of the equation that we had was ax plus by"},{"Start":"00:16.710 ","End":"00:22.980","Text":"plus cz equals d."},{"Start":"00:22.980 ","End":"00:29.095","Text":"I\u0027d like to rewrite it in a slightly different form,"},{"Start":"00:29.095 ","End":"00:32.120","Text":"where I just made the letters capital and little"},{"Start":"00:32.120 ","End":"00:35.090","Text":"d is minus big D and I put everything on the left."},{"Start":"00:35.090 ","End":"00:39.905","Text":"Now, this is basically a linear equation with 3 variables."},{"Start":"00:39.905 ","End":"00:43.310","Text":"Everything appears most with degree 1."},{"Start":"00:43.310 ","End":"00:47.660","Text":"Now, if I allow degree 2 terms,"},{"Start":"00:47.660 ","End":"00:51.650","Text":"then we will get something known as quadric surfaces."},{"Start":"00:51.650 ","End":"00:56.424","Text":"Quadric like quadratic, meaning degree 2."},{"Start":"00:56.424 ","End":"01:01.460","Text":"In this case, we have a lot of terms in the general equation,"},{"Start":"01:01.460 ","End":"01:03.710","Text":"we have degree 2,"},{"Start":"01:03.710 ","End":"01:05.324","Text":"we have x squared,"},{"Start":"01:05.324 ","End":"01:08.680","Text":"we have y squared,"},{"Start":"01:08.680 ","End":"01:11.605","Text":"we have z squared."},{"Start":"01:11.605 ","End":"01:16.320","Text":"Then we have all the mixed terms like xy,"},{"Start":"01:16.320 ","End":"01:19.180","Text":"it\u0027s also a degree 2 term."},{"Start":"01:19.180 ","End":"01:29.055","Text":"We\u0027re going to get xz and also another letter with the yz."},{"Start":"01:29.055 ","End":"01:35.350","Text":"Then of course, we also have the linear terms A,"},{"Start":"01:35.350 ","End":"01:39.805","Text":"B, C, D, E, F, Gx."},{"Start":"01:39.805 ","End":"01:41.710","Text":"I will get rid of this."},{"Start":"01:41.710 ","End":"01:46.885","Text":"Plus Hy, let\u0027s see, G, H,"},{"Start":"01:46.885 ","End":"01:52.140","Text":"Iz, after I comes J,"},{"Start":"01:52.140 ","End":"01:55.470","Text":"a constant, equals 0."},{"Start":"01:55.470 ","End":"01:57.405","Text":"What an equation."},{"Start":"01:57.405 ","End":"02:00.665","Text":"We\u0027re not going to deal with this in all its generality,"},{"Start":"02:00.665 ","End":"02:05.470","Text":"but this is the general definition of a quadric surface."},{"Start":"02:05.470 ","End":"02:09.000","Text":"I guess it has to have at least one of the degree 2 terms,"},{"Start":"02:09.000 ","End":"02:10.455","Text":"which is from here to here."},{"Start":"02:10.455 ","End":"02:11.870","Text":"Otherwise, it will just be linear."},{"Start":"02:11.870 ","End":"02:14.240","Text":"If I just take this and change the letter names,"},{"Start":"02:14.240 ","End":"02:15.410","Text":"that will be the plane,"},{"Start":"02:15.410 ","End":"02:16.490","Text":"which is also a surface,"},{"Start":"02:16.490 ","End":"02:18.245","Text":"but it\u0027s not quadric."},{"Start":"02:18.245 ","End":"02:22.610","Text":"I\u0027ll tell you in advance which surfaces we\u0027re going to study."},{"Start":"02:22.610 ","End":"02:27.605","Text":"I\u0027ll just give them names so you can get used to the names."},{"Start":"02:27.605 ","End":"02:30.175","Text":"We\u0027re going to study an ellipsoid."},{"Start":"02:30.175 ","End":"02:33.585","Text":"Next, we\u0027ll study a cone."},{"Start":"02:33.585 ","End":"02:36.935","Text":"Then the cylinder."},{"Start":"02:36.935 ","End":"02:39.260","Text":"They\u0027re all surfaces by you notice."},{"Start":"02:39.260 ","End":"02:44.135","Text":"Then we\u0027re going to have something called a hyperboloid."},{"Start":"02:44.135 ","End":"02:48.365","Text":"But a hyperboloid is going to come in 2 varieties."},{"Start":"02:48.365 ","End":"02:54.170","Text":"We\u0027re going to have the kind that\u0027s called 1 sheet and this kind"},{"Start":"02:54.170 ","End":"03:01.155","Text":"that\u0027s called 2 sheets or 2-sheeted hyperboloid."},{"Start":"03:01.155 ","End":"03:05.855","Text":"Then we\u0027ll have a paraboloid."},{"Start":"03:05.855 ","End":"03:10.115","Text":"There\u0027s also going to be 2 varieties of that."},{"Start":"03:10.115 ","End":"03:19.570","Text":"There\u0027s going to be the elliptic variety and there\u0027s going to be the hyperbolic variety,"},{"Start":"03:19.570 ","End":"03:24.185","Text":"so I would say hyperbolic paraboloid or elliptic paraboloid."},{"Start":"03:24.185 ","End":"03:25.895","Text":"That\u0027s the agenda."},{"Start":"03:25.895 ","End":"03:29.540","Text":"We\u0027re going to start with the ellipsoid."},{"Start":"03:29.540 ","End":"03:35.295","Text":"The equation of the ellipsoid is as follows: x squared over"},{"Start":"03:35.295 ","End":"03:41.440","Text":"a squared, plus y squared over b squared,"},{"Start":"03:41.440 ","End":"03:48.270","Text":"plus z squared over c squared equals 1."},{"Start":"03:48.270 ","End":"03:50.680","Text":"It doesn\u0027t quite look like this,"},{"Start":"03:50.680 ","End":"03:52.030","Text":"but if you think about it,"},{"Start":"03:52.030 ","End":"03:58.135","Text":"if I let capital A be 1 over a squared and capital B 1 over b squared and so on,"},{"Start":"03:58.135 ","End":"04:00.520","Text":"and I let J be minus 1,"},{"Start":"04:00.520 ","End":"04:01.900","Text":"which I can bring over to the other side,"},{"Start":"04:01.900 ","End":"04:03.995","Text":"then this really is of this form."},{"Start":"04:03.995 ","End":"04:13.030","Text":"It\u0027s basically what we get if we don\u0027t have all the mixed and linear terms are missing."},{"Start":"04:13.030 ","End":"04:16.195","Text":"We just have these terms and this one."},{"Start":"04:16.195 ","End":"04:20.560","Text":"These are all positive and J is negative on the other side,"},{"Start":"04:20.560 ","End":"04:23.810","Text":"it\u0027s positive, then we get an ellipsoid."},{"Start":"04:24.950 ","End":"04:30.295","Text":"I\u0027ll show you a picture of what it looks like."},{"Start":"04:30.295 ","End":"04:35.600","Text":"I\u0027d like to compare it with the 2D equivalent,"},{"Start":"04:35.600 ","End":"04:39.105","Text":"which is an ellipse. Here\u0027s an ellipse."},{"Start":"04:39.105 ","End":"04:48.110","Text":"I\u0027m just assuming that this is part of the y-axis and this is part of the x-axis,"},{"Start":"04:48.110 ","End":"04:52.430","Text":"and let\u0027s say this is the origin."},{"Start":"04:52.430 ","End":"04:55.010","Text":"This is the point to where x is a."},{"Start":"04:55.010 ","End":"04:57.500","Text":"This is the point where y is b."},{"Start":"04:57.500 ","End":"05:00.290","Text":"Then we get the equation,"},{"Start":"05:00.290 ","End":"05:08.860","Text":"x squared over a squared plus y squared over b squared equals 1."},{"Start":"05:08.860 ","End":"05:13.025","Text":"There is a geometric meaning to a and b."},{"Start":"05:13.025 ","End":"05:18.950","Text":"A is like half of the long diameter,"},{"Start":"05:18.950 ","End":"05:22.080","Text":"and it\u0027s called the major axis, the minor axis."},{"Start":"05:22.080 ","End":"05:23.800","Text":"If b was bigger than a,"},{"Start":"05:23.800 ","End":"05:26.350","Text":"then it would be more vertical."},{"Start":"05:26.350 ","End":"05:30.120","Text":"Anyway, we have a meaning for a and a."},{"Start":"05:30.120 ","End":"05:32.585","Text":"The same thing applies here,"},{"Start":"05:32.585 ","End":"05:34.700","Text":"that the a, b, and c,"},{"Start":"05:34.700 ","End":"05:36.230","Text":"it\u0027s harder to see in 3D,"},{"Start":"05:36.230 ","End":"05:42.185","Text":"but a would be the point at which it cuts the x-axis."},{"Start":"05:42.185 ","End":"05:45.005","Text":"This would be where x is a,"},{"Start":"05:45.005 ","End":"05:49.515","Text":"and this would be where y is b,"},{"Start":"05:49.515 ","End":"05:54.485","Text":"and this here would be where z is equal to"},{"Start":"05:54.485 ","End":"06:01.860","Text":"c. That\u0027s really it."},{"Start":"06:01.860 ","End":"06:05.285","Text":"This assumes that it\u0027s centered at the origin, like here."},{"Start":"06:05.285 ","End":"06:07.655","Text":"There is a variation on this."},{"Start":"06:07.655 ","End":"06:09.499","Text":"In all what follows,"},{"Start":"06:09.499 ","End":"06:12.709","Text":"we\u0027re going to assume that things are centered at the origin."},{"Start":"06:12.709 ","End":"06:17.000","Text":"If you want to shift things from the origin to a new point,"},{"Start":"06:17.000 ","End":"06:20.430","Text":"for example, I\u0027ve thought off in 2D."},{"Start":"06:20.470 ","End":"06:25.655","Text":"In the 2D case, if I wanted to move the origin to another point,"},{"Start":"06:25.655 ","End":"06:28.375","Text":"h, k, let\u0027s say,"},{"Start":"06:28.375 ","End":"06:31.545","Text":"then I would get the equation,"},{"Start":"06:31.545 ","End":"06:34.980","Text":"we\u0027d replace the x by x minus h,"},{"Start":"06:34.980 ","End":"06:37.170","Text":"we\u0027d replace the y by y minus k,"},{"Start":"06:37.170 ","End":"06:38.880","Text":"and essentially the same thing."},{"Start":"06:38.880 ","End":"06:43.730","Text":"This also works in 3D if we didn\u0027t want"},{"Start":"06:43.730 ","End":"06:50.615","Text":"the center at the origin and we wanted it at the point let\u0027s say,"},{"Start":"06:50.615 ","End":"06:58.640","Text":"x_0, y_0, z_0, then we\u0027d get an adapted form of this."},{"Start":"06:58.640 ","End":"07:01.400","Text":"I just squeeze it in here."},{"Start":"07:01.400 ","End":"07:04.070","Text":"Again, you just replace x by x minus x_0,"},{"Start":"07:04.070 ","End":"07:05.690","Text":"y by y minus y_0,"},{"Start":"07:05.690 ","End":"07:07.430","Text":"and everything else holds."},{"Start":"07:07.430 ","End":"07:10.730","Text":"In the future, after the ellipsoid,"},{"Start":"07:10.730 ","End":"07:13.460","Text":"I won\u0027t be talking about transferring"},{"Start":"07:13.460 ","End":"07:18.440","Text":"the surface as it\u0027s not centered at the origin but centered somewhere else,"},{"Start":"07:18.440 ","End":"07:21.920","Text":"we just do the same trick, replacing x, y,"},{"Start":"07:21.920 ","End":"07:26.650","Text":"and z by x minus x_0 and so on in general."},{"Start":"07:26.650 ","End":"07:29.839","Text":"Before we move on, I want to mention a special case."},{"Start":"07:29.839 ","End":"07:36.125","Text":"Again, I\u0027ll take the analogy in 2D that an ellipse can actually also be a circle."},{"Start":"07:36.125 ","End":"07:38.450","Text":"If we, for example,"},{"Start":"07:38.450 ","End":"07:42.320","Text":"have from here that a equals b,"},{"Start":"07:42.320 ","End":"07:44.060","Text":"and also let\u0027s rename it."},{"Start":"07:44.060 ","End":"07:46.340","Text":"If a equals b, let\u0027s call that r,"},{"Start":"07:46.340 ","End":"07:49.280","Text":"then we multiply both sides by r squared,"},{"Start":"07:49.280 ","End":"07:55.190","Text":"then we get x squared plus y squared equals r squared."},{"Start":"07:55.190 ","End":"08:00.275","Text":"That\u0027s a circle of radius r. Analogously here,"},{"Start":"08:00.275 ","End":"08:10.350","Text":"if we take this equation and also let a equals b equals c and call that r,"},{"Start":"08:10.350 ","End":"08:12.365","Text":"then we get the equation,"},{"Start":"08:12.365 ","End":"08:18.905","Text":"x squared plus y squared plus z squared equals r squared,"},{"Start":"08:18.905 ","End":"08:23.555","Text":"which we know is a sphere centered at the origin with radius"},{"Start":"08:23.555 ","End":"08:29.590","Text":"r. A sphere is a special case of an ellipsoid."},{"Start":"08:29.590 ","End":"08:33.765","Text":"That\u0027s all I want to say about ellipsoids."},{"Start":"08:33.765 ","End":"08:36.810","Text":"Let\u0027s move on to the next one."},{"Start":"08:36.810 ","End":"08:41.194","Text":"Now, we come to the cone and I\u0027ll start straight away with a diagram."},{"Start":"08:41.194 ","End":"08:44.435","Text":"Here\u0027s what the cone looks like."},{"Start":"08:44.435 ","End":"08:46.490","Text":"It\u0027s not what we usually call a cone."},{"Start":"08:46.490 ","End":"08:48.680","Text":"We usually think of a finite cone."},{"Start":"08:48.680 ","End":"08:51.980","Text":"This cone is different in 2 ways."},{"Start":"08:51.980 ","End":"08:56.960","Text":"First of all, there\u0027s 2 parts to it and there\u0027s tip to tip."},{"Start":"08:56.960 ","End":"08:58.670","Text":"Also, it\u0027s not finite."},{"Start":"08:58.670 ","End":"09:01.489","Text":"The lines go on infinitely,"},{"Start":"09:01.489 ","End":"09:05.230","Text":"so the cone is double infinite if you like."},{"Start":"09:05.230 ","End":"09:09.550","Text":"Also, this particular cone,"},{"Start":"09:09.550 ","End":"09:12.750","Text":"we say opens up in the z-direction."},{"Start":"09:12.750 ","End":"09:14.370","Text":"It\u0027s like swallowed the z-axis."},{"Start":"09:14.370 ","End":"09:19.480","Text":"The z-axis is the axis of symmetry also if you like."},{"Start":"09:19.880 ","End":"09:27.215","Text":"There are others which open up along the x-axis or the y-axis."},{"Start":"09:27.215 ","End":"09:29.780","Text":"I\u0027ll say something in a moment about them."},{"Start":"09:29.780 ","End":"09:31.850","Text":"Also, there\u0027s no good 2D analogy,"},{"Start":"09:31.850 ","End":"09:33.940","Text":"so I won\u0027t bring a 2D analogy."},{"Start":"09:33.940 ","End":"09:35.195","Text":"But I will give you the equation,"},{"Start":"09:35.195 ","End":"09:39.455","Text":"the equation I\u0027m going to give you is of a more general cone"},{"Start":"09:39.455 ","End":"09:44.090","Text":"called elliptic because there is also a circular cone."},{"Start":"09:44.090 ","End":"09:45.710","Text":"Usually, when we think of a cone,"},{"Start":"09:45.710 ","End":"09:50.390","Text":"we would say that the cross-section with a horizontal plane should be a circle."},{"Start":"09:50.390 ","End":"09:52.340","Text":"But I\u0027m going to give you the general case for"},{"Start":"09:52.340 ","End":"09:56.030","Text":"ellipse and show you how it could become a circle."},{"Start":"09:56.030 ","End":"10:03.875","Text":"The equation is x squared over a squared plus y squared over"},{"Start":"10:03.875 ","End":"10:11.300","Text":"b squared equals z squared over c squared,"},{"Start":"10:11.300 ","End":"10:16.385","Text":"where a, b, c are some positive numbers."},{"Start":"10:16.385 ","End":"10:21.230","Text":"As I said, z is the exception because x and y are on one side,"},{"Start":"10:21.230 ","End":"10:24.695","Text":"z is on the other side and it opens up in the z-direction."},{"Start":"10:24.695 ","End":"10:27.320","Text":"If you just did the other x squared over"},{"Start":"10:27.320 ","End":"10:31.130","Text":"a squared plus z squared over c squared equals y squared over b squared,"},{"Start":"10:31.130 ","End":"10:34.830","Text":"you\u0027d get it opening in the y-direction."},{"Start":"10:34.830 ","End":"10:37.190","Text":"In the next sections,"},{"Start":"10:37.190 ","End":"10:39.440","Text":"I\u0027m not going to repeat all that."},{"Start":"10:39.440 ","End":"10:45.980","Text":"That we can change the order of the variables or the names or the positions to"},{"Start":"10:45.980 ","End":"10:51.860","Text":"get to adapt it to orient in other orientations."},{"Start":"10:51.860 ","End":"10:57.510","Text":"Here we took z as the center axis or it opens up in the z-direction."},{"Start":"10:57.520 ","End":"11:05.099","Text":"As for circular, if we have that a equals b,"},{"Start":"11:05.680 ","End":"11:09.410","Text":"then we get the circular cone."},{"Start":"11:09.410 ","End":"11:15.780","Text":"That\u0027s the classic cone where the cross-section is a circle."},{"Start":"11:17.380 ","End":"11:26.225","Text":"That\u0027s about it for elliptic cones and circular cones."},{"Start":"11:26.225 ","End":"11:29.190","Text":"Let\u0027s move on."}],"ID":9762},{"Watched":false,"Name":"Exercises 11","Duration":"6m 15s","ChapterTopicVideoID":9811,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9811.jpeg","UploadDate":"2017-08-01T05:02:43.0000000","DurationForVideoObject":"PT6M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.360","Text":"In this exercise, we have a plane described by 3 points on the plane,"},{"Start":"00:06.360 ","End":"00:09.689","Text":"but we want to find the equation of the plane."},{"Start":"00:09.689 ","End":"00:11.580","Text":"To find the equation of the plane,"},{"Start":"00:11.580 ","End":"00:14.040","Text":"we usually require 2 quantities."},{"Start":"00:14.040 ","End":"00:16.619","Text":"1 is a point on the plane."},{"Start":"00:16.619 ","End":"00:18.780","Text":"We certainly have that."},{"Start":"00:18.780 ","End":"00:21.135","Text":"We have 3 of them we just have to pick 1."},{"Start":"00:21.135 ","End":"00:25.470","Text":"The second is a normal vector to the plane."},{"Start":"00:25.470 ","End":"00:29.700","Text":"Let\u0027s start working on the normal vector."},{"Start":"00:29.700 ","End":"00:36.135","Text":"Now, the idea is to find 2 vectors in the plane and"},{"Start":"00:36.135 ","End":"00:42.600","Text":"take a vector that\u0027s perpendicular to those using the cross-product."},{"Start":"00:42.600 ","End":"00:46.310","Text":"There\u0027s any number of combinations I could use."},{"Start":"00:46.310 ","End":"00:54.135","Text":"What I\u0027m going to do is I\u0027m going to compute the vectors PQ and say PR."},{"Start":"00:54.135 ","End":"00:57.435","Text":"Or you could have made different choices,"},{"Start":"00:57.435 ","End":"00:59.870","Text":"and you might get a slightly different answer,"},{"Start":"00:59.870 ","End":"01:03.485","Text":"which would also be correct anyway."},{"Start":"01:03.485 ","End":"01:12.905","Text":"PQ is, we just subtract coordinates of P from the coordinates of Q."},{"Start":"01:12.905 ","End":"01:21.895","Text":"1 vector we get in brackets notation will be 1 minus 0 is 1,"},{"Start":"01:21.895 ","End":"01:25.800","Text":"0 minus 1, minus 1,"},{"Start":"01:25.800 ","End":"01:28.275","Text":"1 minus 1 is 0."},{"Start":"01:28.275 ","End":"01:30.465","Text":"That\u0027s the PQ vector."},{"Start":"01:30.465 ","End":"01:33.360","Text":"The other one, the PR,"},{"Start":"01:33.360 ","End":"01:37.230","Text":"same thing, same method,"},{"Start":"01:37.230 ","End":"01:40.710","Text":"1 minus 0 is 1,"},{"Start":"01:40.710 ","End":"01:46.020","Text":"minus 3 minus 1 is minus 4,"},{"Start":"01:46.020 ","End":"01:52.080","Text":"and minus 1 minus 1 is minus 2."},{"Start":"01:52.080 ","End":"01:57.970","Text":"These are 2 vectors parallel to the plane."},{"Start":"01:58.400 ","End":"02:03.995","Text":"If I take something that\u0027s perpendicular or orthogonal, normal,"},{"Start":"02:03.995 ","End":"02:07.460","Text":"whatever to these 2 vectors,"},{"Start":"02:07.460 ","End":"02:14.340","Text":"then it\u0027ll be orthogonal to the plane."},{"Start":"02:14.340 ","End":"02:22.805","Text":"I would like to take my normal vector to equal this first one,"},{"Start":"02:22.805 ","End":"02:28.530","Text":"cross-product with the second one."},{"Start":"02:30.520 ","End":"02:35.690","Text":"I\u0027m just going to give you the answer to this because you know how to do this thing."},{"Start":"02:35.690 ","End":"02:38.479","Text":"It\u0027s just the mechanical exercise,"},{"Start":"02:38.479 ","End":"02:40.375","Text":"plugging into the formula."},{"Start":"02:40.375 ","End":"02:42.690","Text":"There\u0027s more than 1 way of doing it but anyway,"},{"Start":"02:42.690 ","End":"02:44.805","Text":"this is the answer."},{"Start":"02:44.805 ","End":"02:49.805","Text":"We have a normal vector and I mentioned we also need a point on the plane."},{"Start":"02:49.805 ","End":"02:52.460","Text":"This is the position vector of the point."},{"Start":"02:52.460 ","End":"02:54.890","Text":"I\u0027ll just take the first one."},{"Start":"02:54.890 ","End":"02:57.740","Text":"The first one, position vector,"},{"Start":"02:57.740 ","End":"02:59.450","Text":"we\u0027ll call it r naught,"},{"Start":"02:59.450 ","End":"03:06.860","Text":"is 0, 1, 1."},{"Start":"03:06.860 ","End":"03:09.980","Text":"Then the formula for the plane is"},{"Start":"03:09.980 ","End":"03:17.585","Text":"that n dot product"},{"Start":"03:17.585 ","End":"03:27.160","Text":"with r minus r naught is equal to 0."},{"Start":"03:27.160 ","End":"03:34.395","Text":"We get that 2, 2, 3, sorry,"},{"Start":"03:34.395 ","End":"03:38.760","Text":"minus 3 dot-product with,"},{"Start":"03:38.760 ","End":"03:42.030","Text":"now, r is x, y, z."},{"Start":"03:42.030 ","End":"03:47.190","Text":"Here we have x minus 0."},{"Start":"03:47.190 ","End":"03:49.425","Text":"I\u0027m putting the 0 in for emphasis."},{"Start":"03:49.425 ","End":"03:53.010","Text":"y minus 1, that\u0027s the 1 from here,"},{"Start":"03:53.010 ","End":"03:59.595","Text":"and z minus 1 is the 1 from there, equals 0."},{"Start":"03:59.595 ","End":"04:09.435","Text":"If we expand, we get 2x plus"},{"Start":"04:09.435 ","End":"04:14.430","Text":"2y minus 1 minus"},{"Start":"04:14.430 ","End":"04:20.100","Text":"3 z minus 1 equals 0."},{"Start":"04:20.100 ","End":"04:28.840","Text":"Then that gives us that 2x plus 2y minus 3z."},{"Start":"04:30.300 ","End":"04:35.680","Text":"Some folks like to leave the number on the left and make it equal to 0."},{"Start":"04:35.680 ","End":"04:38.770","Text":"Some folks like to bring the numbers to the right."},{"Start":"04:38.770 ","End":"04:41.630","Text":"I\u0027ll bring the numbers to the right."},{"Start":"04:42.470 ","End":"04:51.605","Text":"On the left I have minus 2 plus 3 is 1."},{"Start":"04:51.605 ","End":"04:55.230","Text":"On the other side it\u0027s minus 1."},{"Start":"04:56.530 ","End":"04:59.180","Text":"That\u0027s the answer."},{"Start":"04:59.180 ","End":"05:02.214","Text":"That\u0027s the equation of the plane."},{"Start":"05:02.214 ","End":"05:06.320","Text":"I\u0027ll highlight it and I\u0027ll just mention that it\u0027s a good idea"},{"Start":"05:06.320 ","End":"05:09.890","Text":"if you have time to substitute each of"},{"Start":"05:09.890 ","End":"05:17.680","Text":"these 3 points in the plane equation and see if it works out right."},{"Start":"05:17.680 ","End":"05:21.350","Text":"For example, if I put in 0, 1, 1,"},{"Start":"05:21.350 ","End":"05:27.180","Text":"I\u0027ve got here 0 plus 2 minus 3."},{"Start":"05:27.180 ","End":"05:28.590","Text":"Yep, it works."},{"Start":"05:28.590 ","End":"05:30.000","Text":"Let\u0027s try the last one."},{"Start":"05:30.000 ","End":"05:32.280","Text":"1 minus 3 minus 1."},{"Start":"05:32.280 ","End":"05:34.920","Text":"I\u0027ve got minus 2,"},{"Start":"05:34.920 ","End":"05:41.610","Text":"and then minus 6 is minus 8."},{"Start":"05:41.610 ","End":"05:52.785","Text":"Two minus 6 is minus 4."},{"Start":"05:52.785 ","End":"05:56.520","Text":"Minus 4 minus 3 [inaudible] is plus 3,"},{"Start":"05:56.520 ","End":"05:59.025","Text":"minus 4 plus 3 is minus 1."},{"Start":"05:59.025 ","End":"06:02.100","Text":"The middle one also works."},{"Start":"06:02.100 ","End":"06:03.510","Text":"One, 0, 1."},{"Start":"06:03.510 ","End":"06:07.185","Text":"I\u0027ve got 2, nothing minus 3."},{"Start":"06:07.185 ","End":"06:12.515","Text":"Yep. That\u0027s an optional step but recommended if you have time."},{"Start":"06:12.515 ","End":"06:15.210","Text":"Okay. We\u0027re done."}],"ID":9763},{"Watched":false,"Name":"Exercises 12","Duration":"4m 16s","ChapterTopicVideoID":9812,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9812.jpeg","UploadDate":"2017-08-01T05:02:56.6300000","DurationForVideoObject":"PT4M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we need the equation of a plane."},{"Start":"00:04.080 ","End":"00:07.080","Text":"It passes through a given point,"},{"Start":"00:07.080 ","End":"00:13.200","Text":"and it\u0027s also orthogonal/perpendicular to this line,"},{"Start":"00:13.200 ","End":"00:15.300","Text":"which is given in vector form."},{"Start":"00:15.300 ","End":"00:16.860","Text":"Normally for a plane,"},{"Start":"00:16.860 ","End":"00:18.930","Text":"we need a point and a normal vector."},{"Start":"00:18.930 ","End":"00:20.360","Text":"Well, the point we have."},{"Start":"00:20.360 ","End":"00:24.149","Text":"What about a normal vector to the plane?"},{"Start":"00:24.149 ","End":"00:33.470","Text":"Well, if we take a direction vector for this line and the direction vector is gotten by"},{"Start":"00:33.470 ","End":"00:43.770","Text":"the coefficients of the t. If I take the vector v as 1,"},{"Start":"00:43.770 ","End":"00:49.545","Text":"3, 4 from the coefficients of t,"},{"Start":"00:49.545 ","End":"00:54.780","Text":"this vector is parallel to the line,"},{"Start":"00:54.780 ","End":"00:58.175","Text":"and if it\u0027s parallel to the line,"},{"Start":"00:58.175 ","End":"01:05.750","Text":"then it\u0027s going to be orthogonal to the plane."},{"Start":"01:05.750 ","End":"01:08.310","Text":"Let me rephrase."},{"Start":"01:08.310 ","End":"01:15.030","Text":"If I say that something is orthogonal to this line,"},{"Start":"01:15.030 ","End":"01:20.105","Text":"it\u0027s exactly the same as saying orthogonal to this vector,"},{"Start":"01:20.105 ","End":"01:25.780","Text":"which means that this vector will be our normal vector for the plane."},{"Start":"01:25.780 ","End":"01:28.180","Text":"I\u0027ll just take my n to be the same thing,"},{"Start":"01:28.180 ","End":"01:31.399","Text":"just using a different letter,"},{"Start":"01:31.399 ","End":"01:35.630","Text":"and then I have now a normal."},{"Start":"01:35.630 ","End":"01:44.545","Text":"In other words, a vector which is perpendicular or orthogonal to the plane,"},{"Start":"01:44.545 ","End":"01:50.715","Text":"and I have the point 0, 2, 1."},{"Start":"01:50.715 ","End":"01:57.540","Text":"This is my n. This is my r naught,"},{"Start":"01:57.540 ","End":"02:03.470","Text":"we called it, which is the position vector of that 0, 2, minus 1."},{"Start":"02:03.470 ","End":"02:09.240","Text":"When I have the position vector of a point and the normal,"},{"Start":"02:09.240 ","End":"02:16.110","Text":"then the equation is that r minus"},{"Start":"02:16.110 ","End":"02:26.100","Text":"r naught dot n is equal to 0."},{"Start":"02:26.100 ","End":"02:28.350","Text":"Now, r is vector x,"},{"Start":"02:28.350 ","End":"02:30.465","Text":"y, z in general."},{"Start":"02:30.465 ","End":"02:36.000","Text":"What we get from here is that x, y,"},{"Start":"02:36.000 ","End":"02:39.810","Text":"z minus, well, maybe I\u0027ll write that,"},{"Start":"02:39.810 ","End":"02:44.970","Text":"r in general is x, y, z."},{"Start":"02:44.970 ","End":"02:49.515","Text":"So r minus r naught is going to be x,"},{"Start":"02:49.515 ","End":"02:54.645","Text":"I\u0027ll write minus 0 to emphasize,"},{"Start":"02:54.645 ","End":"02:59.000","Text":"y minus 2, z minus minus 1,"},{"Start":"02:59.000 ","End":"03:05.029","Text":"z plus 1 dot with the normal vector,"},{"Start":"03:05.029 ","End":"03:13.300","Text":"which we got as I said from the direction vector of the line 1,"},{"Start":"03:13.300 ","End":"03:17.129","Text":"3, 4, is equal to 0."},{"Start":"03:17.129 ","End":"03:18.970","Text":"Okay. The dot product,"},{"Start":"03:18.970 ","End":"03:23.330","Text":"we just multiply each pair and then we add."},{"Start":"03:23.330 ","End":"03:28.165","Text":"So this times this is x, this times this,"},{"Start":"03:28.165 ","End":"03:31.940","Text":"3 times y minus 2,"},{"Start":"03:31.940 ","End":"03:36.380","Text":"and then 4 times z plus 1."},{"Start":"03:36.380 ","End":"03:38.890","Text":"This is equal to 0,"},{"Start":"03:38.890 ","End":"03:42.210","Text":"and then I can expand. Lets see."},{"Start":"03:42.210 ","End":"03:45.945","Text":"I have x and I have 3y,"},{"Start":"03:45.945 ","End":"03:48.780","Text":"and from here I have 4z."},{"Start":"03:48.780 ","End":"03:54.030","Text":"Now the numbers, we have to decide on the left or the right."},{"Start":"03:54.030 ","End":"03:55.470","Text":"Some people prefer this,"},{"Start":"03:55.470 ","End":"03:58.260","Text":"but that I\u0027ll put them on the right."},{"Start":"03:58.260 ","End":"04:03.995","Text":"On the left, I have minus 6 plus 4,"},{"Start":"04:03.995 ","End":"04:05.885","Text":"which is minus 2,"},{"Start":"04:05.885 ","End":"04:09.570","Text":"so on the right it becomes 2."},{"Start":"04:10.160 ","End":"04:15.700","Text":"That is the answer and we\u0027re done."}],"ID":9764},{"Watched":false,"Name":"Exercises 13","Duration":"4m 38s","ChapterTopicVideoID":9813,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9813.jpeg","UploadDate":"2017-08-01T05:03:09.6870000","DurationForVideoObject":"PT4M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.690","Text":"In this exercise, we have to find the equation of the plane containing this point,"},{"Start":"00:06.690 ","End":"00:12.135","Text":"and the other condition is that it\u0027s going to be parallel to this plane."},{"Start":"00:12.135 ","End":"00:17.550","Text":"Now, the best conditions that we can get for a plane"},{"Start":"00:17.550 ","End":"00:21.345","Text":"is a point on the plane and a normal vector."},{"Start":"00:21.345 ","End":"00:24.780","Text":"Now, we don\u0027t have a normal vector."},{"Start":"00:24.780 ","End":"00:27.780","Text":"We have the condition that it\u0027s parallel to this plane,"},{"Start":"00:27.780 ","End":"00:33.705","Text":"but that\u0027s actually very close because we know the normal vector for this plane."},{"Start":"00:33.705 ","End":"00:43.285","Text":"The normal vector for this plane is just gotten from the coefficients 4, 8, minus 2."},{"Start":"00:43.285 ","End":"00:45.860","Text":"Actually, I should say a normal vector,"},{"Start":"00:45.860 ","End":"00:48.110","Text":"not the normal vector because there\u0027s many of them,"},{"Start":"00:48.110 ","End":"00:51.830","Text":"many non-zero scalar times."},{"Start":"00:51.830 ","End":"00:53.660","Text":"A normal vector is a normal vector."},{"Start":"00:53.660 ","End":"00:57.055","Text":"Anyway, this is the most obvious 1."},{"Start":"00:57.055 ","End":"00:59.690","Text":"Now, if the planes are parallel,"},{"Start":"00:59.690 ","End":"01:02.225","Text":"normal to 1 is going to be a normal to the other."},{"Start":"01:02.225 ","End":"01:05.825","Text":"They\u0027re parallel so anything orthogonal to 1 is going to be orthogonal to the other."},{"Start":"01:05.825 ","End":"01:07.955","Text":"We have the normal vector,"},{"Start":"01:07.955 ","End":"01:10.485","Text":"so this is vector n,"},{"Start":"01:10.485 ","End":"01:14.854","Text":"and then we have our point on the plane."},{"Start":"01:14.854 ","End":"01:18.740","Text":"Usually we take its position vector"},{"Start":"01:18.740 ","End":"01:21.090","Text":"rather than the point itself when we call it our naught,"},{"Start":"01:21.090 ","End":"01:25.645","Text":"and that would be these numbers but as a vector."},{"Start":"01:25.645 ","End":"01:29.750","Text":"Position vector means the vector connecting it from the origin to the point."},{"Start":"01:29.750 ","End":"01:32.130","Text":"Anyway, we know all this."},{"Start":"01:32.900 ","End":"01:35.870","Text":"Now we have the normal and we have the point,"},{"Start":"01:35.870 ","End":"01:38.690","Text":"and now we just apply the equation."},{"Start":"01:38.690 ","End":"01:50.190","Text":"Let me just remind you that we use x, y, z to represent in the formula r."},{"Start":"01:50.190 ","End":"01:59.759","Text":"The formula is that r minus r naught dot n is equal to 0,"},{"Start":"01:59.759 ","End":"02:02.765","Text":"but I need vector signs above all of these."},{"Start":"02:02.765 ","End":"02:05.100","Text":"This is dot product."},{"Start":"02:05.270 ","End":"02:08.415","Text":"What we get is if we just substitute,"},{"Start":"02:08.415 ","End":"02:10.500","Text":"r is this and our naught is that."},{"Start":"02:10.500 ","End":"02:16.720","Text":"So x minus minus 7 is x plus 7,"},{"Start":"02:17.030 ","End":"02:21.310","Text":"where is n, here, times 4."},{"Start":"02:22.610 ","End":"02:27.870","Text":"Then the next coordinate, let\u0027s see,"},{"Start":"02:27.870 ","End":"02:33.850","Text":"it\u0027s going to be y minus 3 and then times 8."},{"Start":"02:35.090 ","End":"02:37.580","Text":"I never can remember the formula."},{"Start":"02:37.580 ","End":"02:39.710","Text":"Sometimes we put the n in front or after,"},{"Start":"02:39.710 ","End":"02:41.540","Text":"it doesn\u0027t make any real difference."},{"Start":"02:41.540 ","End":"02:43.775","Text":"Put it after."},{"Start":"02:43.775 ","End":"02:47.650","Text":"I think it\u0027s usually put before, whatever."},{"Start":"02:47.650 ","End":"02:49.020","Text":"Okay."},{"Start":"02:49.020 ","End":"02:59.770","Text":"Last 1 is z minus 9 times the last coordinate of the normal vector,"},{"Start":"02:59.770 ","End":"03:02.245","Text":"which is minus 2,"},{"Start":"03:02.245 ","End":"03:06.590","Text":"I\u0027ll just put a minus 2 here, equals 0."},{"Start":"03:06.590 ","End":"03:09.580","Text":"Okay. I\u0027ll collect the variables."},{"Start":"03:09.580 ","End":"03:11.290","Text":"I\u0027ve got 4x."},{"Start":"03:11.290 ","End":"03:18.395","Text":"From here, I have 8y minus 2z."},{"Start":"03:18.395 ","End":"03:21.340","Text":"Then you just have to make a choice."},{"Start":"03:21.340 ","End":"03:23.980","Text":"Do you want to put the numbers on the left or on the right?"},{"Start":"03:23.980 ","End":"03:26.710","Text":"They\u0027re both acceptable standard forms."},{"Start":"03:26.710 ","End":"03:30.230","Text":"I\u0027ll go with the numbers on the right."},{"Start":"03:30.230 ","End":"03:31.510","Text":"Let\u0027s see."},{"Start":"03:31.510 ","End":"03:41.025","Text":"On the left, we had 28 minus 24 is 4, plus 18 is 22."},{"Start":"03:41.025 ","End":"03:43.225","Text":"But when I bring it over to the right,"},{"Start":"03:43.225 ","End":"03:47.090","Text":"it will be minus 22."},{"Start":"03:47.760 ","End":"03:53.935","Text":"Something I do sometimes just as a check."},{"Start":"03:53.935 ","End":"03:56.050","Text":"If it\u0027s a plane with 3 points,"},{"Start":"03:56.050 ","End":"03:57.310","Text":"I can do 3 checks here."},{"Start":"03:57.310 ","End":"03:58.075","Text":"We have a point."},{"Start":"03:58.075 ","End":"04:04.080","Text":"At least I can check that this is on the plane,"},{"Start":"04:04.080 ","End":"04:06.700","Text":"so let\u0027s just mentally do it quickly."},{"Start":"04:06.700 ","End":"04:11.170","Text":"4 times minus 7 is minus 28."},{"Start":"04:11.170 ","End":"04:14.110","Text":"This and this is plus 24,"},{"Start":"04:14.110 ","End":"04:16.895","Text":"so we\u0027re down to minus 4."},{"Start":"04:16.895 ","End":"04:20.475","Text":"Minus 2z is minus 18,"},{"Start":"04:20.475 ","End":"04:21.705","Text":"yeah, minus 22."},{"Start":"04:21.705 ","End":"04:23.505","Text":"This point works."},{"Start":"04:23.505 ","End":"04:27.800","Text":"We can also see that we have 4, 8, minus 2"},{"Start":"04:27.800 ","End":"04:30.080","Text":"and we have here for 4, 8, minus 2."},{"Start":"04:30.080 ","End":"04:33.650","Text":"Okay. This is looking good."},{"Start":"04:33.650 ","End":"04:38.220","Text":"I\u0027ll highlight it and we\u0027ll declare that that\u0027s the answer and we\u0027re done."}],"ID":9765},{"Watched":false,"Name":"Exercises 14","Duration":"4m 47s","ChapterTopicVideoID":9814,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9814.jpeg","UploadDate":"2017-08-01T05:03:23.5330000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this exercise, we have 2 different planes."},{"Start":"00:04.440 ","End":"00:11.115","Text":"Often we use the Greek letter Pi for a plane."},{"Start":"00:11.115 ","End":"00:14.190","Text":"Roman P is used for points,"},{"Start":"00:14.190 ","End":"00:15.720","Text":"so use Greek letters,"},{"Start":"00:15.720 ","End":"00:18.195","Text":"the Greek P, that\u0027s Pi."},{"Start":"00:18.195 ","End":"00:23.100","Text":"Plane Pi 1 is given with this formula and"},{"Start":"00:23.100 ","End":"00:28.980","Text":"the 2nd plane Pi 2 is given with this formula."},{"Start":"00:28.980 ","End":"00:32.240","Text":"Just don\u0027t think it has anything to do with Pi from"},{"Start":"00:32.240 ","End":"00:39.600","Text":"the circle circumference over diameter or whatever."},{"Start":"00:39.600 ","End":"00:41.640","Text":"Just a Greek letter."},{"Start":"00:41.640 ","End":"00:46.110","Text":"What we want to know is are these 2 planes parallel,"},{"Start":"00:46.110 ","End":"00:50.230","Text":"are they orthogonal, meaning perpendicular or normal?"},{"Start":"00:50.230 ","End":"00:52.715","Text":"Neither this nor that."},{"Start":"00:52.715 ","End":"00:58.755","Text":"Let\u0027s see. What are going to help us here are the normals."},{"Start":"00:58.755 ","End":"01:03.300","Text":"Let\u0027s call n_1 the normal for the 1st plane."},{"Start":"01:03.300 ","End":"01:06.165","Text":"We can get that from the coefficients."},{"Start":"01:06.165 ","End":"01:13.170","Text":"That would be vector 4, 8, minus 2."},{"Start":"01:13.170 ","End":"01:18.570","Text":"The normal vector for the 2nd plane or"},{"Start":"01:18.570 ","End":"01:24.150","Text":"a normal vector is gotten from the coefficients here."},{"Start":"01:24.150 ","End":"01:27.300","Text":"I could take 2, 1,"},{"Start":"01:27.300 ","End":"01:30.870","Text":"8. Now, here\u0027s the thing."},{"Start":"01:30.870 ","End":"01:35.170","Text":"The plane\u0027s relative mutual position"},{"Start":"01:35.170 ","End":"01:39.685","Text":"is very tied in with the mutual position of the normals."},{"Start":"01:39.685 ","End":"01:42.700","Text":"Specifically, if the planes are parallel,"},{"Start":"01:42.700 ","End":"01:43.780","Text":"just think about it,"},{"Start":"01:43.780 ","End":"01:46.195","Text":"the normals are also going to be parallel."},{"Start":"01:46.195 ","End":"01:50.415","Text":"They\u0027re perpendicular to parallel planes,"},{"Start":"01:50.415 ","End":"01:52.930","Text":"going to be the same."},{"Start":"01:54.500 ","End":"01:56.850","Text":"Likewise, orthogonal."},{"Start":"01:56.850 ","End":"01:59.470","Text":"If 2 planes are orthogonal,"},{"Start":"01:59.470 ","End":"02:05.435","Text":"then the normal vectors are also going to be orthogonal."},{"Start":"02:05.435 ","End":"02:11.190","Text":"All we have to do is check these 2 normals to see are they parallel?"},{"Start":"02:11.190 ","End":"02:13.245","Text":"Are they orthogonal?"},{"Start":"02:13.245 ","End":"02:18.835","Text":"For parallel, we need for 1 of them to be a multiple of the other."},{"Start":"02:18.835 ","End":"02:21.030","Text":"These are not 0 vectors,"},{"Start":"02:21.030 ","End":"02:24.640","Text":"so 1 of them has to be a non-zero scalar times the other."},{"Start":"02:24.640 ","End":"02:30.680","Text":"Let\u0027s say that we have to have n_2 is equal to some scalar k,"},{"Start":"02:30.680 ","End":"02:37.550","Text":"not 0, times n_1 vector."},{"Start":"02:39.690 ","End":"02:45.140","Text":"Well, there\u0027s many ways to see this is not going to work out."},{"Start":"02:45.140 ","End":"02:49.910","Text":"For 1 thing, if k is positive and k times n_1,"},{"Start":"02:49.910 ","End":"02:55.004","Text":"it\u0027s going to be plus, plus, minus."},{"Start":"02:55.004 ","End":"02:56.480","Text":"If k is negative,"},{"Start":"02:56.480 ","End":"02:58.910","Text":"I\u0027m gonna get minus, minus, plus."},{"Start":"02:58.910 ","End":"03:02.320","Text":"But there\u0027s no way I\u0027m going to get plus, plus, plus."},{"Start":"03:02.320 ","End":"03:06.230","Text":"There is no such k. I could also try doing computations."},{"Start":"03:06.230 ","End":"03:09.620","Text":"If there was a k, k times 4 is 2,"},{"Start":"03:09.620 ","End":"03:11.875","Text":"so k would have to be a 1/2."},{"Start":"03:11.875 ","End":"03:14.705","Text":"Then we\u0027d have to have a 1/2 times 8 is 1,"},{"Start":"03:14.705 ","End":"03:16.295","Text":"so it doesn\u0027t work out."},{"Start":"03:16.295 ","End":"03:18.815","Text":"Many ways to see it doesn\u0027t work out."},{"Start":"03:18.815 ","End":"03:22.940","Text":"This does not work out,"},{"Start":"03:22.940 ","End":"03:26.560","Text":"and so these 2 normal vectors are not parallel."},{"Start":"03:26.560 ","End":"03:29.340","Text":"If they\u0027re not parallel,"},{"Start":"03:29.340 ","End":"03:32.790","Text":"then neither are the planes."},{"Start":"03:32.790 ","End":"03:36.020","Text":"Now let\u0027s go rule the parallel out."},{"Start":"03:36.020 ","End":"03:38.690","Text":"Now let\u0027s go for orthogonal."},{"Start":"03:38.690 ","End":"03:41.255","Text":"Orthogonal, as I say,"},{"Start":"03:41.255 ","End":"03:44.135","Text":"planes are orthogonal, normals are orthogonal,"},{"Start":"03:44.135 ","End":"03:50.000","Text":"orthogonal or perpendicular can be checked by dot product if top product is 0."},{"Start":"03:50.000 ","End":"03:55.230","Text":"Let\u0027s see what is n_1.n_ 2,"},{"Start":"03:55.230 ","End":"03:58.679","Text":"and see is this equal to 0?"},{"Start":"03:58.679 ","End":"04:01.025","Text":"Well, it\u0027s a simple computation."},{"Start":"04:01.025 ","End":"04:04.070","Text":"Multiply component-wise and add."},{"Start":"04:04.070 ","End":"04:10.310","Text":"This is equal to 4 times"},{"Start":"04:10.310 ","End":"04:18.500","Text":"2 plus 8 times 1 plus negative 2 times 8."},{"Start":"04:18.500 ","End":"04:29.280","Text":"Let\u0027s see. This is 8 plus 8 minus 16."},{"Start":"04:29.280 ","End":"04:33.045","Text":"Yes, bingo, we have 0,"},{"Start":"04:33.045 ","End":"04:36.615","Text":"so they are orthogonal."},{"Start":"04:36.615 ","End":"04:44.535","Text":"Yes. That\u0027s the answer that the 2 planes are orthogonal,"},{"Start":"04:44.535 ","End":"04:47.740","Text":"and we are done."}],"ID":9766},{"Watched":false,"Name":"Exercises 15","Duration":"11m 33s","ChapterTopicVideoID":9815,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9815.jpeg","UploadDate":"2017-08-01T05:03:55.7970000","DurationForVideoObject":"PT11M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise we have 2 planes,"},{"Start":"00:03.480 ","End":"00:05.925","Text":"Pi 1 and Pi 2,"},{"Start":"00:05.925 ","End":"00:09.480","Text":"and each is given in a different form."},{"Start":"00:09.480 ","End":"00:13.845","Text":"Pi 1 is conveniently given by an equation."},{"Start":"00:13.845 ","End":"00:22.630","Text":"Pi 2, we just have the hint that it goes through these 3 points."},{"Start":"00:22.630 ","End":"00:25.405","Text":"We\u0027re going to have to work a bit harder."},{"Start":"00:25.405 ","End":"00:29.040","Text":"The question here asks us about whether"},{"Start":"00:29.040 ","End":"00:35.170","Text":"the 2 planes are parallel or orthogonal or neither of these."},{"Start":"00:35.930 ","End":"00:41.460","Text":"We had a similar question like this but it\u0027s okay if you don\u0027t remember."},{"Start":"00:41.460 ","End":"00:45.140","Text":"What we have to do is get normal vectors for each of"},{"Start":"00:45.140 ","End":"00:47.960","Text":"these planes and then ask these questions"},{"Start":"00:47.960 ","End":"00:51.049","Text":"about the normal vectors: are the normal vectors parallel,"},{"Start":"00:51.049 ","End":"00:52.730","Text":"or orthogonal, or neither?"},{"Start":"00:52.730 ","End":"00:56.300","Text":"That will answer the original question."},{"Start":"00:56.300 ","End":"00:59.085","Text":"Now, for the plane Pi 1,"},{"Start":"00:59.085 ","End":"01:01.240","Text":"that\u0027s called the normal n_1,"},{"Start":"01:01.240 ","End":"01:06.920","Text":"that\u0027s easy to get because you can get it from the coefficients of the x y z."},{"Start":"01:06.920 ","End":"01:10.610","Text":"So a normal vector we could take as"},{"Start":"01:10.610 ","End":"01:17.310","Text":"2, minus 3, 4."},{"Start":"01:17.310 ","End":"01:24.145","Text":"The question is, how do we get a normal vector n_2 for the second plane?"},{"Start":"01:24.145 ","End":"01:26.930","Text":"We\u0027re just given hints."},{"Start":"01:26.930 ","End":"01:29.520","Text":"Here\u0027s what we can do."},{"Start":"01:29.520 ","End":"01:33.230","Text":"These points, maybe I\u0027ll name them."},{"Start":"01:33.230 ","End":"01:35.390","Text":"We could call this one P,"},{"Start":"01:35.390 ","End":"01:37.460","Text":"we could call this one Q,"},{"Start":"01:37.460 ","End":"01:43.530","Text":"we could call this one R. If I have 3 points in a plane,"},{"Start":"01:43.530 ","End":"01:47.350","Text":"say P, Q, R,"},{"Start":"01:50.930 ","End":"01:54.810","Text":"this P doesn\u0027t look very good, there it is."},{"Start":"01:54.810 ","End":"02:01.330","Text":"If I want to get a normal vector to the plane,"},{"Start":"02:01.330 ","End":"02:04.900","Text":"what I can do is take for example,"},{"Start":"02:04.900 ","End":"02:11.385","Text":"the vector that\u0027s parallel to PQ."},{"Start":"02:11.385 ","End":"02:18.150","Text":"I could take the vector any combination, let\u0027s say PR."},{"Start":"02:18.150 ","End":"02:25.570","Text":"Now, these 2 vectors are parallel to the plane containing P, Q,"},{"Start":"02:25.570 ","End":"02:33.254","Text":"and R. If I have 2 vectors that are parallel to the plane,"},{"Start":"02:33.254 ","End":"02:39.065","Text":"then their cross-product will be perpendicular to each of these."},{"Start":"02:39.065 ","End":"02:43.745","Text":"So it will be a normal vector to the plane."},{"Start":"02:43.745 ","End":"02:49.320","Text":"What I have to do is take PQ,"},{"Start":"02:49.760 ","End":"02:57.315","Text":"and then cross that vector cross product with PR,"},{"Start":"02:57.315 ","End":"03:03.170","Text":"and that should give me a normal to this second plane."},{"Start":"03:03.170 ","End":"03:09.530","Text":"This whole plane is the plane Pi 2."},{"Start":"03:09.530 ","End":"03:16.770","Text":"What we\u0027ll do is say PQ,"},{"Start":"03:16.770 ","End":"03:18.555","Text":"we can get this,"},{"Start":"03:18.555 ","End":"03:21.930","Text":"it\u0027s a displacement vector sometimes called,"},{"Start":"03:21.930 ","End":"03:26.770","Text":"the one that takes me from P to Q, subtract components,"},{"Start":"03:26.770 ","End":"03:30.065","Text":"2 minus 1 is 1,"},{"Start":"03:30.065 ","End":"03:33.335","Text":"2 minus 2 is 0,"},{"Start":"03:33.335 ","End":"03:36.605","Text":"3 minus 2 is 1."},{"Start":"03:36.605 ","End":"03:38.780","Text":"That\u0027s the PQ part."},{"Start":"03:38.780 ","End":"03:41.619","Text":"Now the PR part,"},{"Start":"03:41.619 ","End":"03:47.085","Text":"I\u0027ll take the components of R and subtract the components of P,"},{"Start":"03:47.085 ","End":"03:49.950","Text":"and I want the cross product."},{"Start":"03:49.950 ","End":"04:00.369","Text":"That will be minus 3 minus 1 is minus 4,"},{"Start":"04:00.369 ","End":"04:12.165","Text":"minus 2 minus 2 is minus 4,"},{"Start":"04:12.165 ","End":"04:22.605","Text":"and minus 6 minus 2 would be minus 8."},{"Start":"04:22.605 ","End":"04:28.540","Text":"Now like I said,"},{"Start":"04:29.030 ","End":"04:31.320","Text":"there\u0027s not a unique normal."},{"Start":"04:31.320 ","End":"04:40.610","Text":"Multiplying by a non-zero scalar will also give me a parallel vector."},{"Start":"04:40.610 ","End":"04:46.250","Text":"Reason I\u0027m saying this is I see all these are divisible by 4."},{"Start":"04:46.250 ","End":"04:50.705","Text":"I also see that they\u0027re all negative and these are both nuisances."},{"Start":"04:50.705 ","End":"04:59.480","Text":"So what I\u0027m going do is replace this vector by a more convenient vector,"},{"Start":"04:59.480 ","End":"05:02.840","Text":"which will be dividing by minus 4,"},{"Start":"05:02.840 ","End":"05:08.075","Text":"I\u0027ll get 1, 1, 2."},{"Start":"05:08.075 ","End":"05:09.770","Text":"That\u0027s much nicer."},{"Start":"05:09.770 ","End":"05:11.600","Text":"It\u0027s a scalar times this,"},{"Start":"05:11.600 ","End":"05:16.790","Text":"so it will give me a multiple of the normal,"},{"Start":"05:16.790 ","End":"05:19.050","Text":"which is still a normal."},{"Start":"05:20.090 ","End":"05:24.065","Text":"There are several ways to compute cross-product."},{"Start":"05:24.065 ","End":"05:27.995","Text":"I\u0027ll use the one with the determinant."},{"Start":"05:27.995 ","End":"05:31.910","Text":"What we do is we take a 3-by-3 determinant,"},{"Start":"05:31.910 ","End":"05:37.545","Text":"on the top row we put i, j, k,"},{"Start":"05:37.545 ","End":"05:44.010","Text":"then we put one of the first one, 1, 0, 1,"},{"Start":"05:44.010 ","End":"05:46.380","Text":"and then the other one, 1,"},{"Start":"05:46.380 ","End":"05:52.680","Text":"1, 2, and we compute this determinant."},{"Start":"05:52.730 ","End":"05:58.490","Text":"What we get, I\u0027m assuming you know determinants if"},{"Start":"05:58.490 ","End":"06:04.385","Text":"not the formula that just gives you the 3 components."},{"Start":"06:04.385 ","End":"06:06.620","Text":"Like in the first component,"},{"Start":"06:06.620 ","End":"06:12.485","Text":"it might be this times this minus this times this."},{"Start":"06:12.485 ","End":"06:14.675","Text":"There\u0027s another way of doing it."},{"Start":"06:14.675 ","End":"06:18.365","Text":"I\u0027ll do it this way. So we look at the i,"},{"Start":"06:18.365 ","End":"06:25.460","Text":"and we remove the row and column with the i,"},{"Start":"06:25.460 ","End":"06:27.560","Text":"and we get a 2-by-2 determinant,"},{"Start":"06:27.560 ","End":"06:31.939","Text":"which is this 0, 1,"},{"Start":"06:31.939 ","End":"06:38.985","Text":"1, 2, and then that\u0027s the coefficient of i."},{"Start":"06:38.985 ","End":"06:42.940","Text":"Then for the j,"},{"Start":"06:43.070 ","End":"06:45.210","Text":"well actually it\u0027s a minus."},{"Start":"06:45.210 ","End":"06:50.760","Text":"The j1 gets a negative determinant,"},{"Start":"06:50.760 ","End":"06:53.715","Text":"something j, I\u0027ll fill it in a moment."},{"Start":"06:53.715 ","End":"06:56.580","Text":"The k gets a plus again,"},{"Start":"06:56.580 ","End":"07:01.240","Text":"a 2-by-2 determinant with a k. Now,"},{"Start":"07:01.240 ","End":"07:05.210","Text":"for the j, I just cross off the row and column with the j."},{"Start":"07:05.210 ","End":"07:07.730","Text":"What we\u0027re left with is 1, 1, 1,"},{"Start":"07:07.730 ","End":"07:12.555","Text":"2, 1, 1, 1,"},{"Start":"07:12.555 ","End":"07:16.560","Text":"2, and for the k,"},{"Start":"07:16.560 ","End":"07:22.840","Text":"we get this determinant 1, 0, 1, 1."},{"Start":"07:23.360 ","End":"07:32.255","Text":"That gives us a 2-by-2 determinant is this diagonal product minus this diagonals product."},{"Start":"07:32.255 ","End":"07:35.785","Text":"So 0 minus 1,"},{"Start":"07:35.785 ","End":"07:44.990","Text":"so that\u0027s minus i. I\u0027ll go back to the angular bracket notation."},{"Start":"07:44.990 ","End":"07:47.595","Text":"So instead of writing minus i,"},{"Start":"07:47.595 ","End":"07:51.120","Text":"I\u0027ll write minus 1, here."},{"Start":"07:51.120 ","End":"07:56.044","Text":"In the next one we have for the j,"},{"Start":"07:56.044 ","End":"08:01.340","Text":"2 minus 1 is 1,"},{"Start":"08:01.340 ","End":"08:03.050","Text":"but there\u0027s a minus."},{"Start":"08:03.050 ","End":"08:07.970","Text":"So it\u0027s minus 1 or minus j,"},{"Start":"08:07.970 ","End":"08:12.335","Text":"which is like this with the bracket notation for vectors."},{"Start":"08:12.335 ","End":"08:17.900","Text":"The last one, 1 minus 0 is 1,"},{"Start":"08:17.900 ","End":"08:19.250","Text":"and there\u0027s a plus,"},{"Start":"08:19.250 ","End":"08:25.170","Text":"so it\u0027s plus k. So I just put plus 1 or plain old 1."},{"Start":"08:25.360 ","End":"08:31.765","Text":"Now I have the 2 normal vectors."},{"Start":"08:31.765 ","End":"08:39.345","Text":"This is n_1 and I\u0027ll write this again as n_2."},{"Start":"08:39.345 ","End":"08:42.705","Text":"Let me just highlight these."},{"Start":"08:42.705 ","End":"08:49.424","Text":"Here is a normal vector to the first plane Pi 1,"},{"Start":"08:49.424 ","End":"08:54.750","Text":"and here\u0027s a normal vector to the second plane Pi 2."},{"Start":"08:54.750 ","End":"08:57.825","Text":"Now we just have to ask,"},{"Start":"08:57.825 ","End":"09:00.785","Text":"are these 2 parallel?"},{"Start":"09:00.785 ","End":"09:06.750","Text":"If these are parallel then the planes are going to be parallel and vice versa."},{"Start":"09:07.520 ","End":"09:12.315","Text":"I\u0027d like using the trick with the sign,"},{"Start":"09:12.315 ","End":"09:19.705","Text":"because if I take a non-zero constant and multiply it by 1 and hope to get to the other,"},{"Start":"09:19.705 ","End":"09:22.610","Text":"the middle one is going to be the odd one out."},{"Start":"09:22.610 ","End":"09:25.520","Text":"If I multiply it by a positive constant,"},{"Start":"09:25.520 ","End":"09:27.890","Text":"I\u0027m going to get plus, minus, plus."},{"Start":"09:27.890 ","End":"09:31.100","Text":"Otherwise I\u0027m going to get minus, plus, minus."},{"Start":"09:31.100 ","End":"09:37.440","Text":"But in no way does it fit this because I can\u0027t have a minus, minus, plus."},{"Start":"09:37.440 ","End":"09:39.080","Text":"It\u0027s the middle one that\u0027s the odd one out."},{"Start":"09:39.080 ","End":"09:44.630","Text":"So no positive or negative scalar can multiply this to give me this,"},{"Start":"09:44.630 ","End":"09:50.315","Text":"so they are not parallel."},{"Start":"09:50.315 ","End":"09:52.930","Text":"The normal vectors aren\u0027t parallel,"},{"Start":"09:52.930 ","End":"09:55.450","Text":"so the planes are not parallel."},{"Start":"09:55.450 ","End":"09:59.435","Text":"That still gives us just chance with the orthogonal."},{"Start":"09:59.435 ","End":"10:06.590","Text":"The planes are orthogonal if and only if the normal vectors are orthogonal."},{"Start":"10:06.870 ","End":"10:15.220","Text":"The test for orthogonal is using the dot-product to see if it\u0027s 0."},{"Start":"10:15.220 ","End":"10:19.480","Text":"So is n_1 dot-product with n_2 equals,"},{"Start":"10:19.480 ","End":"10:22.540","Text":"this is what we need to check, 0?"},{"Start":"10:22.540 ","End":"10:25.985","Text":"Let\u0027s see, just do the computation."},{"Start":"10:25.985 ","End":"10:28.335","Text":"This dot with this,"},{"Start":"10:28.335 ","End":"10:33.720","Text":"we get 2 times minus 1,"},{"Start":"10:33.720 ","End":"10:40.510","Text":"and then minus 3 times minus 1,"},{"Start":"10:40.510 ","End":"10:48.900","Text":"and then 4 times 1."},{"Start":"10:48.900 ","End":"10:51.200","Text":"What does this come out to be?"},{"Start":"10:51.200 ","End":"11:05.090","Text":"Comes out minus 2 plus 3 plus 4."},{"Start":"11:05.090 ","End":"11:10.350","Text":"This is equal to 5,"},{"Start":"11:10.350 ","End":"11:15.875","Text":"and 5 is not equal to 0."},{"Start":"11:15.875 ","End":"11:23.760","Text":"So we\u0027re also not orthogonal, not perpendicular."},{"Start":"11:24.740 ","End":"11:30.140","Text":"I guess that leaves us with the neither or neither,"},{"Start":"11:30.140 ","End":"11:34.290","Text":"however you say it, and we\u0027re done."}],"ID":9767},{"Watched":false,"Name":"Exercises 16","Duration":"3m 58s","ChapterTopicVideoID":9807,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9807.jpeg","UploadDate":"2017-08-01T05:01:11.5230000","DurationForVideoObject":"PT3M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this exercise, we\u0027re given a plane and a line, the plane,"},{"Start":"00:04.200 ","End":"00:09.810","Text":"we\u0027ll call it Pi, is given by typical plane equation."},{"Start":"00:09.810 ","End":"00:13.140","Text":"The line is given in the parametric form,"},{"Start":"00:13.140 ","End":"00:14.640","Text":"x equals, y equals,"},{"Start":"00:14.640 ","End":"00:18.390","Text":"z equals as functions of t."},{"Start":"00:18.390 ","End":"00:21.120","Text":"The question is,"},{"Start":"00:21.120 ","End":"00:24.225","Text":"does the line intersect the plane?"},{"Start":"00:24.225 ","End":"00:28.755","Text":"If it does, let\u0027s find the point of intersection."},{"Start":"00:28.755 ","End":"00:31.830","Text":"The most straightforward thing to do is to say,"},{"Start":"00:31.830 ","End":"00:40.155","Text":"a typical point on the line is given by 1 minus t, 3t,"},{"Start":"00:40.155 ","End":"00:45.560","Text":"1 plus t for x, y, z and to substitute into"},{"Start":"00:45.560 ","End":"00:49.860","Text":"the plane equation and that gives us an equation in t."},{"Start":"00:49.860 ","End":"00:52.445","Text":"If it has a solution then good,"},{"Start":"00:52.445 ","End":"00:54.110","Text":"we have an intersection."},{"Start":"00:54.110 ","End":"00:58.645","Text":"Let\u0027s do that. What I\u0027m going to do is say,"},{"Start":"00:58.645 ","End":"01:01.580","Text":"I\u0027m looking at the equation of the plane,"},{"Start":"01:01.580 ","End":"01:06.490","Text":"x is 1 minus t,"},{"Start":"01:06.490 ","End":"01:13.575","Text":"and then I have minus y is 3t and then"},{"Start":"01:13.575 ","End":"01:18.240","Text":"3 times z is 1 plus t."},{"Start":"01:18.240 ","End":"01:23.245","Text":"This has to equal 6 if it\u0027s going to be on the plane."},{"Start":"01:23.245 ","End":"01:25.730","Text":"Let\u0027s see if it has a solution for t."},{"Start":"01:25.730 ","End":"01:29.930","Text":"We got 2 minus 2t"},{"Start":"01:29.930 ","End":"01:38.300","Text":"minus 3t plus 3 plus 3t equals 6."},{"Start":"01:38.300 ","End":"01:42.065","Text":"Let\u0027s put t\u0027s on the left and numbers on the right."},{"Start":"01:42.065 ","End":"01:51.600","Text":"The minus 3t and the plus 3t cancels so on the left I have minus 2t."},{"Start":"01:51.600 ","End":"01:58.730","Text":"Let\u0027s see, on the right I get 6 minus 3 minus 2,"},{"Start":"01:58.730 ","End":"02:03.495","Text":"that gives me 1 and so,"},{"Start":"02:03.495 ","End":"02:08.680","Text":"t is equal to minus 1/2."},{"Start":"02:08.680 ","End":"02:12.410","Text":"Now I know that they intersect."},{"Start":"02:12.410 ","End":"02:15.640","Text":"What I need is the point of intersection,"},{"Start":"02:15.640 ","End":"02:23.045","Text":"so after just put this value of t into the parametric equation of the line."},{"Start":"02:23.045 ","End":"02:27.065","Text":"This will give us that x is equal to"},{"Start":"02:27.065 ","End":"02:35.070","Text":"1 minus t is 1 minus minus 1/2 is 1 1/2."},{"Start":"02:35.070 ","End":"02:40.110","Text":"y is going to equal 3 times"},{"Start":"02:40.110 ","End":"02:46.425","Text":"t which is minus 3 times 1/2 minus 1 1/2."},{"Start":"02:46.425 ","End":"02:50.505","Text":"I\u0027ll do it as mixed numbers, the fractions."},{"Start":"02:50.505 ","End":"02:54.270","Text":"z is 1 plus t,"},{"Start":"02:54.270 ","End":"02:59.260","Text":"1 minus 1/2 is 1 1/2."},{"Start":"02:59.690 ","End":"03:09.135","Text":"The intersection point is the point 1 1/2,"},{"Start":"03:09.135 ","End":"03:15.315","Text":"minus 1 1/2, 1/2,"},{"Start":"03:15.315 ","End":"03:18.055","Text":"and that\u0027s the answer."},{"Start":"03:18.055 ","End":"03:22.675","Text":"But I\u0027d just like to do a quick check to see,"},{"Start":"03:22.675 ","End":"03:26.840","Text":"for example, if this point really is on the plane."},{"Start":"03:26.840 ","End":"03:28.960","Text":"You don\u0027t have to do this,"},{"Start":"03:28.960 ","End":"03:32.690","Text":"but I like to sometimes verify things."},{"Start":"03:33.330 ","End":"03:39.085","Text":"2x would be 3 minus y,"},{"Start":"03:39.085 ","End":"03:41.530","Text":"so it\u0027s minus minus 1 1/2,"},{"Start":"03:41.530 ","End":"03:44.050","Text":"sum up to 4 1/2."},{"Start":"03:44.050 ","End":"03:48.840","Text":"4 1/2 and 3z is another 1 1/2."},{"Start":"03:48.840 ","End":"03:52.990","Text":"Yeah, that gives me 6 and so I\u0027m more confident"},{"Start":"03:52.990 ","End":"03:57.860","Text":"that this is the right answer. That\u0027s all."}],"ID":9768},{"Watched":false,"Name":"Exercises 17","Duration":"3m ","ChapterTopicVideoID":9808,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9808.jpeg","UploadDate":"2017-08-01T05:01:19.3800000","DurationForVideoObject":"PT3M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.315","Text":"This question is similar to the previous question."},{"Start":"00:03.315 ","End":"00:05.295","Text":"We\u0027re given the plane,"},{"Start":"00:05.295 ","End":"00:07.245","Text":"we call the plane Pi,"},{"Start":"00:07.245 ","End":"00:09.270","Text":"not to be confused with the number Pi,"},{"Start":"00:09.270 ","End":"00:11.955","Text":"it\u0027s just a Greek letter,"},{"Start":"00:11.955 ","End":"00:17.160","Text":"and its equation is x minus y plus z equals 3."},{"Start":"00:17.160 ","End":"00:24.780","Text":"Then we have a line L given in vector form as follows,"},{"Start":"00:24.780 ","End":"00:31.200","Text":"as a function of t. We want to know if the line and the plane intersect,"},{"Start":"00:31.200 ","End":"00:32.760","Text":"and if so, where?"},{"Start":"00:32.760 ","End":"00:35.950","Text":"Meaning, what\u0027s the point of intersection?"},{"Start":"00:35.960 ","End":"00:44.210","Text":"What we do is we see that this is like x, y,"},{"Start":"00:44.210 ","End":"00:49.190","Text":"and z, and we want to know for some t, if x, y,"},{"Start":"00:49.190 ","End":"00:54.484","Text":"and z are such that they will satisfy the equation of the plane."},{"Start":"00:54.484 ","End":"00:56.375","Text":"That\u0027s what I\u0027m going to do."},{"Start":"00:56.375 ","End":"01:01.340","Text":"We\u0027re going to figure out x minus y plus z for a point on the line."},{"Start":"01:01.340 ","End":"01:10.280","Text":"X is 5 plus 2_t minus y,"},{"Start":"01:10.280 ","End":"01:15.800","Text":"which is 1 minus 5_t, and then,"},{"Start":"01:15.800 ","End":"01:23.230","Text":"plus z, which is 3_t has got to equal 3."},{"Start":"01:23.230 ","End":"01:25.400","Text":"If we find such a t,"},{"Start":"01:25.400 ","End":"01:28.830","Text":"then we know that there is an intersection."},{"Start":"01:28.840 ","End":"01:33.950","Text":"Let\u0027s see if we can find t. Opening up,"},{"Start":"01:33.950 ","End":"01:40.520","Text":"we have 5 plus 2_t minus 1 plus 5_t"},{"Start":"01:40.520 ","End":"01:48.900","Text":"plus 3_t is equal to 3."},{"Start":"01:52.190 ","End":"01:56.495","Text":"I just noticed that I miscopied the question."},{"Start":"01:56.495 ","End":"02:00.290","Text":"This was actually a plus,"},{"Start":"02:00.290 ","End":"02:04.640","Text":"and so this here is also a plus."},{"Start":"02:04.640 ","End":"02:10.895","Text":"That means that here is a minus."},{"Start":"02:10.895 ","End":"02:17.195","Text":"Yeah, I knew something was wrong because I knew it was suppose to cancel."},{"Start":"02:17.195 ","End":"02:24.370","Text":"Look, 2_t and 3_t is 5_t with minus 5_t, the t\u0027s canceled."},{"Start":"02:24.370 ","End":"02:27.090","Text":"5 minus 1 is 4,"},{"Start":"02:27.090 ","End":"02:31.500","Text":"so we get the equation, 4 equals 3."},{"Start":"02:31.500 ","End":"02:34.035","Text":"Now, that\u0027s impossible."},{"Start":"02:34.035 ","End":"02:42.685","Text":"That cannot be, and that means that for no value of t will this point be on the plane."},{"Start":"02:42.685 ","End":"02:46.640","Text":"No point on the line is also on the plane,"},{"Start":"02:46.640 ","End":"02:48.650","Text":"so they don\u0027t intersect."},{"Start":"02:48.650 ","End":"02:50.750","Text":"Do L and Pi intersect?"},{"Start":"02:50.750 ","End":"02:55.100","Text":"No, and so we don\u0027t answer the second part."},{"Start":"02:55.100 ","End":"03:00.150","Text":"If so, where? Because they don\u0027t intersect. That\u0027s it."}],"ID":9769},{"Watched":false,"Name":"Exercises 18","Duration":"19m 19s","ChapterTopicVideoID":9809,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9809.jpeg","UploadDate":"2017-08-01T05:02:13.1630000","DurationForVideoObject":"PT19M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we\u0027re given 2 planes,"},{"Start":"00:03.630 ","End":"00:10.755","Text":"Pi1 and Pi2 in the regular linear equation form."},{"Start":"00:10.755 ","End":"00:14.700","Text":"This is the equation of Pi1,"},{"Start":"00:14.700 ","End":"00:18.490","Text":"this is the equation of Pi2."},{"Start":"00:19.240 ","End":"00:25.965","Text":"These 2 planes we\u0027re told intersect in a line."},{"Start":"00:25.965 ","End":"00:29.160","Text":"If 2 planes are not parallel,"},{"Start":"00:29.160 ","End":"00:32.740","Text":"they intersect in a line."},{"Start":"00:33.820 ","End":"00:38.135","Text":"Actually they could be identical, in which case,"},{"Start":"00:38.135 ","End":"00:39.620","Text":"they intersect in a plane,"},{"Start":"00:39.620 ","End":"00:41.885","Text":"but those are crazy cases."},{"Start":"00:41.885 ","End":"00:46.115","Text":"Usually 2 planes unless parallel will intersect in a line."},{"Start":"00:46.115 ","End":"00:51.720","Text":"Let\u0027s call that line L. In set theory notation,"},{"Start":"00:51.720 ","End":"00:54.910","Text":"it\u0027s Pi1 intersection with Pi2."},{"Start":"00:54.910 ","End":"00:57.985","Text":"But if you\u0027re not familiar with that, never mind."},{"Start":"00:57.985 ","End":"01:04.345","Text":"We want to know the equation of the intersection line in the vector form."},{"Start":"01:04.345 ","End":"01:06.890","Text":"The 3 kinds of equations of lines,"},{"Start":"01:06.890 ","End":"01:09.800","Text":"vector, parametric, and symmetric,"},{"Start":"01:09.800 ","End":"01:12.095","Text":"we want the vector form."},{"Start":"01:12.095 ","End":"01:19.250","Text":"Here\u0027s the trick, any line in"},{"Start":"01:19.250 ","End":"01:25.805","Text":"the plane has to intersect at least 1 of the coordinate planes."},{"Start":"01:25.805 ","End":"01:28.980","Text":"It either has to hit the x,"},{"Start":"01:28.980 ","End":"01:30.420","Text":"y plane or the x,"},{"Start":"01:30.420 ","End":"01:33.090","Text":"z plane or the z, y plane."},{"Start":"01:33.090 ","End":"01:36.200","Text":"It can\u0027t miss all 3 coordinate planes."},{"Start":"01:36.200 ","End":"01:38.540","Text":"If you just think about it, I\u0027m not going to give you a proof."},{"Start":"01:38.540 ","End":"01:47.110","Text":"But, no way an infinite line cannot hit at least 1 of those 3 coordinate planes."},{"Start":"01:48.590 ","End":"01:53.480","Text":"Let\u0027s try the x, y plane."},{"Start":"01:53.480 ","End":"02:00.210","Text":"If that doesn\u0027t work, we\u0027ve got 2 more chances to see where it hits the x,"},{"Start":"02:00.210 ","End":"02:03.105","Text":"z plane or z, y plane."},{"Start":"02:03.105 ","End":"02:08.100","Text":"Let\u0027s see where, my question is,"},{"Start":"02:08.100 ","End":"02:14.880","Text":"where does L cross"},{"Start":"02:14.880 ","End":"02:20.980","Text":"the x, y plane?"},{"Start":"02:21.860 ","End":"02:26.550","Text":"If I don\u0027t get an answer,"},{"Start":"02:26.550 ","End":"02:31.395","Text":"it doesn\u0027t cross then I\u0027ll try afterwards the x,"},{"Start":"02:31.395 ","End":"02:35.925","Text":"z plane and I\u0027ll try the y, z plane."},{"Start":"02:35.925 ","End":"02:40.090","Text":"It\u0027s got to across 1 of these."},{"Start":"02:40.310 ","End":"02:44.990","Text":"In fact, I know it\u0027s going to cross the x, y plane."},{"Start":"02:44.990 ","End":"02:48.395","Text":"It\u0027s just that we get first-time lucky."},{"Start":"02:48.395 ","End":"02:50.585","Text":"How do we do that?"},{"Start":"02:50.585 ","End":"02:55.735","Text":"The x, y plane is where z equals 0."},{"Start":"02:55.735 ","End":"03:01.250","Text":"What I do is put z equals 0 in both of"},{"Start":"03:01.250 ","End":"03:09.020","Text":"these plane equations and then solve and then I\u0027ll get 2 equations in 2 unknowns,"},{"Start":"03:09.020 ","End":"03:12.440","Text":"x and y, and I\u0027ll solve for x and y."},{"Start":"03:12.440 ","End":"03:17.265","Text":"What we get is 2 equations."},{"Start":"03:17.265 ","End":"03:21.645","Text":"We get minus x plus"},{"Start":"03:21.645 ","End":"03:30.270","Text":"7y and then the z part is 0 equals 24."},{"Start":"03:30.270 ","End":"03:36.135","Text":"The other equation, minus 5x plus 6y,"},{"Start":"03:36.135 ","End":"03:41.700","Text":"the z is 0 so we get equals minus 3."},{"Start":"03:41.700 ","End":"03:45.300","Text":"Then I want to solve x and y."},{"Start":"03:45.300 ","End":"03:50.770","Text":"Then we\u0027ll add the condition that z is 0 and we\u0027ll get the intersection."},{"Start":"03:51.320 ","End":"03:55.270","Text":"Let\u0027s see how should we do this."},{"Start":"03:56.170 ","End":"04:02.120","Text":"I see that I can extract x from here easily,"},{"Start":"04:02.120 ","End":"04:03.695","Text":"so let\u0027s do that."},{"Start":"04:03.695 ","End":"04:05.480","Text":"We\u0027ll get that x,"},{"Start":"04:05.480 ","End":"04:07.445","Text":"if I take it to the right,"},{"Start":"04:07.445 ","End":"04:14.385","Text":"will equal 7y minus 24."},{"Start":"04:14.385 ","End":"04:19.490","Text":"Then if I put that x in here,"},{"Start":"04:19.490 ","End":"04:23.460","Text":"I\u0027ll get minus 5,"},{"Start":"04:23.860 ","End":"04:34.595","Text":"7y minus 24 plus 6y equals minus 3."},{"Start":"04:34.595 ","End":"04:36.875","Text":"Multiplying out."},{"Start":"04:36.875 ","End":"04:38.900","Text":"Let\u0027s see if we can do it all in 1."},{"Start":"04:38.900 ","End":"04:46.390","Text":"We\u0027ve got minus 35y plus 6y."},{"Start":"04:46.390 ","End":"04:51.570","Text":"On this side we have minus 29y."},{"Start":"04:51.570 ","End":"04:55.830","Text":"I said minus 35 plus 6."},{"Start":"04:55.830 ","End":"05:00.630","Text":"Numbers go on the right so what I\u0027ll get."},{"Start":"05:00.630 ","End":"05:03.350","Text":"This times, this is,"},{"Start":"05:03.350 ","End":"05:06.545","Text":"5 times 24 is a 120,"},{"Start":"05:06.545 ","End":"05:10.700","Text":"it\u0027s a plus because of minus, minus, 120."},{"Start":"05:10.700 ","End":"05:14.985","Text":"But on the other side, it\u0027s minus 120."},{"Start":"05:14.985 ","End":"05:19.250","Text":"I\u0027ve got minus 123."},{"Start":"05:19.630 ","End":"05:23.135","Text":"Y is equal"},{"Start":"05:23.135 ","End":"05:33.330","Text":"to 123 over 29."},{"Start":"05:33.330 ","End":"05:36.235","Text":"Now that we have y,"},{"Start":"05:36.235 ","End":"05:40.945","Text":"we can substitute in here and get what x is."},{"Start":"05:40.945 ","End":"05:44.950","Text":"X is equal to 7 times"},{"Start":"05:44.950 ","End":"05:54.860","Text":"123 over 29 minus 24."},{"Start":"05:55.050 ","End":"06:02.660","Text":"I make it 165 over 29."},{"Start":"06:02.670 ","End":"06:06.380","Text":"We have a point p,"},{"Start":"06:08.670 ","End":"06:15.055","Text":"it coordinates are where is the x that\u0027s here,"},{"Start":"06:15.055 ","End":"06:19.930","Text":"165 over 29,"},{"Start":"06:19.930 ","End":"06:27.095","Text":"the y is 123 over 29,"},{"Start":"06:27.095 ","End":"06:29.945","Text":"and the z is 0."},{"Start":"06:29.945 ","End":"06:39.665","Text":"This point p is on L. I have a point on the line."},{"Start":"06:39.665 ","End":"06:42.420","Text":"Now for the vector equation of the line,"},{"Start":"06:42.420 ","End":"06:43.565","Text":"I need a point,"},{"Start":"06:43.565 ","End":"06:46.550","Text":"and I need a direction vector,"},{"Start":"06:46.550 ","End":"06:49.925","Text":"a vector parallel to the line."},{"Start":"06:49.925 ","End":"06:59.370","Text":"Now, here\u0027s the idea of how to find such a vector."},{"Start":"07:03.490 ","End":"07:09.435","Text":"L lies in the plane Pi1."},{"Start":"07:09.435 ","End":"07:11.535","Text":"We\u0027ll get to Pi2 in a moment."},{"Start":"07:11.535 ","End":"07:13.690","Text":"L lies in Pi1,"},{"Start":"07:13.690 ","End":"07:20.805","Text":"so it\u0027s going to be perpendicular or orthogonal to the normal to that plane."},{"Start":"07:20.805 ","End":"07:30.630","Text":"I\u0027m saying that L is the symbol for perpendicular or orthogonally looks like this."},{"Start":"07:30.920 ","End":"07:39.200","Text":"The normal we\u0027re talking about for the first plane is vector minus 1,"},{"Start":"07:39.200 ","End":"07:44.870","Text":"7, minus 2, just the coefficients of x, y, and z."},{"Start":"07:44.870 ","End":"07:50.660","Text":"Hence as I said, it\u0027s just a shorthand way of writing this is orthogonal to this."},{"Start":"07:50.660 ","End":"07:56.105","Text":"Now, same thing with plane Pi2, the second plane."},{"Start":"07:56.105 ","End":"07:59.965","Text":"L lies in that one also."},{"Start":"07:59.965 ","End":"08:01.510","Text":"I mean it lies in the intersection,"},{"Start":"08:01.510 ","End":"08:02.980","Text":"it lies in each of them."},{"Start":"08:02.980 ","End":"08:08.575","Text":"It\u0027s also going to be orthogonal to the normal to this plane."},{"Start":"08:08.575 ","End":"08:16.845","Text":"L has got to be perpendicular also to minus 5,"},{"Start":"08:16.845 ","End":"08:22.085","Text":"6, 3, the coefficients here."},{"Start":"08:22.085 ","End":"08:32.090","Text":"Now how do I find something perpendicular to 2 given vectors?"},{"Start":"08:32.400 ","End":"08:37.120","Text":"One way and the most obvious way is to use the cross-product."},{"Start":"08:37.120 ","End":"08:45.950","Text":"Remember that the cross product of 2 vectors is perpendicular or orthogonal to them both."},{"Start":"08:45.950 ","End":"08:47.810","Text":"For a direction vector,"},{"Start":"08:47.810 ","End":"08:52.410","Text":"all I would have to do is take this cross-product."},{"Start":"08:52.410 ","End":"08:58.280","Text":"Let\u0027s call v the direction vector"},{"Start":"08:58.280 ","End":"09:04.225","Text":"of the line L. As v I could take minus 1,"},{"Start":"09:04.225 ","End":"09:10.640","Text":"7 minus 2 vector cross"},{"Start":"09:10.640 ","End":"09:18.825","Text":"product minus 5, 6, 3."},{"Start":"09:18.825 ","End":"09:23.140","Text":"Let\u0027s do this computation using the method with determinants."},{"Start":"09:23.140 ","End":"09:24.565","Text":"I\u0027ll do it over here."},{"Start":"09:24.565 ","End":"09:28.885","Text":"We write i, j,"},{"Start":"09:28.885 ","End":"09:32.875","Text":"k, then we write minus 1,"},{"Start":"09:32.875 ","End":"09:37.284","Text":"7, minus 2, is the first vector,"},{"Start":"09:37.284 ","End":"09:38.710","Text":"and then the second one,"},{"Start":"09:38.710 ","End":"09:42.955","Text":"minus 5, 6, 3."},{"Start":"09:42.955 ","End":"09:49.310","Text":"Then I\u0027m going to use the method of the co-factors."},{"Start":"09:50.340 ","End":"09:53.125","Text":"What we will get,"},{"Start":"09:53.125 ","End":"09:59.410","Text":"now for the first place that\u0027s the place of i,"},{"Start":"09:59.410 ","End":"10:05.110","Text":"we mentally delete the row"},{"Start":"10:05.110 ","End":"10:10.420","Text":"and column with the i and we just look at this determinant,"},{"Start":"10:10.420 ","End":"10:17.395","Text":"this 2 by 2 determinant and it\u0027s equal to this diagonal less this diagonal."},{"Start":"10:17.395 ","End":"10:24.790","Text":"21 minus minus 12 is"},{"Start":"10:24.790 ","End":"10:30.290","Text":"21 plus 12 is 33."},{"Start":"10:33.000 ","End":"10:37.660","Text":"Then the j is slightly different,"},{"Start":"10:37.660 ","End":"10:41.020","Text":"it gets a minus sign."},{"Start":"10:41.020 ","End":"10:45.520","Text":"But we also, mentally cross off"},{"Start":"10:45.520 ","End":"10:51.880","Text":"the row and column and we\u0027re left with a determinant of 4 numbers minus 1,"},{"Start":"10:51.880 ","End":"10:54.595","Text":"minus 2, minus 5, 3."},{"Start":"10:54.595 ","End":"10:57.190","Text":"Again, we do the diagonals."},{"Start":"10:57.190 ","End":"11:01.120","Text":"Minus 1 times 3 is minus 3,"},{"Start":"11:01.120 ","End":"11:07.940","Text":"minus 3 takeaway 10 is minus 13."},{"Start":"11:07.980 ","End":"11:15.145","Text":"But remember I said this one\u0027s a minus so it\u0027s actually plus 13."},{"Start":"11:15.145 ","End":"11:22.670","Text":"It\u0027s like a plus minus plus when we do this method with the co-factors."},{"Start":"11:22.830 ","End":"11:27.070","Text":"We\u0027re left with the last one which is going to be a plus."},{"Start":"11:27.070 ","End":"11:32.545","Text":"We get the determinant of this square here."},{"Start":"11:32.545 ","End":"11:36.880","Text":"Minus 1 times 6 is minus 6."},{"Start":"11:36.880 ","End":"11:42.910","Text":"Minus 6 less minus 35 is"},{"Start":"11:42.910 ","End":"11:49.885","Text":"minus 6 plus 35 is 29."},{"Start":"11:49.885 ","End":"11:55.240","Text":"This vector is perpendicular to"},{"Start":"11:55.240 ","End":"12:01.580","Text":"the 2 normals of the plains and therefore parallel to the line of intersection."},{"Start":"12:01.980 ","End":"12:11.440","Text":"This here is a direction vector"},{"Start":"12:11.440 ","End":"12:16.645","Text":"of L. Now we have the 2 bits of information we need."},{"Start":"12:16.645 ","End":"12:23.469","Text":"We have a point on the line and we have a direction vector for the line."},{"Start":"12:23.469 ","End":"12:31.150","Text":"From these, it\u0027s easy to get the vector equation of the line, let\u0027s see."},{"Start":"12:31.150 ","End":"12:41.440","Text":"We get that r of t vector equals r-naught,"},{"Start":"12:41.440 ","End":"12:45.310","Text":"r-naught is a particular point that would be"},{"Start":"12:45.310 ","End":"12:50.935","Text":"just the position vector of this point plus tv."},{"Start":"12:50.935 ","End":"13:01.280","Text":"In other words, we have 165 over 29,"},{"Start":"13:01.920 ","End":"13:13.645","Text":"123 over 29, 0 plus t"},{"Start":"13:13.645 ","End":"13:30.760","Text":"times vector 33, 13, 29."},{"Start":"13:30.760 ","End":"13:35.290","Text":"I could stop here or maybe just combine"},{"Start":"13:35.290 ","End":"13:40.540","Text":"into 1 factor these 2 but there is some simplification."},{"Start":"13:40.540 ","End":"13:42.520","Text":"I\u0027m just going to show you how I do this,"},{"Start":"13:42.520 ","End":"13:45.379","Text":"I wouldn\u0027t expect you to know."},{"Start":"13:45.420 ","End":"13:51.090","Text":"I noticed that 33 times 5 is"},{"Start":"13:51.090 ","End":"13:58.710","Text":"165 and I\u0027m just wondering what would happen if I multiplied this vector here,"},{"Start":"13:58.710 ","End":"14:03.555","Text":"this v by 5 over 29."},{"Start":"14:03.555 ","End":"14:14.115","Text":"Notice that 5 over 29v is 5 over 29 times this,"},{"Start":"14:14.115 ","End":"14:21.925","Text":"which would be 165 over 29."},{"Start":"14:21.925 ","End":"14:32.020","Text":"Then 5 over 29 times this would be 65 over 29,"},{"Start":"14:32.020 ","End":"14:37.405","Text":"and 5 over 29 times this would be 5."},{"Start":"14:37.405 ","End":"14:43.000","Text":"Now what I\u0027m going to do is decompose this into 2 bits so that"},{"Start":"14:43.000 ","End":"14:49.285","Text":"this is now equal to,"},{"Start":"14:49.285 ","End":"14:52.495","Text":"just continue over here."},{"Start":"14:52.495 ","End":"15:02.920","Text":"This bit I can say is 5 over 29 times v,"},{"Start":"15:02.920 ","End":"15:08.935","Text":"which is 33, 13, 29."},{"Start":"15:08.935 ","End":"15:13.614","Text":"Plus, now the difference from here to here,"},{"Start":"15:13.614 ","End":"15:21.265","Text":"I have to add 58 over 29."},{"Start":"15:21.265 ","End":"15:23.545","Text":"Let me just highlight the vector."},{"Start":"15:23.545 ","End":"15:28.435","Text":"What I\u0027m doing is, subtracting"},{"Start":"15:28.435 ","End":"15:36.670","Text":"this minus this so let\u0027s see what I have to add to make them equal."},{"Start":"15:36.670 ","End":"15:41.275","Text":"The first bit is just 0 and then"},{"Start":"15:41.275 ","End":"15:48.370","Text":"this minus this is 58 over 29,"},{"Start":"15:48.370 ","End":"15:57.775","Text":"which is 2 and this minus this is minus 5."},{"Start":"15:57.775 ","End":"16:08.570","Text":"Then plus t times 33, 13, 29."},{"Start":"16:08.670 ","End":"16:18.069","Text":"Now this, if I replace instead of t plus 5 over 29,"},{"Start":"16:18.069 ","End":"16:20.785","Text":"I could call that s, so I\u0027m writing this here."},{"Start":"16:20.785 ","End":"16:29.420","Text":"The side substitute s equals t plus 5 over 29,"},{"Start":"16:29.820 ","End":"16:37.810","Text":"and just as t runs overall number so does s. What I get is from this and this,"},{"Start":"16:37.810 ","End":"16:41.455","Text":"I get, I\u0027ll copy this bit first."},{"Start":"16:41.455 ","End":"16:49.210","Text":"It\u0027s 0, 2 minus 5 plus s"},{"Start":"16:49.210 ","End":"16:58.285","Text":"times 33, 13, 29."},{"Start":"16:58.285 ","End":"17:01.615","Text":"If I want to combine it all into 1,"},{"Start":"17:01.615 ","End":"17:05.980","Text":"I could say the equation of the line"},{"Start":"17:05.980 ","End":"17:14.830","Text":"L is the r of t. Let\u0027s make it r of s. Of course,"},{"Start":"17:14.830 ","End":"17:18.294","Text":"the particular dummy parameter doesn\u0027t matter."},{"Start":"17:18.294 ","End":"17:22.300","Text":"Could be s, could change it back to t at the end if I wanted to."},{"Start":"17:22.300 ","End":"17:26.410","Text":"R of s is equal to,"},{"Start":"17:26.410 ","End":"17:35.080","Text":"the first component is 33s plus 0."},{"Start":"17:35.080 ","End":"17:38.080","Text":"The s looks a bit like a 5,"},{"Start":"17:38.080 ","End":"17:47.000","Text":"let\u0027s see if we can make it like an s. Then 2 plus 13s,"},{"Start":"17:48.600 ","End":"17:57.050","Text":"and then minus 5 plus 29s."},{"Start":"17:57.600 ","End":"18:04.510","Text":"This would be a perfectly good vector,"},{"Start":"18:04.510 ","End":"18:08.785","Text":"they\u0027re very similar, the parametric and the vector form of the line."},{"Start":"18:08.785 ","End":"18:13.675","Text":"I could leave this as the answer and you know what?"},{"Start":"18:13.675 ","End":"18:15.910","Text":"We can just replace s with t at the end,"},{"Start":"18:15.910 ","End":"18:23.995","Text":"so let\u0027s write the final answer as r of t is equal to 33t,"},{"Start":"18:23.995 ","End":"18:33.710","Text":"2 plus 13t, minus 5 plus 29t,"},{"Start":"18:34.740 ","End":"18:40.870","Text":"and highlight that but like I said,"},{"Start":"18:40.870 ","End":"18:45.325","Text":"if you didn\u0027t know all these fancy simplification tricks,"},{"Start":"18:45.325 ","End":"18:49.645","Text":"we could just as well have left the answer."},{"Start":"18:49.645 ","End":"18:55.330","Text":"Let\u0027s see it could have said r of t is equal"},{"Start":"18:55.330 ","End":"19:05.400","Text":"to this here and I have these ugly fractions in it."},{"Start":"19:06.770 ","End":"19:13.080","Text":"But yeah, anyway, so you could rather,"},{"Start":"19:13.080 ","End":"19:15.900","Text":"say stop here or continue and simplify,"},{"Start":"19:15.900 ","End":"19:20.700","Text":"and I think we\u0027ve had enough with this exercise."}],"ID":9770},{"Watched":false,"Name":"Exercises 19","Duration":"4m 44s","ChapterTopicVideoID":9810,"CourseChapterTopicPlaylistID":57476,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9810.jpeg","UploadDate":"2017-08-01T05:02:26.1200000","DurationForVideoObject":"PT4M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we have a plane in the line."},{"Start":"00:03.120 ","End":"00:04.905","Text":"The plane is Pi,"},{"Start":"00:04.905 ","End":"00:08.865","Text":"that\u0027s its name, and it\u0027s given by this equation."},{"Start":"00:08.865 ","End":"00:11.540","Text":"The line is given by this,"},{"Start":"00:11.540 ","End":"00:15.060","Text":"it\u0027s the vector equation of the line."},{"Start":"00:15.060 ","End":"00:18.720","Text":"What we want to know about the line and the plane is"},{"Start":"00:18.720 ","End":"00:23.139","Text":"whether they\u0027re parallel or perpendicular or neither."},{"Start":"00:23.330 ","End":"00:26.385","Text":"Now for the plane Pi,"},{"Start":"00:26.385 ","End":"00:29.070","Text":"I know what the normal vector is."},{"Start":"00:29.070 ","End":"00:31.920","Text":"I can take the coefficients of x, y,"},{"Start":"00:31.920 ","End":"00:35.505","Text":"z as a vector 5,"},{"Start":"00:35.505 ","End":"00:42.570","Text":"minus 3, minus 6 is"},{"Start":"00:42.570 ","End":"00:50.670","Text":"normal orthogonal or perpendicular to the plane Pi."},{"Start":"00:50.720 ","End":"00:54.395","Text":"If I look at the line equation,"},{"Start":"00:54.395 ","End":"00:57.830","Text":"I can get a direction vector for the line."},{"Start":"00:57.830 ","End":"00:59.914","Text":"So if I take v,"},{"Start":"00:59.914 ","End":"01:03.445","Text":"just the coefficients also of the t,"},{"Start":"01:03.445 ","End":"01:09.380","Text":"I can say that minus 10, 6,"},{"Start":"01:09.380 ","End":"01:15.660","Text":"12 is not a normal,"},{"Start":"01:15.660 ","End":"01:20.830","Text":"but is the direction vector"},{"Start":"01:23.480 ","End":"01:28.940","Text":"to the line,"},{"Start":"01:28.940 ","End":"01:33.150","Text":"so it\u0027s parallel to the line."},{"Start":"01:34.970 ","End":"01:39.380","Text":"I have a vector that\u0027s parallel to the line and I"},{"Start":"01:39.380 ","End":"01:44.010","Text":"have a vector that\u0027s orthogonal to the plane."},{"Start":"01:44.770 ","End":"01:50.045","Text":"Now, because n is normal to the plane,"},{"Start":"01:50.045 ","End":"01:58.204","Text":"the condition on being parallel to the plane is to be perpendicular to the normal."},{"Start":"01:58.204 ","End":"02:02.600","Text":"In other words, if n. something is 0,"},{"Start":"02:02.600 ","End":"02:06.035","Text":"then that something will be parallel to the plane."},{"Start":"02:06.035 ","End":"02:08.940","Text":"Let\u0027s check for this v,"},{"Start":"02:10.120 ","End":"02:14.010","Text":"n.v is, let\u0027s see,"},{"Start":"02:14.010 ","End":"02:17.940","Text":"5 times 10 is minus 50,"},{"Start":"02:17.940 ","End":"02:23.900","Text":"minus 18, minus 72."},{"Start":"02:23.900 ","End":"02:26.975","Text":"Whatever it is, it\u0027s not 0,"},{"Start":"02:26.975 ","End":"02:33.710","Text":"which means that v is not parallel to the plane,"},{"Start":"02:33.710 ","End":"02:37.115","Text":"which means that l is not parallel to the plane."},{"Start":"02:37.115 ","End":"02:43.735","Text":"So v, and then hence l,"},{"Start":"02:43.735 ","End":"02:53.715","Text":"is not parallel to the plane which we called Pi."},{"Start":"02:53.715 ","End":"03:00.510","Text":"This is a no. Let\u0027s try perpendicular."},{"Start":"03:00.510 ","End":"03:04.725","Text":"To be perpendicular to the plane,"},{"Start":"03:04.725 ","End":"03:11.630","Text":"we just have to have v parallel to n. If the line is parallel to the normal,"},{"Start":"03:11.630 ","End":"03:15.005","Text":"then it\u0027s going to be perpendicular to the plane."},{"Start":"03:15.005 ","End":"03:22.865","Text":"What I want to ask is v parallel"},{"Start":"03:22.865 ","End":"03:31.265","Text":"to n. Parallel means that 1 is going to be some scalar times the other."},{"Start":"03:31.265 ","End":"03:35.675","Text":"Now if we look at these or even just looking at the first coefficient,"},{"Start":"03:35.675 ","End":"03:40.965","Text":"you see that minus 2 times 5 is minus 10."},{"Start":"03:40.965 ","End":"03:44.010","Text":"If I continue with this minus 2,"},{"Start":"03:44.010 ","End":"03:48.120","Text":"it will bring me from minus 3-6 and from minus 6-12."},{"Start":"03:48.120 ","End":"03:56.469","Text":"So the answer is yes because v is minus 2"},{"Start":"03:57.260 ","End":"04:05.340","Text":"times n, 1 is the scalar times the other"},{"Start":"04:05.340 ","End":"04:13.685","Text":"and so l is perpendicular or orthogonal normal."},{"Start":"04:13.685 ","End":"04:15.740","Text":"You can also write it like this,"},{"Start":"04:15.740 ","End":"04:18.530","Text":"l is perpendicular to Pi."},{"Start":"04:18.530 ","End":"04:20.195","Text":"Or maybe even write that."},{"Start":"04:20.195 ","End":"04:21.875","Text":"I\u0027ll use orthogonal."},{"Start":"04:21.875 ","End":"04:25.660","Text":"I\u0027ve been using that most, orthogonal."},{"Start":"04:26.060 ","End":"04:34.480","Text":"So l and Pi are orthogonal."},{"Start":"04:34.760 ","End":"04:38.910","Text":"I use the word perpendicular, that same thing."},{"Start":"04:38.910 ","End":"04:43.570","Text":"Perpendicular. So that\u0027s the answer."}],"ID":9771}],"Thumbnail":null,"ID":57476},{"Name":"Quadratic Surfaces","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Quadric Surfaces","Duration":"14m 25s","ChapterTopicVideoID":9877,"CourseChapterTopicPlaylistID":57477,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9877.jpeg","UploadDate":"2020-01-16T07:12:55.1730000","DurationForVideoObject":"PT14M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.690","Text":"Done with the cone now the cylinder."},{"Start":"00:03.690 ","End":"00:05.720","Text":"I guess I should have mentioned it earlier,"},{"Start":"00:05.720 ","End":"00:09.105","Text":"there are actually several kinds of cylinders."},{"Start":"00:09.105 ","End":"00:12.810","Text":"The 2 main kinds, well,"},{"Start":"00:12.810 ","End":"00:15.460","Text":"just like with the paraboloid,"},{"Start":"00:17.510 ","End":"00:19.844","Text":"just a quick copy-paste,"},{"Start":"00:19.844 ","End":"00:22.410","Text":"there\u0027s elliptic and hyperbolic."},{"Start":"00:22.410 ","End":"00:27.325","Text":"Actually, you can get more specialized in the elliptic."},{"Start":"00:27.325 ","End":"00:31.220","Text":"As you already know, a special case of an ellipse is a circle,"},{"Start":"00:31.220 ","End":"00:34.790","Text":"so we also have a circular cylinder,"},{"Start":"00:34.790 ","End":"00:36.705","Text":"which is the usual cylinder,"},{"Start":"00:36.705 ","End":"00:39.050","Text":"and later on when we get to the paraboloid."},{"Start":"00:39.050 ","End":"00:40.520","Text":"But in case I forget,"},{"Start":"00:40.520 ","End":"00:44.180","Text":"I\u0027ll say elliptical and circular."},{"Start":"00:44.180 ","End":"00:52.455","Text":"In fact, we\u0027ve been seeing this all along even with the ellipsoid we\u0027ve had the sphere."},{"Start":"00:52.455 ","End":"00:57.180","Text":"Certainly with the cone, we\u0027ve had elliptical and circular."},{"Start":"00:58.000 ","End":"01:02.484","Text":"Let\u0027s first of all start with a picture."},{"Start":"01:02.484 ","End":"01:05.365","Text":"I brought all 3 pictures at once."},{"Start":"01:05.365 ","End":"01:12.375","Text":"This is the elliptic cylinder"},{"Start":"01:12.375 ","End":"01:15.540","Text":"and this is the circular,"},{"Start":"01:15.540 ","End":"01:18.159","Text":"which is a special case,"},{"Start":"01:18.590 ","End":"01:22.030","Text":"just like a circle is a special case of an ellipse,"},{"Start":"01:22.030 ","End":"01:29.305","Text":"and this one is the hyperbolic cylinder less commonly heard about."},{"Start":"01:29.305 ","End":"01:39.325","Text":"Actually, they have unusual equations in 3D because if we take the 2D equations,"},{"Start":"01:39.325 ","End":"01:42.790","Text":"in fact, I just have the idea of making a table."},{"Start":"01:42.790 ","End":"01:47.280","Text":"Let me write here ellipse, circle,"},{"Start":"01:47.280 ","End":"01:54.550","Text":"and hyperbola and then I write the equation for each of these."},{"Start":"01:54.550 ","End":"01:57.910","Text":"The ellipse we already did this before,"},{"Start":"01:57.910 ","End":"02:00.340","Text":"would be, let\u0027s see,"},{"Start":"02:00.340 ","End":"02:08.829","Text":"it was x squared over a squared plus y squared over b squared equals 1."},{"Start":"02:08.829 ","End":"02:14.290","Text":"Then we said that if a equals b and we call that r, and then simplify,"},{"Start":"02:14.290 ","End":"02:20.455","Text":"we get x squared plus y squared equals r-squared."},{"Start":"02:20.455 ","End":"02:26.854","Text":"The hyperbola, you may or may not have studied it in 2D,"},{"Start":"02:26.854 ","End":"02:31.760","Text":"the hyperbola has the equation x squared"},{"Start":"02:31.760 ","End":"02:37.880","Text":"over a squared minus y squared over b squared equals 1."},{"Start":"02:37.880 ","End":"02:39.965","Text":"In case you haven\u0027t encountered it,"},{"Start":"02:39.965 ","End":"02:43.970","Text":"maybe I\u0027ll just show you what this looks like."},{"Start":"02:43.970 ","End":"02:46.220","Text":"I\u0027ll delete this again in a moment,"},{"Start":"02:46.220 ","End":"02:48.305","Text":"so don\u0027t worry if it\u0027s covering everything."},{"Start":"02:48.305 ","End":"02:51.440","Text":"But what I\u0027m talking about is this x"},{"Start":"02:51.440 ","End":"02:54.770","Text":"squared over a squared minus y squared over b squared equals 1."},{"Start":"02:54.770 ","End":"02:57.740","Text":"You can see that here and that\u0027s this hyperbola,"},{"Start":"02:57.740 ","End":"03:02.955","Text":"that\u0027s this curve here that has 2 branches."},{"Start":"03:02.955 ","End":"03:06.280","Text":"I think they\u0027re called sheets here."},{"Start":"03:06.410 ","End":"03:11.825","Text":"Anyway, the reason there\u0027s another one is that if I reverse"},{"Start":"03:11.825 ","End":"03:17.840","Text":"the subtraction sign and it was the y squared minus the x squared,"},{"Start":"03:17.840 ","End":"03:19.940","Text":"then it would be the other way."},{"Start":"03:19.940 ","End":"03:24.680","Text":"But we\u0027ve already talked about in general all these things."},{"Start":"03:24.680 ","End":"03:29.430","Text":"We bring them in 1 particular orientation like this and"},{"Start":"03:29.430 ","End":"03:35.900","Text":"we\u0027ll just interchange the variables to get them in the other orientations."},{"Start":"03:35.900 ","End":"03:39.035","Text":"I just wanted to give you an idea that this"},{"Start":"03:39.035 ","End":"03:45.040","Text":"is a hyperbola and now I\u0027m coming to the point,"},{"Start":"03:45.040 ","End":"03:52.255","Text":"these 2D shapes, and the thing is that these exact same equations,"},{"Start":"03:52.255 ","End":"03:56.575","Text":"but in 3D have a different interpretation."},{"Start":"03:56.575 ","End":"03:59.170","Text":"How can that be? Well, there\u0027s a missing z."},{"Start":"03:59.170 ","End":"04:01.570","Text":"In 3D, this is an equation in x, y,"},{"Start":"04:01.570 ","End":"04:04.660","Text":"and z, which just happens to be without z,"},{"Start":"04:04.660 ","End":"04:09.405","Text":"and then these 3 things become the elliptic,"},{"Start":"04:09.405 ","End":"04:13.965","Text":"circular, and hyperbolic cylinder."},{"Start":"04:13.965 ","End":"04:17.655","Text":"There, I wrote that. The reason that there\u0027s no z,"},{"Start":"04:17.655 ","End":"04:21.790","Text":"the implication is, that if we take a curve in 2 dimensions,"},{"Start":"04:21.790 ","End":"04:23.550","Text":"I\u0027ll just do a quick freehand,"},{"Start":"04:23.550 ","End":"04:25.125","Text":"whether we have the ellipse,"},{"Start":"04:25.125 ","End":"04:27.215","Text":"or whether we have the circle,"},{"Start":"04:27.215 ","End":"04:32.885","Text":"or whether we have the hyperbola in the plane."},{"Start":"04:32.885 ","End":"04:35.390","Text":"Well, these don\u0027t look the greatest,"},{"Start":"04:35.390 ","End":"04:36.995","Text":"but you can see it here."},{"Start":"04:36.995 ","End":"04:42.230","Text":"This is an ellipse and where it cuts the x,"},{"Start":"04:42.230 ","End":"04:44.465","Text":"y plane is an ellipse."},{"Start":"04:44.465 ","End":"04:46.940","Text":"It\u0027s just that we raise vertical lines through."},{"Start":"04:46.940 ","End":"04:50.240","Text":"Z is unbounded, which means like taking a shape"},{"Start":"04:50.240 ","End":"04:56.240","Text":"in 2 dimensions and then drawing vertical lines parallel through each point,"},{"Start":"04:56.240 ","End":"04:57.530","Text":"and then we get a cylinder."},{"Start":"04:57.530 ","End":"04:59.855","Text":"In fact, that\u0027s generally what a cylinder is."},{"Start":"04:59.855 ","End":"05:01.955","Text":"We have a curve."},{"Start":"05:01.955 ","End":"05:03.350","Text":"You can even generalize it."},{"Start":"05:03.350 ","End":"05:09.980","Text":"Any curve in the plane and then we just give it a third dimension by raising the z."},{"Start":"05:10.370 ","End":"05:14.460","Text":"The same equations, in 2D they mean 1 thing,"},{"Start":"05:14.460 ","End":"05:21.290","Text":"and then 3D they mean something else and that\u0027s basically it."},{"Start":"05:21.290 ","End":"05:24.739","Text":"I just want to remind you again that the missing variable,"},{"Start":"05:24.739 ","End":"05:26.525","Text":"in this case, it\u0027s a missing z,"},{"Start":"05:26.525 ","End":"05:29.885","Text":"that would be the z is the axis,"},{"Start":"05:29.885 ","End":"05:34.540","Text":"which this shape is centered around, so to speak."},{"Start":"05:34.540 ","End":"05:37.330","Text":"Of course, as we said earlier,"},{"Start":"05:37.330 ","End":"05:41.060","Text":"we can change the variables around to have it in the direction of"},{"Start":"05:41.060 ","End":"05:45.440","Text":"y or x and even the hyperbola has 2 forms."},{"Start":"05:45.440 ","End":"05:48.745","Text":"One like this and the other like this, anyway."},{"Start":"05:48.745 ","End":"05:50.830","Text":"We\u0027re done with the cylinder."},{"Start":"05:50.830 ","End":"05:53.990","Text":"We\u0027re going to move on next to the hyperboloid."},{"Start":"05:53.990 ","End":"05:57.470","Text":"No, wait, there\u0027s one kind I forgot. I suddenly remembered."},{"Start":"05:57.470 ","End":"05:58.805","Text":"It\u0027s not very common,"},{"Start":"05:58.805 ","End":"06:02.840","Text":"but there is actually a thing called a parabolic cylinder."},{"Start":"06:02.840 ","End":"06:04.430","Text":"If I take the equation,"},{"Start":"06:04.430 ","End":"06:11.600","Text":"any equation of a parabola in the plane y equals x squared,"},{"Start":"06:11.600 ","End":"06:15.180","Text":"I prefer to take x equals y squared,"},{"Start":"06:15.580 ","End":"06:20.650","Text":"have it on the side and without any z,"},{"Start":"06:20.650 ","End":"06:23.719","Text":"and then this is a parabola in the plane,"},{"Start":"06:23.719 ","End":"06:30.230","Text":"but here it becomes a parabolic cylinder."},{"Start":"06:30.230 ","End":"06:33.400","Text":"I\u0027ll throw in the picture."},{"Start":"06:33.400 ","End":"06:35.480","Text":"This is what it looks like,"},{"Start":"06:35.480 ","End":"06:38.240","Text":"parabolic, not that common,"},{"Start":"06:38.240 ","End":"06:40.205","Text":"but just for completeness sake."},{"Start":"06:40.205 ","End":"06:43.540","Text":"Now we can go on to the hyperboloid."},{"Start":"06:43.540 ","End":"06:47.190","Text":"Here we are. The next one is hyperboloid."},{"Start":"06:47.190 ","End":"06:49.380","Text":"There are 2 kinds, 1 sheet,"},{"Start":"06:49.380 ","End":"06:53.380","Text":"and 2 sheets and actually,"},{"Start":"06:53.380 ","End":"06:56.390","Text":"it can also be elliptic or circular,"},{"Start":"06:56.390 ","End":"07:00.020","Text":"I\u0027m just assuming the elliptic and the circular will be a special case."},{"Start":"07:00.020 ","End":"07:02.675","Text":"I\u0027ll write that here, we have room."},{"Start":"07:02.675 ","End":"07:05.105","Text":"Also can be elliptical, circular."},{"Start":"07:05.105 ","End":"07:09.350","Text":"Now before I give you the pictures\u0027 diagrams for each of these,"},{"Start":"07:09.350 ","End":"07:12.920","Text":"I want to just look again in 2 dimensions."},{"Start":"07:12.920 ","End":"07:14.770","Text":"We\u0027ve had this picture before,"},{"Start":"07:14.770 ","End":"07:16.450","Text":"there\u0027s way too much information on it."},{"Start":"07:16.450 ","End":"07:18.250","Text":"I just want you to look at the general shape."},{"Start":"07:18.250 ","End":"07:20.980","Text":"But there were 2 variations on the plane,"},{"Start":"07:20.980 ","End":"07:24.590","Text":"we had this and this."},{"Start":"07:25.260 ","End":"07:28.330","Text":"Well, we discussed this."},{"Start":"07:28.330 ","End":"07:36.760","Text":"Now, I\u0027d like to imagine that this is not like the x-axis and this is the y-axis."},{"Start":"07:36.760 ","End":"07:39.520","Text":"I want you to imagine this as being"},{"Start":"07:39.520 ","End":"07:46.615","Text":"the whole xy plane and that this will be the z-axis in 3D."},{"Start":"07:46.615 ","End":"07:47.830","Text":"Looking at the side,"},{"Start":"07:47.830 ","End":"07:49.750","Text":"this whole thing is a plane."},{"Start":"07:49.750 ","End":"07:56.965","Text":"The same here, that this will be the xy plane and this will be the z-axis."},{"Start":"07:56.965 ","End":"08:03.070","Text":"Now imagine that we rotate this shape around this axis."},{"Start":"08:03.070 ","End":"08:05.214","Text":"If you rotate this shape,"},{"Start":"08:05.214 ","End":"08:11.290","Text":"we will get the 1 sheeted hyperboloid that you will see in a moment,"},{"Start":"08:11.290 ","End":"08:12.970","Text":"but it only has 1 bit."},{"Start":"08:12.970 ","End":"08:14.215","Text":"It\u0027s all connected."},{"Start":"08:14.215 ","End":"08:19.570","Text":"But if I rotate this about the z-axis,"},{"Start":"08:19.570 ","End":"08:22.645","Text":"I\u0027ll have 1 bit here and 1 bit here."},{"Start":"08:22.645 ","End":"08:25.885","Text":"Let\u0027s see how this looks in 3D."},{"Start":"08:25.885 ","End":"08:30.730","Text":"Push this to the side and here they are."},{"Start":"08:30.730 ","End":"08:34.030","Text":"The 2 hyperboloids."},{"Start":"08:34.030 ","End":"08:37.390","Text":"Obviously this is the 1 sheeted and this is the 2 sheeted."},{"Start":"08:37.390 ","End":"08:40.000","Text":"We\u0027ll assume that this is the elliptical."},{"Start":"08:40.000 ","End":"08:42.340","Text":"Not necessarily circular like I implied,"},{"Start":"08:42.340 ","End":"08:47.995","Text":"if we just did a rotation and both of them,"},{"Start":"08:47.995 ","End":"08:49.840","Text":"the center is the z-axis."},{"Start":"08:49.840 ","End":"08:59.590","Text":"The z-axis is the exceptional one and the equation is that x squared over"},{"Start":"08:59.590 ","End":"09:03.475","Text":"a squared plus y squared over b squared"},{"Start":"09:03.475 ","End":"09:10.300","Text":"minus z squared over c squared equals 1."},{"Start":"09:10.300 ","End":"09:12.700","Text":"See the odd one out is the z,"},{"Start":"09:12.700 ","End":"09:14.335","Text":"it\u0027s the one with the minus."},{"Start":"09:14.335 ","End":"09:18.505","Text":"That\u0027s the one where the hyperboloid is centered around."},{"Start":"09:18.505 ","End":"09:21.070","Text":"Now for the 2 sheeted one,"},{"Start":"09:21.070 ","End":"09:26.320","Text":"actually the equation is not hard to remember."},{"Start":"09:26.320 ","End":"09:29.650","Text":"Wherever you see on the left a plus you put a minus and vice versa."},{"Start":"09:29.650 ","End":"09:36.085","Text":"We have minus x squared over a squared minus y squared"},{"Start":"09:36.085 ","End":"09:44.405","Text":"over b squared plus z squared over c squared equals 1."},{"Start":"09:44.405 ","End":"09:46.300","Text":"Once again, the odd one out,"},{"Start":"09:46.300 ","End":"09:51.830","Text":"I mean this time it\u0027s the one with the plus is the one where it\u0027s centered around."},{"Start":"09:53.400 ","End":"09:56.215","Text":"To get it to be circular,"},{"Start":"09:56.215 ","End":"09:59.630","Text":"if we have a equals b,"},{"Start":"10:00.840 ","End":"10:08.935","Text":"then it becomes a circular hyperboloid."},{"Start":"10:08.935 ","End":"10:12.530","Text":"Otherwise it\u0027s just elliptic."},{"Start":"10:12.690 ","End":"10:16.390","Text":"I don\u0027t think I want to say anything more about hyperboloids and"},{"Start":"10:16.390 ","End":"10:20.005","Text":"I think I want to go straight on to the paraboloid."},{"Start":"10:20.005 ","End":"10:23.769","Text":"Paraboloid, let\u0027s start with the elliptic circular."},{"Start":"10:23.769 ","End":"10:31.165","Text":"I already put the picture here and let me give you the equation."},{"Start":"10:31.165 ","End":"10:38.350","Text":"The equation for this will be that x squared over"},{"Start":"10:38.350 ","End":"10:44.500","Text":"a squared plus y squared over b"},{"Start":"10:44.500 ","End":"10:52.435","Text":"squared equals z over c. Now,"},{"Start":"10:52.435 ","End":"10:58.645","Text":"z is the exceptional one and it certainly is centered around the z-axis."},{"Start":"10:58.645 ","End":"11:01.780","Text":"As usual, we can rearrange the letters if you want it differently."},{"Start":"11:01.780 ","End":"11:06.265","Text":"The thing is that it actually makes a difference if c is positive or negative."},{"Start":"11:06.265 ","End":"11:10.150","Text":"I just mentioned that if c is bigger than 0,"},{"Start":"11:10.150 ","End":"11:13.405","Text":"it\u0027s like this and if c is less than 0,"},{"Start":"11:13.405 ","End":"11:15.025","Text":"it just faces down,"},{"Start":"11:15.025 ","End":"11:20.270","Text":"just like we have a parabola that\u0027s concave up or concave down."},{"Start":"11:22.290 ","End":"11:26.770","Text":"This is just a mnemonic to indicate that depending on the sign of"},{"Start":"11:26.770 ","End":"11:31.400","Text":"c is we\u0027ll get different facing up, facing down."},{"Start":"11:31.410 ","End":"11:34.435","Text":"If a equals b,"},{"Start":"11:34.435 ","End":"11:36.850","Text":"it\u0027s going to be circular."},{"Start":"11:36.850 ","End":"11:39.640","Text":"Otherwise the cross-section is an ellipse."},{"Start":"11:39.640 ","End":"11:45.880","Text":"If a equals b is equal to r,"},{"Start":"11:45.880 ","End":"11:51.175","Text":"then it\u0027s a circular rather than"},{"Start":"11:51.175 ","End":"11:59.060","Text":"an elliptic circular paraboloid."},{"Start":"12:00.270 ","End":"12:02.890","Text":"That\u0027s it for elliptic circular."},{"Start":"12:02.890 ","End":"12:04.885","Text":"Now the hyperbolic."},{"Start":"12:04.885 ","End":"12:07.885","Text":"Here\u0027s the picture and it\u0027s a funny shape,"},{"Start":"12:07.885 ","End":"12:10.540","Text":"a bit like a saddle, I would say."},{"Start":"12:10.540 ","End":"12:18.700","Text":"Notice that if I take sections with vertical planes,"},{"Start":"12:18.700 ","End":"12:28.630","Text":"if I take sections parallel to the xz plane,"},{"Start":"12:28.630 ","End":"12:30.070","Text":"I\u0027m going to get parabolas."},{"Start":"12:30.070 ","End":"12:33.040","Text":"All these lines are parabolas that are facing up."},{"Start":"12:33.040 ","End":"12:37.660","Text":"On the other hand, if I take planes parallel to the zy plane,"},{"Start":"12:37.660 ","End":"12:39.310","Text":"I\u0027ll get down with parabolas."},{"Start":"12:39.310 ","End":"12:40.735","Text":"This is a downward one,"},{"Start":"12:40.735 ","End":"12:43.270","Text":"and this one at the end is a downward parabola."},{"Start":"12:43.270 ","End":"12:46.630","Text":"Actually if I take horizontal cross-sections,"},{"Start":"12:46.630 ","End":"12:50.290","Text":"I mean parallel to the xy plane, I\u0027ll get hyperbolas."},{"Start":"12:50.290 ","End":"12:53.995","Text":"On the top half of the space above,"},{"Start":"12:53.995 ","End":"12:55.210","Text":"when z is positive,"},{"Start":"12:55.210 ","End":"12:58.075","Text":"I\u0027ll get the sheet\u0027s going one way,"},{"Start":"12:58.075 ","End":"13:00.520","Text":"the branches and on the other way,"},{"Start":"13:00.520 ","End":"13:02.650","Text":"if I take it below the xy plane,"},{"Start":"13:02.650 ","End":"13:04.850","Text":"they\u0027ll go the other way."},{"Start":"13:04.890 ","End":"13:08.485","Text":"Well, anyway, you just have to imagine it."},{"Start":"13:08.485 ","End":"13:12.850","Text":"I\u0027ll give you the equation of this one."},{"Start":"13:12.850 ","End":"13:15.850","Text":"It\u0027s actually very similar to this one."},{"Start":"13:15.850 ","End":"13:21.190","Text":"In fact, it\u0027s x squared over a squared."},{"Start":"13:21.190 ","End":"13:22.990","Text":"The only differences is a minus here,"},{"Start":"13:22.990 ","End":"13:32.155","Text":"minus y squared over b squared equals z over c. Once again,"},{"Start":"13:32.155 ","End":"13:37.420","Text":"it makes a difference if c is positive or c is negative."},{"Start":"13:37.420 ","End":"13:43.285","Text":"In each of these cases, the picture is for c positive."},{"Start":"13:43.285 ","End":"13:45.430","Text":"In this case if c is negative,"},{"Start":"13:45.430 ","End":"13:49.795","Text":"the way it reverses instead of going a parabola up"},{"Start":"13:49.795 ","End":"13:55.330","Text":"in the xz plane and a downward parabola in the yz plane,"},{"Start":"13:55.330 ","End":"13:56.845","Text":"it will be the other way round,"},{"Start":"13:56.845 ","End":"14:05.035","Text":"but the general shape is the same and the z is the odd one out because it centers round,"},{"Start":"14:05.035 ","End":"14:09.950","Text":"close to the center of symmetry is the z-axis."},{"Start":"14:11.310 ","End":"14:13.720","Text":"I think we\u0027re done."},{"Start":"14:13.720 ","End":"14:19.420","Text":"The only intention was a preliminary acquaintance with all these quadric surfaces,"},{"Start":"14:19.420 ","End":"14:22.555","Text":"and they\u0027re actually even others which are variations on this."},{"Start":"14:22.555 ","End":"14:26.510","Text":"I\u0027m declaring this subject to be done."}],"ID":9742},{"Watched":false,"Name":"Exercise 20","Duration":"3m 41s","ChapterTopicVideoID":9804,"CourseChapterTopicPlaylistID":57477,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9804.jpeg","UploadDate":"2017-08-01T05:02:32.4170000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"We\u0027re in the exercises on quadric surfaces,"},{"Start":"00:03.630 ","End":"00:10.125","Text":"and they\u0027re not really intended for you to actually sketch them in 3D."},{"Start":"00:10.125 ","End":"00:14.054","Text":"For that, you need software and the 3D is complicated."},{"Start":"00:14.054 ","End":"00:16.800","Text":"There are exercises, but really more for me"},{"Start":"00:16.800 ","End":"00:19.710","Text":"to give you an idea of what these things look like,"},{"Start":"00:19.710 ","End":"00:22.260","Text":"that you\u0027d be familiar with them."},{"Start":"00:22.260 ","End":"00:25.955","Text":"Still I\u0027d like to give you a little bit of intuition."},{"Start":"00:25.955 ","End":"00:28.435","Text":"Take this one for example."},{"Start":"00:28.435 ","End":"00:30.810","Text":"Normally, that\u0027d be x, y, and z."},{"Start":"00:30.810 ","End":"00:32.940","Text":"But here there\u0027s only y and z."},{"Start":"00:32.940 ","End":"00:34.915","Text":"There\u0027s only 2 variables."},{"Start":"00:34.915 ","End":"00:37.670","Text":"When this happens, it\u0027s typically a cylinder."},{"Start":"00:37.670 ","End":"00:43.250","Text":"Now, let me show you a sketch I borrowed from another clip."},{"Start":"00:43.250 ","End":"00:51.050","Text":"This is a sketch of a 2D ellipse in x and y."},{"Start":"00:51.050 ","End":"00:56.750","Text":"I want to use this to give you an idea of how to draw something like this."},{"Start":"00:56.750 ","End":"00:58.070","Text":"Now for one thing,"},{"Start":"00:58.070 ","End":"00:59.990","Text":"it\u0027s the wrong variables."},{"Start":"00:59.990 ","End":"01:04.525","Text":"Let\u0027s switch x and y to y and z."},{"Start":"01:04.525 ","End":"01:07.610","Text":"Here, I change this to y and z,"},{"Start":"01:07.610 ","End":"01:09.725","Text":"and here to y and z,"},{"Start":"01:09.725 ","End":"01:12.600","Text":"and this is the equation of an ellipse,"},{"Start":"01:12.600 ","End":"01:14.374","Text":"and in our case,"},{"Start":"01:14.374 ","End":"01:19.360","Text":"we have, well, I could write this as z squared over 1 squared,"},{"Start":"01:19.360 ","End":"01:21.610","Text":"so it looks even more like this,"},{"Start":"01:21.610 ","End":"01:26.939","Text":"and a is the big radius,"},{"Start":"01:26.939 ","End":"01:29.370","Text":"and b is the small radius."},{"Start":"01:29.370 ","End":"01:35.080","Text":"The ellipse has it\u0027s like 2 radii."},{"Start":"01:36.770 ","End":"01:41.520","Text":"This a is 3, and this b is 1,"},{"Start":"01:41.520 ","End":"01:45.690","Text":"and this is what it looks like in 2D,"},{"Start":"01:45.690 ","End":"01:47.780","Text":"but remember we\u0027re now in 3D"},{"Start":"01:47.780 ","End":"01:50.240","Text":"and just x happens to be missing."},{"Start":"01:50.240 ","End":"01:54.531","Text":"All you have to do, is take this ellipse"},{"Start":"01:54.531 ","End":"01:57.860","Text":"and extend it upwards and downwards"},{"Start":"01:57.860 ","End":"02:05.515","Text":"infinitely like in vertical straight lines vertical to the plane of the drawing,"},{"Start":"02:05.515 ","End":"02:09.170","Text":"and best thing I can do is just give you"},{"Start":"02:09.170 ","End":"02:15.095","Text":"a 3-dimensional sketch that was made by computer,"},{"Start":"02:15.095 ","End":"02:17.180","Text":"and this is what it looks like."},{"Start":"02:17.180 ","End":"02:20.270","Text":"As you can see, if we just look at the z, y plane,"},{"Start":"02:20.270 ","End":"02:24.425","Text":"these 2 points, I marked these 2 points here."},{"Start":"02:24.425 ","End":"02:28.715","Text":"This is where y is 3,"},{"Start":"02:28.715 ","End":"02:38.415","Text":"this point here would be the point x is 0, y is 3, z is 0,"},{"Start":"02:38.415 ","End":"02:41.230","Text":"and this point would be the point."},{"Start":"02:41.230 ","End":"02:45.785","Text":"Well, x is again 0 because there were in the zy plane."},{"Start":"02:45.785 ","End":"02:55.140","Text":"This time y is 0 and z is 1, and then he place,"},{"Start":"02:55.140 ","End":"03:01.235","Text":"we can make a cross-section parallel to the zy plane and get this ellipse."},{"Start":"03:01.235 ","End":"03:07.380","Text":"The ellipse extends infinitely in the x-direction."},{"Start":"03:07.380 ","End":"03:11.395","Text":"This like, the x-axis is perpendicular to this."},{"Start":"03:11.395 ","End":"03:15.050","Text":"Anyway, It\u0027s just to give you an idea"},{"Start":"03:15.050 ","End":"03:19.940","Text":"and the most important thing here was that,"},{"Start":"03:19.940 ","End":"03:22.970","Text":"because we had a variable missing x,"},{"Start":"03:22.970 ","End":"03:27.455","Text":"we could sketch it in the plane of the other 2 variables, y, z,"},{"Start":"03:27.455 ","End":"03:31.670","Text":"and then extended infinitely in"},{"Start":"03:31.670 ","End":"03:37.280","Text":"the direction perpendicular to the plane of the sketch and like so."},{"Start":"03:37.280 ","End":"03:40.260","Text":"That\u0027s all for this one."}],"ID":9743},{"Watched":false,"Name":"Exercise 21","Duration":"3m 2s","ChapterTopicVideoID":9805,"CourseChapterTopicPlaylistID":57477,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9805.jpeg","UploadDate":"2017-08-01T05:02:41.2870000","DurationForVideoObject":"PT3M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.870","Text":"This exercise is not really an exercise,"},{"Start":"00:03.870 ","End":"00:08.730","Text":"I don\u0027t expect you to graph surfaces 3D."},{"Start":"00:08.730 ","End":"00:14.830","Text":"Just getting you more familiar with these quadratic surfaces."},{"Start":"00:14.930 ","End":"00:21.690","Text":"A look at this formula and it reminds me of something more general,"},{"Start":"00:21.690 ","End":"00:25.240","Text":"and this is what I\u0027m talking about."},{"Start":"00:25.430 ","End":"00:28.325","Text":"It definitely fits this,"},{"Start":"00:28.325 ","End":"00:34.535","Text":"where a is equal to square root of 4 is 2,"},{"Start":"00:34.535 ","End":"00:41.970","Text":"b is 3, and c is the square root of"},{"Start":"00:41.970 ","End":"00:51.640","Text":"6 and this equation is the equation of an ellipsoid."},{"Start":"00:51.640 ","End":"00:55.710","Text":"In general, if a,"},{"Start":"00:55.710 ","End":"00:58.100","Text":"b, and c were all the same,"},{"Start":"00:58.100 ","End":"01:04.580","Text":"then it would be a sphere even though a sphere is also a special kind of ellipsoid."},{"Start":"01:04.580 ","End":"01:08.330","Text":"But in this case, these numbers are not all the same,"},{"Start":"01:08.330 ","End":"01:11.495","Text":"so it\u0027s a real ellipsoid."},{"Start":"01:11.495 ","End":"01:16.435","Text":"Then I\u0027ll just show you what it looks like."},{"Start":"01:16.435 ","End":"01:25.415","Text":"These numbers 2, 3 square root of 6 can be seen in the sketch."},{"Start":"01:25.415 ","End":"01:34.265","Text":"The 2 would be where the ellipsoid cuts the x-axis."},{"Start":"01:34.265 ","End":"01:37.285","Text":"So this point here,"},{"Start":"01:37.285 ","End":"01:39.735","Text":"use a better color,"},{"Start":"01:39.735 ","End":"01:48.630","Text":"this would be 2,0,0 that\u0027s where the x, that\u0027s the a."},{"Start":"01:48.630 ","End":"01:58.020","Text":"This point here would be 0,3,0,"},{"Start":"01:58.020 ","End":"02:04.820","Text":"and this point here would be 0,0 root 6."},{"Start":"02:04.820 ","End":"02:06.860","Text":"That\u0027s the meaning of these a, b,"},{"Start":"02:06.860 ","End":"02:13.260","Text":"and c. It\u0027s where the ellipsoid cuts the axis,"},{"Start":"02:13.260 ","End":"02:16.710","Text":"each one of its own value,"},{"Start":"02:16.710 ","End":"02:20.170","Text":"a, b, or c respectively."},{"Start":"02:20.780 ","End":"02:27.680","Text":"Nothing much more to say except perhaps I\u0027ll note that these lines that"},{"Start":"02:27.680 ","End":"02:34.025","Text":"are drawn on the ellipsoid are actually contours or level curves."},{"Start":"02:34.025 ","End":"02:42.950","Text":"The ones going this way where the value of z is constant and the ones going this"},{"Start":"02:42.950 ","End":"02:48.275","Text":"way where y is constant and"},{"Start":"02:48.275 ","End":"02:55.930","Text":"the ones like this where the values of x are constant."},{"Start":"02:55.930 ","End":"03:02.280","Text":"Anyway, that\u0027s neither here nor there that\u0027s it."}],"ID":9744},{"Watched":false,"Name":"Exercise 22","Duration":"4m 27s","ChapterTopicVideoID":9806,"CourseChapterTopicPlaylistID":57477,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9806.jpeg","UploadDate":"2017-08-01T05:02:55.6400000","DurationForVideoObject":"PT4M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.035","Text":"In this exercise, we\u0027re asked to sketch the graph of the following quadric surface."},{"Start":"00:07.035 ","End":"00:10.755","Text":"You\u0027re not really expected to sketch 3D."},{"Start":"00:10.755 ","End":"00:17.775","Text":"It\u0027s really just a solved example to get you more familiar with quadric surfaces."},{"Start":"00:17.775 ","End":"00:19.590","Text":"Just have to follow."},{"Start":"00:19.590 ","End":"00:22.755","Text":"You wouldn\u0027t get something like this in an exam."},{"Start":"00:22.755 ","End":"00:29.520","Text":"Sketches are usually computer aided anyway."},{"Start":"00:29.520 ","End":"00:38.020","Text":"Look through the list of quadric surfaces and let\u0027s see which 1 it fits most closely."},{"Start":"00:38.090 ","End":"00:49.310","Text":"I find this 1 which really fits this except for the bit with the minus 6."},{"Start":"00:49.310 ","End":"00:52.820","Text":"Let\u0027s for a moment ignore the minus 6."},{"Start":"00:52.820 ","End":"00:57.465","Text":"Then what we have here is exactly what we have here with"},{"Start":"00:57.465 ","End":"01:02.370","Text":"a equals 2 because this 4 is a squared,"},{"Start":"01:02.370 ","End":"01:08.040","Text":"so a is 2 and b is 2 and c,"},{"Start":"01:08.040 ","End":"01:09.810","Text":"well, this is like z over 1,"},{"Start":"01:09.810 ","End":"01:13.600","Text":"so c equals 1."},{"Start":"01:14.210 ","End":"01:19.724","Text":"In general, the shape of this graph,"},{"Start":"01:19.724 ","End":"01:22.720","Text":"I\u0027ll show you what it is."},{"Start":"01:23.170 ","End":"01:27.140","Text":"Here\u0027s what it generally looks like."},{"Start":"01:27.140 ","End":"01:34.580","Text":"The z is the 1 which is the odd 1 out amongst the 3 variables."},{"Start":"01:34.580 ","End":"01:39.750","Text":"If you saw something with different order,"},{"Start":"01:39.750 ","End":"01:42.410","Text":"like maybe it would be z squared over"},{"Start":"01:42.410 ","End":"01:46.680","Text":"a squared plus x squared over b squared equals y over c,"},{"Start":"01:46.680 ","End":"01:49.205","Text":"then y would be the odd 1 out."},{"Start":"01:49.205 ","End":"01:51.680","Text":"The odd 1 out is the axis that it\u0027s"},{"Start":"01:51.680 ","End":"01:54.680","Text":"centered along and in this case z is the odd 1 out,"},{"Start":"01:54.680 ","End":"01:58.115","Text":"so z is the axis it\u0027s centered on."},{"Start":"01:58.115 ","End":"02:01.970","Text":"The sign of c, in this case it\u0027s positive,"},{"Start":"02:01.970 ","End":"02:07.205","Text":"tells us whether it\u0027s facing upwards or downwards."},{"Start":"02:07.205 ","End":"02:08.765","Text":"But in our case,"},{"Start":"02:08.765 ","End":"02:13.415","Text":"this is what it\u0027s going to look like with the z and a positive"},{"Start":"02:13.415 ","End":"02:22.715","Text":"c. This shape is called an elliptic paraboloid."},{"Start":"02:22.715 ","End":"02:34.170","Text":"I\u0027ll write that. Elliptic and paraboloid."},{"Start":"02:34.170 ","End":"02:38.975","Text":"The reason it\u0027s elliptic is because"},{"Start":"02:38.975 ","End":"02:47.460","Text":"cross-sections parallel to the x-y plane would cut this,"},{"Start":"02:47.460 ","End":"02:50.505","Text":"you can see in ellipses."},{"Start":"02:50.505 ","End":"02:54.240","Text":"In our particular case,"},{"Start":"02:54.240 ","End":"02:58.080","Text":"we have that a equals b."},{"Start":"02:58.080 ","End":"03:00.970","Text":"If a equals b,"},{"Start":"03:00.970 ","End":"03:04.310","Text":"they won\u0027t be ellipses, they\u0027ll be circles."},{"Start":"03:04.310 ","End":"03:10.970","Text":"Well, a circle is a special case of an ellipse but then it\u0027s circular cross-sections."},{"Start":"03:10.970 ","End":"03:14.150","Text":"But I don\u0027t know if there\u0027s a word circular paraboloid."},{"Start":"03:14.150 ","End":"03:18.480","Text":"I suppose it could be and that\u0027s what we have in our case."},{"Start":"03:18.480 ","End":"03:19.970","Text":"The cross sections are circles."},{"Start":"03:19.970 ","End":"03:22.820","Text":"There\u0027s still the matter of the minus 6."},{"Start":"03:22.820 ","End":"03:25.490","Text":"Whenever I have z as a function of x, y,"},{"Start":"03:25.490 ","End":"03:29.120","Text":"if I had a minus 6,"},{"Start":"03:29.120 ","End":"03:34.910","Text":"it just means that I lowered the whole thing 6 units"},{"Start":"03:34.910 ","End":"03:42.525","Text":"downwards and so what I would get would be this."},{"Start":"03:42.525 ","End":"03:44.930","Text":"The differences are, first of all,"},{"Start":"03:44.930 ","End":"03:49.895","Text":"that the cross-sections are circular because a equals b,"},{"Start":"03:49.895 ","End":"03:53.045","Text":"although a circle is a special case of an ellipse."},{"Start":"03:53.045 ","End":"03:55.310","Text":"The other thing is that the tip,"},{"Start":"03:55.310 ","End":"04:02.120","Text":"normally the apex is at the origin but here the apex"},{"Start":"04:02.120 ","End":"04:09.485","Text":"has moved 6 units down so it\u0027s at the point where x and y are still 0,"},{"Start":"04:09.485 ","End":"04:13.560","Text":"but z is minus 6."},{"Start":"04:13.900 ","End":"04:18.425","Text":"Yeah, I don\u0027t want to get into any more detail than that."},{"Start":"04:18.425 ","End":"04:22.370","Text":"Here we have a elliptical circular"},{"Start":"04:22.370 ","End":"04:28.650","Text":"paraboloid that\u0027s been shifted down 6 units. That\u0027s it."}],"ID":9745},{"Watched":false,"Name":"Exercise 23","Duration":"2m 42s","ChapterTopicVideoID":9802,"CourseChapterTopicPlaylistID":57477,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9802.jpeg","UploadDate":"2017-08-01T05:02:08.0500000","DurationForVideoObject":"PT2M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:06.945","Text":"You\u0027re not really expected to sketch graphs of 3D surfaces."},{"Start":"00:06.945 ","End":"00:13.125","Text":"This is more of an exercise for me to get you familiarized with quadric surfaces."},{"Start":"00:13.125 ","End":"00:19.140","Text":"Now, I look among the list of surfaces that we\u0027ve studied and the"},{"Start":"00:19.140 ","End":"00:24.855","Text":"closest 1 to this is the following and that\u0027s this 1."},{"Start":"00:24.855 ","End":"00:29.325","Text":"But the main difference between this and what we have is that here,"},{"Start":"00:29.325 ","End":"00:31.680","Text":"it\u0027s the y variable,"},{"Start":"00:31.680 ","End":"00:34.440","Text":"that\u0027s the odd 1 out and that\u0027s on the other side."},{"Start":"00:34.440 ","End":"00:37.095","Text":"Whereas here it\u0027s the z variable."},{"Start":"00:37.095 ","End":"00:45.620","Text":"This would be actually a cone that is centered along the z-axis."},{"Start":"00:45.620 ","End":"00:48.950","Text":"We\u0027ll have to rotate the variables because we"},{"Start":"00:48.950 ","End":"00:54.335","Text":"have y here and it will has to be centered on the y-axis."},{"Start":"00:54.335 ","End":"01:01.460","Text":"This 1 looks in general something like this and as I said,"},{"Start":"01:01.460 ","End":"01:05.960","Text":"it\u0027s centered along the z-axis because that\u0027s the odd 1 out variable."},{"Start":"01:05.960 ","End":"01:10.790","Text":"The other thing is that if a and b happen to be equal,"},{"Start":"01:10.790 ","End":"01:15.280","Text":"then these cross sections are circles, otherwise they\u0027re ellipses."},{"Start":"01:15.280 ","End":"01:21.485","Text":"You could say that this in general is an elliptic cone or a circular cone."},{"Start":"01:21.485 ","End":"01:29.510","Text":"In our case, what we have is x squared over a squared."},{"Start":"01:29.510 ","End":"01:32.195","Text":"In our case, a would be 1/2."},{"Start":"01:32.195 ","End":"01:34.955","Text":"We could write this x squared over 1/2 squared,"},{"Start":"01:34.955 ","End":"01:37.190","Text":"putting the 4 in the denominator,"},{"Start":"01:37.190 ","End":"01:42.610","Text":"plus z squared over 1/4"},{"Start":"01:42.610 ","End":"01:48.620","Text":"squared equals y squared over 1 squared."},{"Start":"01:48.620 ","End":"01:50.270","Text":"We have our a, b, and c,"},{"Start":"01:50.270 ","End":"01:53.005","Text":"except that y is the odd 1 out."},{"Start":"01:53.005 ","End":"01:55.950","Text":"These 2 are different, like the a and the b,"},{"Start":"01:55.950 ","End":"01:58.849","Text":"so the cross sections are going to be elliptic."},{"Start":"01:58.849 ","End":"02:03.920","Text":"As I mentioned, this is centered on the z-axis."},{"Start":"02:03.920 ","End":"02:06.275","Text":"So this 1 is centered along the y-axis."},{"Start":"02:06.275 ","End":"02:09.350","Text":"and I\u0027ll want to show you what this looks like."},{"Start":"02:09.350 ","End":"02:13.400","Text":"Here it is with the y-axis being here,"},{"Start":"02:13.400 ","End":"02:16.130","Text":"that\u0027s what it\u0027s centered on and these are not circles,"},{"Start":"02:16.130 ","End":"02:18.980","Text":"they are ellipses because 1/2 is not equal to 1/4,"},{"Start":"02:18.980 ","End":"02:24.830","Text":"a is not equal to b. I\u0027m expected to go any deeper into this just to have"},{"Start":"02:24.830 ","End":"02:32.850","Text":"a general idea of changing the variables round and just recognizing the basic form."},{"Start":"02:33.410 ","End":"02:35.975","Text":"Once again it\u0027s a cone,"},{"Start":"02:35.975 ","End":"02:39.559","Text":"but with elliptic cross-sections, not circular."},{"Start":"02:39.559 ","End":"02:42.900","Text":"Okay. Done with this 1."}],"ID":9746},{"Watched":false,"Name":"Exercise 24","Duration":"4m 58s","ChapterTopicVideoID":9803,"CourseChapterTopicPlaylistID":57477,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9803.jpeg","UploadDate":"2017-08-01T05:02:21.6770000","DurationForVideoObject":"PT4M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.485","Text":"In this exercise, we have to sketch the following quadric surface."},{"Start":"00:07.485 ","End":"00:10.680","Text":"But as in the other exercises of this kind,"},{"Start":"00:10.680 ","End":"00:12.795","Text":"you\u0027re not really expected to sketch."},{"Start":"00:12.795 ","End":"00:16.054","Text":"These are more exercises to familiarize you"},{"Start":"00:16.054 ","End":"00:21.060","Text":"with the quadric surfaces and the variations on them."},{"Start":"00:21.060 ","End":"00:28.625","Text":"What I do is I look for the closest one on the list that we studied,"},{"Start":"00:28.625 ","End":"00:30.860","Text":"what\u0027s closest to this one,"},{"Start":"00:30.860 ","End":"00:35.290","Text":"and I\u0027ll produce the following."},{"Start":"00:35.540 ","End":"00:40.355","Text":"Here it is, although it doesn\u0027t really look like this,"},{"Start":"00:40.355 ","End":"00:44.375","Text":"but if you make a few changes then it would."},{"Start":"00:44.375 ","End":"00:47.450","Text":"For one thing, here,"},{"Start":"00:47.450 ","End":"00:50.060","Text":"z is the odd one out variable,"},{"Start":"00:50.060 ","End":"00:52.115","Text":"x and y have similar roles,"},{"Start":"00:52.115 ","End":"00:54.170","Text":"and z is the different one."},{"Start":"00:54.170 ","End":"00:57.575","Text":"In our case, x is the odd one out."},{"Start":"00:57.575 ","End":"01:00.680","Text":"The other thing is that there are signs here."},{"Start":"01:00.680 ","End":"01:02.540","Text":"Here there are minuses,"},{"Start":"01:02.540 ","End":"01:05.430","Text":"and here there are pluses."},{"Start":"01:05.570 ","End":"01:08.815","Text":"The denominator is another problem,"},{"Start":"01:08.815 ","End":"01:10.640","Text":"could easily rewrite that."},{"Start":"01:10.640 ","End":"01:15.019","Text":"In fact, why don\u0027t I just rewrite this a little bit with x on this side."},{"Start":"01:15.019 ","End":"01:21.650","Text":"I could write that x over negative 1."},{"Start":"01:21.650 ","End":"01:29.280","Text":"If I do that, then I can make these like plus is equal to,"},{"Start":"01:29.650 ","End":"01:36.785","Text":"it will be minus 4 which is not really part of this."},{"Start":"01:36.785 ","End":"01:44.280","Text":"But plus y squared over root 5."},{"Start":"01:45.020 ","End":"01:50.500","Text":"Sorry, 1 over root 5 squared."},{"Start":"01:50.500 ","End":"01:54.580","Text":"I\u0027m just forcing this to be y squared over something squared."},{"Start":"01:54.580 ","End":"01:59.965","Text":"Here, I\u0027m forcing it to be z squared over something squared."},{"Start":"01:59.965 ","End":"02:03.865","Text":"For the 9, I\u0027ll put it in the denominator as 1/9,"},{"Start":"02:03.865 ","End":"02:06.345","Text":"which is 1/3 squared."},{"Start":"02:06.345 ","End":"02:09.660","Text":"I have my c, my a, and my b."},{"Start":"02:09.660 ","End":"02:12.580","Text":"I also see that x is the odd one out"},{"Start":"02:12.580 ","End":"02:17.995","Text":"and they\u0027re still the matter of the minus 4."},{"Start":"02:17.995 ","End":"02:20.725","Text":"The shape of this,"},{"Start":"02:20.725 ","End":"02:23.890","Text":"if c is positive,"},{"Start":"02:23.890 ","End":"02:27.145","Text":"then it looks like this."},{"Start":"02:27.145 ","End":"02:31.575","Text":"It\u0027s an elliptic paraboloid."},{"Start":"02:31.575 ","End":"02:38.555","Text":"It\u0027s elliptic when a is not the same as b and circular when a is the same as b."},{"Start":"02:38.555 ","End":"02:40.945","Text":"Here our a and b are different,"},{"Start":"02:40.945 ","End":"02:43.295","Text":"so it\u0027s going to be elliptical."},{"Start":"02:43.295 ","End":"02:46.520","Text":"The other thing is that this is for c is positive."},{"Start":"02:46.520 ","End":"02:48.290","Text":"When c is negative,"},{"Start":"02:48.290 ","End":"02:50.660","Text":"it goes the other way around,"},{"Start":"02:50.660 ","End":"02:54.355","Text":"so it goes downwards."},{"Start":"02:54.355 ","End":"02:58.670","Text":"In our case, we also have to remember several changes."},{"Start":"02:58.670 ","End":"03:01.055","Text":"We have to make x the center."},{"Start":"03:01.055 ","End":"03:06.409","Text":"It\u0027s going to face in the negative x direction because of this minus."},{"Start":"03:06.409 ","End":"03:08.000","Text":"But the other thing is,"},{"Start":"03:08.000 ","End":"03:12.830","Text":"the minus 4 will just mean that we have to shift it 4 units"},{"Start":"03:12.830 ","End":"03:17.705","Text":"along the x-axis in the negative x direction."},{"Start":"03:17.705 ","End":"03:22.680","Text":"What we end up with is something like this."},{"Start":"03:22.680 ","End":"03:30.435","Text":"Note, the center is the x-axis and the x-axis,"},{"Start":"03:30.435 ","End":"03:36.150","Text":"this is the positive x-axis."},{"Start":"03:36.150 ","End":"03:45.335","Text":"It goes in the direction of the negative x-axis."},{"Start":"03:45.335 ","End":"03:51.745","Text":"It also has the minus 4."},{"Start":"03:51.745 ","End":"03:56.975","Text":"I think I may have confused you earlier with the minus 4."},{"Start":"03:56.975 ","End":"04:00.859","Text":"Really, what I should have been doing is ignore this 4,"},{"Start":"04:00.859 ","End":"04:04.060","Text":"draw the shape of this and then,"},{"Start":"04:04.060 ","End":"04:07.070","Text":"when I look at x not minus x,"},{"Start":"04:07.070 ","End":"04:08.900","Text":"I actually add 4."},{"Start":"04:08.900 ","End":"04:11.910","Text":"This actually is 4."},{"Start":"04:12.250 ","End":"04:16.350","Text":"This dot here is actually plus 4."},{"Start":"04:16.350 ","End":"04:20.805","Text":"It\u0027s 4, 0, 0, not minus 4."},{"Start":"04:20.805 ","End":"04:23.610","Text":"It\u0027s plus 4 in the x-direction because of this,"},{"Start":"04:23.610 ","End":"04:25.950","Text":"but it still goes,"},{"Start":"04:25.950 ","End":"04:28.985","Text":"it opens up in the negative x-direction."},{"Start":"04:28.985 ","End":"04:33.170","Text":"Anyway, just think about it and you\u0027ll see that this is roughly what we get."},{"Start":"04:33.170 ","End":"04:39.780","Text":"The cross-sections are elliptic because these 2 numbers are different."},{"Start":"04:39.780 ","End":"04:41.750","Text":"That\u0027s all you\u0027re expected."},{"Start":"04:41.750 ","End":"04:44.030","Text":"Just follow the general idea."},{"Start":"04:44.030 ","End":"04:46.580","Text":"When you encounter these quadric surfaces,"},{"Start":"04:46.580 ","End":"04:48.785","Text":"you will not be asked to sketch these."},{"Start":"04:48.785 ","End":"04:51.170","Text":"These were computer-aided,"},{"Start":"04:51.170 ","End":"04:54.910","Text":"not something you would do on your own with a table or something."},{"Start":"04:54.910 ","End":"04:57.290","Text":"That\u0027s all I want to say on this."},{"Start":"04:57.290 ","End":"04:59.070","Text":"We\u0027re done."}],"ID":9747}],"Thumbnail":null,"ID":57477},{"Name":"Functions of Several Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Functions of Several Variables","Duration":"9m 36s","ChapterTopicVideoID":9878,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9878.jpeg","UploadDate":"2022-09-06T23:11:15.8130000","DurationForVideoObject":"PT9M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"We\u0027re still in the chapter on 3D coordinates but this"},{"Start":"00:04.050 ","End":"00:08.650","Text":"is a section called functions of several variables."},{"Start":"00:09.020 ","End":"00:12.240","Text":"It\u0027s not really about several it\u0027s mostly"},{"Start":"00:12.240 ","End":"00:18.180","Text":"about 2 variables but it can be generalized to several variables."},{"Start":"00:18.180 ","End":"00:21.750","Text":"There\u0027s really 2 main things I\u0027m going to be talking about."},{"Start":"00:21.750 ","End":"00:26.730","Text":"1 of them is domains and"},{"Start":"00:26.730 ","End":"00:32.070","Text":"the other is graphs."},{"Start":"00:32.070 ","End":"00:34.109","Text":"If not by graph, how to represent,"},{"Start":"00:34.109 ","End":"00:36.450","Text":"how to draw functions of several variables."},{"Start":"00:36.450 ","End":"00:40.950","Text":"Turns out that the graph is not the best or not the only way."},{"Start":"00:40.950 ","End":"00:43.490","Text":"Because its functions of 2 variables,"},{"Start":"00:43.490 ","End":"00:48.050","Text":"turns out the graphs will be in 3D and that ties in with the 3D coordinates."},{"Start":"00:48.050 ","End":"00:51.980","Text":"Anyway, you may have seen some of these topics before."},{"Start":"00:51.980 ","End":"00:56.620","Text":"If so it\u0027ll just be a review but if not then it\u0027s new."},{"Start":"00:56.620 ","End":"01:00.730","Text":"Let\u0027s talk about domains first."},{"Start":"01:00.880 ","End":"01:05.630","Text":"A domain is also called the domain of definition that\u0027s"},{"Start":"01:05.630 ","End":"01:11.000","Text":"the full name of a function and 2 variables."},{"Start":"01:11.000 ","End":"01:13.160","Text":"It\u0027s pretty much the same as in 1 variable."},{"Start":"01:13.160 ","End":"01:18.080","Text":"It\u0027s just a question of what is valid input to the function."},{"Start":"01:18.080 ","End":"01:19.280","Text":"Well, let\u0027s be specific."},{"Start":"01:19.280 ","End":"01:20.870","Text":"I\u0027ll take an example."},{"Start":"01:20.870 ","End":"01:24.630","Text":"Suppose I have f of x,"},{"Start":"01:24.630 ","End":"01:30.820","Text":"y, and I\u0027m assuming that you know what a function of 2 variables is."},{"Start":"01:30.820 ","End":"01:36.125","Text":"This will be the square root of x plus y."},{"Start":"01:36.125 ","End":"01:38.735","Text":"Now how do I find its domain?"},{"Start":"01:38.735 ","End":"01:43.549","Text":"The domain is very similar to 1 variable."},{"Start":"01:43.549 ","End":"01:48.995","Text":"We just say that the restriction is only because of the square root."},{"Start":"01:48.995 ","End":"01:52.160","Text":"What\u0027s under the square root sign has to be non-negative,"},{"Start":"01:52.160 ","End":"01:58.560","Text":"so I get that x plus y has to be bigger or equal to 0."},{"Start":"01:58.970 ","End":"02:02.630","Text":"Let\u0027s say we want to draw this domain often you might say,"},{"Start":"02:02.630 ","End":"02:05.854","Text":"show this because this is a region"},{"Start":"02:05.854 ","End":"02:10.650","Text":"in the plane of all the x and y that satisfy this condition."},{"Start":"02:10.690 ","End":"02:18.285","Text":"Here\u0027s a sketch of the line x plus y equals 0,"},{"Start":"02:18.285 ","End":"02:22.730","Text":"and all we have to do is find out on which of the side of the line it\u0027s positive"},{"Start":"02:22.730 ","End":"02:27.460","Text":"and on which side it\u0027s negative and then we can get the greater than or equal to 0."},{"Start":"02:27.460 ","End":"02:30.475","Text":"Just plug in any point on,"},{"Start":"02:30.475 ","End":"02:32.180","Text":"let\u0027s say on this side,"},{"Start":"02:32.180 ","End":"02:33.560","Text":"if I plug in the point,"},{"Start":"02:33.560 ","End":"02:37.050","Text":"maybe 1,1 I\u0027ll get 1 plus 1 is 2,"},{"Start":"02:37.050 ","End":"02:39.040","Text":"which is bigger or equal to 0."},{"Start":"02:39.040 ","End":"02:41.720","Text":"On this side it\u0027s bigger than 0,"},{"Start":"02:41.720 ","End":"02:44.150","Text":"on this side it\u0027s less than 0,"},{"Start":"02:44.150 ","End":"02:51.770","Text":"so what I\u0027m going to get basically is all the half plane that\u0027s here."},{"Start":"02:51.770 ","End":"02:54.140","Text":"This is what it\u0027s like after we\u0027ve shaded it,"},{"Start":"02:54.140 ","End":"02:56.490","Text":"so it\u0027s the green portion."},{"Start":"02:56.530 ","End":"02:59.245","Text":"Let\u0027s do another example,"},{"Start":"02:59.245 ","End":"03:02.980","Text":"and again, I\u0027m going to use f of x, y."},{"Start":"03:03.620 ","End":"03:07.100","Text":"This time a little bit different."},{"Start":"03:07.100 ","End":"03:11.870","Text":"The square root of x plus the square root of y."},{"Start":"03:11.870 ","End":"03:16.610","Text":"Again, using the square root and it provides 2 restrictions"},{"Start":"03:16.610 ","End":"03:22.445","Text":"because there\u0027s 2 square roots so I have to have that x is bigger or equal to 0,"},{"Start":"03:22.445 ","End":"03:27.155","Text":"and that y is bigger or equal to 0."},{"Start":"03:27.155 ","End":"03:31.160","Text":"If you think about it, it\u0027s just the first quadrant,"},{"Start":"03:31.160 ","End":"03:34.270","Text":"including the axis and I\u0027ll show you the sketch."},{"Start":"03:34.270 ","End":"03:37.940","Text":"Here it is where x is bigger or equal to 0 and y is bigger or"},{"Start":"03:37.940 ","End":"03:41.120","Text":"equal to 0 including these. Let me just label them."},{"Start":"03:41.120 ","End":"03:46.760","Text":"This was the square root of x plus y bigger or equal to 0, and here,"},{"Start":"03:46.760 ","End":"03:49.445","Text":"square root of x is bigger or equal to 0,"},{"Start":"03:49.445 ","End":"03:55.925","Text":"and the square root of y bigger or equal to 0 with an and in between."},{"Start":"03:55.925 ","End":"03:58.730","Text":"Let\u0027s do another example."},{"Start":"03:58.730 ","End":"04:05.615","Text":"In this case, I have f of x and y is the natural logarithm"},{"Start":"04:05.615 ","End":"04:12.540","Text":"of 9 minus x squared minus 9 y squared."},{"Start":"04:12.540 ","End":"04:18.395","Text":"In here, we had a problem with the square root being restrictive,"},{"Start":"04:18.395 ","End":"04:22.250","Text":"here the natural logarithm is restrictive but this time,"},{"Start":"04:22.250 ","End":"04:26.720","Text":"the argument of the natural logarithm has to be strictly bigger than 0."},{"Start":"04:26.720 ","End":"04:28.840","Text":"Here we had bigger or equal to 0,"},{"Start":"04:28.840 ","End":"04:33.420","Text":"so here we have to have the 9 minus x."}],"ID":9781},{"Watched":false,"Name":"Functions of Two Variables - Domains and Graphs","Duration":"15m 8s","ChapterTopicVideoID":9884,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9884.jpeg","UploadDate":"2020-01-16T07:12:34.1070000","DurationForVideoObject":"PT15M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.175","Text":"Now on to graphs."},{"Start":"00:02.175 ","End":"00:05.535","Text":"I\u0027ll just do this briefly because I want to get into another way of"},{"Start":"00:05.535 ","End":"00:11.340","Text":"representing functions of 2 variables"},{"Start":"00:11.340 ","End":"00:13.830","Text":"but we\u0027ll take an example."},{"Start":"00:13.830 ","End":"00:18.795","Text":"I\u0027ll take an example which is z"},{"Start":"00:18.795 ","End":"00:26.399","Text":"equals 2x squared plus 2y squared minus 4."},{"Start":"00:26.399 ","End":"00:31.470","Text":"Now, it reminds me it\u0027s not quite,"},{"Start":"00:31.470 ","End":"00:34.525","Text":"but it reminds me of a paraboloid."},{"Start":"00:34.525 ","End":"00:36.500","Text":"Just to refresh your memory,"},{"Start":"00:36.500 ","End":"00:42.495","Text":"I copied the picture from the previous clip."},{"Start":"00:42.495 ","End":"00:44.850","Text":"This is a paraboloid,"},{"Start":"00:44.850 ","End":"00:49.205","Text":"this is the form of it where z is positive it faces upwards."},{"Start":"00:49.205 ","End":"00:51.260","Text":"Now this is not quite that,"},{"Start":"00:51.260 ","End":"00:57.780","Text":"but suppose I wrote that instead of that,"},{"Start":"00:57.780 ","End":"01:07.610","Text":"that x squared over 1 squared plus y squared"},{"Start":"01:07.610 ","End":"01:11.750","Text":"over 1 squared equals"},{"Start":"01:11.750 ","End":"01:18.710","Text":"z over 2."},{"Start":"01:18.710 ","End":"01:22.289","Text":"Well, that wouldn\u0027t quite be this,"},{"Start":"01:22.720 ","End":"01:26.935","Text":"but if I ignore for the moment,"},{"Start":"01:26.935 ","End":"01:31.985","Text":"this minus 4, then this is exactly the same as this."},{"Start":"01:31.985 ","End":"01:39.410","Text":"It is a circular because a equals b, circular paraboloid."},{"Start":"01:39.410 ","End":"01:43.820","Text":"The only thing is that it\u0027s not in the standard form where the apex,"},{"Start":"01:43.820 ","End":"01:46.145","Text":"the tip is the origin."},{"Start":"01:46.145 ","End":"01:47.810","Text":"Because of the minus 4,"},{"Start":"01:47.810 ","End":"01:50.270","Text":"I\u0027m going to have to lower it 4 units."},{"Start":"01:50.270 ","End":"01:54.205","Text":"I\u0027ll show you now the graph of this."},{"Start":"01:54.205 ","End":"01:55.910","Text":"Just to be pedantic,"},{"Start":"01:55.910 ","End":"01:58.970","Text":"let\u0027s get this second 1 right and correct it in"},{"Start":"01:58.970 ","End":"02:02.510","Text":"view of the fact that it is that there is a minus 4 here."},{"Start":"02:02.510 ","End":"02:07.380","Text":"I can put z plus 4 over 2 and then that will be right."},{"Start":"02:08.780 ","End":"02:12.810","Text":"Let\u0027s see this here,"},{"Start":"02:12.810 ","End":"02:15.495","Text":"this tip is at minus 4."},{"Start":"02:15.495 ","End":"02:17.870","Text":"If z is 0,"},{"Start":"02:17.870 ","End":"02:20.570","Text":"we get x squared plus y squared is 2,"},{"Start":"02:20.570 ","End":"02:22.300","Text":"which is root 2 squared,"},{"Start":"02:22.300 ","End":"02:25.730","Text":"so this point here and this point here,"},{"Start":"02:25.730 ","End":"02:29.150","Text":"this is where y is the square root of 2,"},{"Start":"02:29.150 ","End":"02:31.310","Text":"and this is where x is the square root of 2."},{"Start":"02:31.310 ","End":"02:33.600","Text":"I don\u0027t want to go into any more detail than that,"},{"Start":"02:33.600 ","End":"02:37.219","Text":"just to remind you that we do have graphs as a way of representing"},{"Start":"02:37.219 ","End":"02:41.645","Text":"surfaces of functions of 2 variables."},{"Start":"02:41.645 ","End":"02:44.150","Text":"But this is not the only way of"},{"Start":"02:44.150 ","End":"02:47.930","Text":"representing because this is difficult and really it has to"},{"Start":"02:47.930 ","End":"02:54.695","Text":"be computer aided or you have to invest in making elaborate drawings."},{"Start":"02:54.695 ","End":"03:00.845","Text":"There is another way of representing functions of 2 variables."},{"Start":"03:00.845 ","End":"03:07.435","Text":"That is the method of contour."},{"Start":"03:07.435 ","End":"03:10.715","Text":"Could be lines or curves."},{"Start":"03:10.715 ","End":"03:14.090","Text":"I\u0027ve seen both used and in economics,"},{"Start":"03:14.090 ","End":"03:18.865","Text":"mostly they\u0027re called the level curves."},{"Start":"03:18.865 ","End":"03:23.270","Text":"There\u0027s a couple of variations of what this thing is."},{"Start":"03:23.270 ","End":"03:25.489","Text":"They\u0027re all the same thing, just different names."},{"Start":"03:25.489 ","End":"03:30.605","Text":"It\u0027s just like when you have a relief map,"},{"Start":"03:30.605 ","End":"03:33.320","Text":"you don\u0027t buy a map book in 3 dimensions,"},{"Start":"03:33.320 ","End":"03:34.660","Text":"it\u0027s in 2 dimensions."},{"Start":"03:34.660 ","End":"03:40.550","Text":"But there are all these lines that show you the contours of points of equal height."},{"Start":"03:40.550 ","End":"03:44.750","Text":"The same idea works here also."},{"Start":"03:44.750 ","End":"03:47.090","Text":"I\u0027ve given you a vague definition."},{"Start":"03:47.090 ","End":"03:49.475","Text":"Let\u0027s give a more formal definition."},{"Start":"03:49.475 ","End":"03:53.375","Text":"A contour curve in 2 dimensions,"},{"Start":"03:53.375 ","End":"03:55.820","Text":"I mean in functions of 2 variables,"},{"Start":"03:55.820 ","End":"03:57.920","Text":"would be where f of x,"},{"Start":"03:57.920 ","End":"04:06.140","Text":"y is equal to some constant k. This gives us an equation in 2 variables,"},{"Start":"04:06.140 ","End":"04:10.850","Text":"x and y, and that would be a curve in the plane."},{"Start":"04:10.850 ","End":"04:13.610","Text":"For different k\u0027s we get different curves."},{"Start":"04:13.610 ","End":"04:15.799","Text":"It\u0027s a whole family of curves."},{"Start":"04:15.799 ","End":"04:18.230","Text":"We\u0027ll give an example in a moment."},{"Start":"04:18.230 ","End":"04:21.125","Text":"I just wanted to say that sometimes if"},{"Start":"04:21.125 ","End":"04:27.500","Text":"you did not get the function in the form z equals f of x,"},{"Start":"04:27.500 ","End":"04:29.810","Text":"y, which is what I\u0027m assuming here."},{"Start":"04:29.810 ","End":"04:32.345","Text":"Sometimes you get it in explicit form,"},{"Start":"04:32.345 ","End":"04:36.380","Text":"which is f of some other function, f of x, y,"},{"Start":"04:36.380 ","End":"04:38.825","Text":"z equals 0 as the explicit form,"},{"Start":"04:38.825 ","End":"04:43.535","Text":"then a level curve would be f of x,"},{"Start":"04:43.535 ","End":"04:46.685","Text":"y, and k equals 0."},{"Start":"04:46.685 ","End":"04:50.375","Text":"This is an explicit function in 2 variables."},{"Start":"04:50.375 ","End":"04:55.320","Text":"It also corresponds to a curve."},{"Start":"04:55.580 ","End":"04:59.030","Text":"That\u0027s what we do in the implicit case anyway."},{"Start":"04:59.030 ","End":"05:02.125","Text":"I\u0027ll show you an example."},{"Start":"05:02.125 ","End":"05:07.640","Text":"Like I said, we only choose a sample values of k. Just like when you buy a map book,"},{"Start":"05:07.640 ","End":"05:11.660","Text":"you get maybe contour line for 100 feet and for"},{"Start":"05:11.660 ","End":"05:16.115","Text":"200 feet and for 300 feet above sea level, and so on."},{"Start":"05:16.115 ","End":"05:22.940","Text":"The first example will be f of x, y,"},{"Start":"05:22.940 ","End":"05:32.360","Text":"which is z is equal to the square root of x squared plus y squared."},{"Start":"05:32.360 ","End":"05:35.690","Text":"I could even write that z is equal to,"},{"Start":"05:35.690 ","End":"05:37.715","Text":"now if I square both sizes,"},{"Start":"05:37.715 ","End":"05:39.500","Text":"just forget about the f part."},{"Start":"05:39.500 ","End":"05:43.460","Text":"Then I get z squared equals x squared plus y squared."},{"Start":"05:43.460 ","End":"05:44.660","Text":"Or if I want,"},{"Start":"05:44.660 ","End":"05:50.389","Text":"x squared plus y squared equals z squared."},{"Start":"05:50.389 ","End":"05:55.070","Text":"This is the equation of a circular cone."},{"Start":"05:55.070 ","End":"05:58.790","Text":"If you go back and look at quadric surfaces,"},{"Start":"05:58.790 ","End":"06:00.260","Text":"this is a circular cone,"},{"Start":"06:00.260 ","End":"06:04.940","Text":"but not quite a cone has 2 parts and they touch tip to tip."},{"Start":"06:04.940 ","End":"06:08.705","Text":"But because this here,"},{"Start":"06:08.705 ","End":"06:12.470","Text":"the square root is only positive or non-negative,"},{"Start":"06:12.470 ","End":"06:17.280","Text":"this is only the top half of the cone."},{"Start":"06:17.320 ","End":"06:20.490","Text":"I\u0027ll just show you the picture."},{"Start":"06:20.540 ","End":"06:24.115","Text":"Here\u0027s our cone, but you might say,"},{"Start":"06:24.115 ","End":"06:28.730","Text":"yes, this is not to level curves, this is just a graph."},{"Start":"06:28.730 ","End":"06:32.150","Text":"This is the graph. I\u0027m coming to the level curves."},{"Start":"06:32.150 ","End":"06:38.030","Text":"Let\u0027s say we take the values of k to be whole numbers."},{"Start":"06:38.030 ","End":"06:41.390","Text":"Like I said, we don\u0027t take every k and you\u0027ll see soon why."},{"Start":"06:41.390 ","End":"06:42.680","Text":"We just take a sample."},{"Start":"06:42.680 ","End":"06:44.450","Text":"Let\u0027s take k equals 1,"},{"Start":"06:44.450 ","End":"06:46.925","Text":"k equals 2, k equals 3,"},{"Start":"06:46.925 ","End":"06:49.310","Text":"and then we get different equations,"},{"Start":"06:49.310 ","End":"06:56.145","Text":"x squared plus y squared equals 1 squared."},{"Start":"06:56.145 ","End":"06:58.500","Text":"X squared plus y squared equals 2 squared."},{"Start":"06:58.500 ","End":"07:01.830","Text":"Each time we let z equals something else."},{"Start":"07:01.830 ","End":"07:07.120","Text":"Notice that x squared plus y squared equals k squared,"},{"Start":"07:07.120 ","End":"07:09.340","Text":"which is what I get when I set z equal k,"},{"Start":"07:09.340 ","End":"07:17.710","Text":"is just a circle with radius k. If I draw for each whole number,"},{"Start":"07:17.710 ","End":"07:21.820","Text":"say k, then I\u0027ll get a series of"},{"Start":"07:21.820 ","End":"07:28.630","Text":"circles and it will look something like this and I\u0027ll scroll down a bit more."},{"Start":"07:28.630 ","End":"07:33.760","Text":"There we are. Actually here,"},{"Start":"07:33.760 ","End":"07:35.485","Text":"you could even draw k equals 0,"},{"Start":"07:35.485 ","End":"07:37.180","Text":"just gives you a point that k equals 1,"},{"Start":"07:37.180 ","End":"07:39.130","Text":"circle of radius 1, 2, 3,"},{"Start":"07:39.130 ","End":"07:41.755","Text":"4, 5 came out very nicely."},{"Start":"07:41.755 ","End":"07:47.200","Text":"Instead of having a 3-dimensional surface,"},{"Start":"07:47.200 ","End":"07:49.855","Text":"which we show in 2-dimensions,"},{"Start":"07:49.855 ","End":"07:53.214","Text":"we just show the level curves"},{"Start":"07:53.214 ","End":"07:59.455","Text":"and usually we label each level curve with the value of k as it is here."},{"Start":"07:59.455 ","End":"08:09.790","Text":"Now this concept of level curves can be generalized to some extent in higher dimensions,"},{"Start":"08:09.790 ","End":"08:13.270","Text":"not usually possibly in 3 dimensions."},{"Start":"08:13.270 ","End":"08:18.255","Text":"You could have a function f of x,"},{"Start":"08:18.255 ","End":"08:21.420","Text":"y, and z and I don\u0027t what letter you would call it,"},{"Start":"08:21.420 ","End":"08:26.749","Text":"maybe w equals doesn\u0027t matter and instead of level curves,"},{"Start":"08:26.749 ","End":"08:30.925","Text":"we could talk of level surfaces."},{"Start":"08:30.925 ","End":"08:32.830","Text":"We would say f of x, y,"},{"Start":"08:32.830 ","End":"08:42.220","Text":"z equals k and that would be a level surface and surfaces even though they\u0027re in 3D,"},{"Start":"08:42.220 ","End":"08:44.680","Text":"we can draw them like we see here,"},{"Start":"08:44.680 ","End":"08:47.440","Text":"like a 2-dimensional picture of a 3-dimensional thing,"},{"Start":"08:47.440 ","End":"08:50.590","Text":"but we don\u0027t really go any higher than 3-dimensions."},{"Start":"08:50.590 ","End":"08:52.960","Text":"I just thought I\u0027d mentioned that."},{"Start":"08:52.960 ","End":"09:01.990","Text":"Now I want to talk about another generalization in 2-dimensions of level curves."},{"Start":"09:01.990 ","End":"09:06.790","Text":"There is a concept of traces,"},{"Start":"09:06.790 ","End":"09:09.880","Text":"which is really a generalization of level curves."},{"Start":"09:09.880 ","End":"09:11.710","Text":"I mean, if you think about it,"},{"Start":"09:11.710 ","End":"09:14.620","Text":"a level curve like f of x,"},{"Start":"09:14.620 ","End":"09:20.170","Text":"y equals k is just the intersection"},{"Start":"09:20.170 ","End":"09:28.225","Text":"of the graph of z equals f of x,"},{"Start":"09:28.225 ","End":"09:34.360","Text":"y with the plane z equals k. Yes,"},{"Start":"09:34.360 ","End":"09:41.485","Text":"this is an equation of a plane that\u0027s parallel to the x, y coordinate plane."},{"Start":"09:41.485 ","End":"09:45.900","Text":"Now, one generalization would be to say, okay,"},{"Start":"09:45.900 ","End":"09:51.580","Text":"why just take the intersection with a horizontal plane?"},{"Start":"09:51.580 ","End":"09:56.830","Text":"Why not with a plane that\u0027s parallel to the other axes and you can do that."},{"Start":"09:56.830 ","End":"10:00.490","Text":"If you take the intersection of f of x,"},{"Start":"10:00.490 ","End":"10:04.570","Text":"y with, say, it could have been instead of this,"},{"Start":"10:04.570 ","End":"10:10.420","Text":"x equals k or y equals k. These"},{"Start":"10:10.420 ","End":"10:16.615","Text":"are all planes that are parallel to the coordinate planes."},{"Start":"10:16.615 ","End":"10:18.670","Text":"Z is parallel to the x, y,"},{"Start":"10:18.670 ","End":"10:22.210","Text":"the x constant is parallel to the yz plane,"},{"Start":"10:22.210 ","End":"10:29.230","Text":"and this is parallel to the xz plane and there\u0027s actually even a greater generalization."},{"Start":"10:29.230 ","End":"10:34.270","Text":"In fact, you could intersect this with any plane and that\u0027s like a cross-section."},{"Start":"10:34.270 ","End":"10:38.575","Text":"I could take it completely diagonally and skew."},{"Start":"10:38.575 ","End":"10:43.330","Text":"But here we\u0027ll stick to just coordinate planes."},{"Start":"10:43.330 ","End":"10:47.680","Text":"Let\u0027s take an example of this."},{"Start":"10:47.680 ","End":"10:50.410","Text":"Let\u0027s take as our function of x,"},{"Start":"10:50.410 ","End":"10:58.480","Text":"y to be 10 minus 4x squared minus y"},{"Start":"10:58.480 ","End":"11:02.410","Text":"squared and I\u0027d like to know what its traces"},{"Start":"11:02.410 ","End":"11:06.715","Text":"when we intersect with the following planes or give actually 3 parts."},{"Start":"11:06.715 ","End":"11:09.550","Text":"The first part is just a level curve,"},{"Start":"11:09.550 ","End":"11:17.575","Text":"and I want to intersect it with z equals 0."},{"Start":"11:17.575 ","End":"11:23.090","Text":"Let\u0027s say this is z and in part b,"},{"Start":"11:23.960 ","End":"11:27.705","Text":"here we\u0027ll take x equals 1,"},{"Start":"11:27.705 ","End":"11:33.430","Text":"the plane parallel to the yz plane and part C,"},{"Start":"11:33.430 ","End":"11:35.350","Text":"we\u0027ll take y equals 2,"},{"Start":"11:35.350 ","End":"11:37.855","Text":"which is parallel to the xz plane."},{"Start":"11:37.855 ","End":"11:42.280","Text":"All 3 of these are traces and the first one is also a level surface."},{"Start":"11:42.280 ","End":"11:45.580","Text":"I\u0027ll do the first one just quickly because we\u0027ve already done that."},{"Start":"11:45.580 ","End":"11:52.450","Text":"If z equals 0, we divide by 10 and switch sides."},{"Start":"11:52.450 ","End":"11:58.165","Text":"We get x squared over"},{"Start":"11:58.165 ","End":"12:08.950","Text":"2.5 minus plus y squared over 10 equals 1 and if you look back,"},{"Start":"12:08.950 ","End":"12:16.994","Text":"this is the equation of an ellipse and I\u0027ll just write the word ellipse."},{"Start":"12:16.994 ","End":"12:21.990","Text":"Let me just wrote that what I want as the question"},{"Start":"12:21.990 ","End":"12:26.860","Text":"is find the traces for the following planes."},{"Start":"12:26.860 ","End":"12:30.160","Text":"Got 1 done now let\u0027s do the other one."},{"Start":"12:30.160 ","End":"12:40.570","Text":"The easiest thing to do is just to substitute x equals 1 in the equation."},{"Start":"12:40.570 ","End":"12:44.815","Text":"What we get is if x equals 1,"},{"Start":"12:44.815 ","End":"12:50.380","Text":"we get 10 minus 4 times 1,"},{"Start":"12:50.380 ","End":"13:00.220","Text":"which is 6 minus y squared and this is equal to z of course and so we get"},{"Start":"13:00.220 ","End":"13:10.210","Text":"an equation in z and y and if we sketch it in the zy plane,"},{"Start":"13:10.210 ","End":"13:11.980","Text":"this will be a parabola."},{"Start":"13:11.980 ","End":"13:18.130","Text":"Because z is a quadratic function of y and I\u0027ll sketch it in a moment."},{"Start":"13:18.130 ","End":"13:21.940","Text":"I\u0027ll show you the pictures and the trace for y equals 2."},{"Start":"13:21.940 ","End":"13:25.540","Text":"Again, we get z equals and if I let y equals 2."},{"Start":"13:25.540 ","End":"13:29.275","Text":"2 squared is 4."},{"Start":"13:29.275 ","End":"13:34.810","Text":"I get almost the same thing this time z equals 6 minus x squared,"},{"Start":"13:34.810 ","End":"13:36.250","Text":"which has the same shape."},{"Start":"13:36.250 ","End":"13:37.540","Text":"It\u0027s just in different variables,"},{"Start":"13:37.540 ","End":"13:41.650","Text":"but the same picture and I\u0027ll show you the diagram."},{"Start":"13:41.650 ","End":"13:45.070","Text":"Here\u0027s several pictures."},{"Start":"13:45.070 ","End":"13:55.450","Text":"This one is an elliptic paraboloid."},{"Start":"13:55.450 ","End":"14:02.665","Text":"As for part B, this is the plane where x equals 1, you can\u0027t see it,"},{"Start":"14:02.665 ","End":"14:07.510","Text":"but it cuts the x axis where x is 1 and it\u0027s parallel to"},{"Start":"14:07.510 ","End":"14:12.620","Text":"the yz plane and we see that what we get is a parabola."},{"Start":"14:12.620 ","End":"14:14.510","Text":"In fact, in both of these cases,"},{"Start":"14:14.510 ","End":"14:15.860","Text":"as I said, we get a parabola,"},{"Start":"14:15.860 ","End":"14:21.620","Text":"it\u0027s the same equation basically and in this one,"},{"Start":"14:21.620 ","End":"14:26.240","Text":"we have the plane where y equals 2 it parallel to"},{"Start":"14:26.240 ","End":"14:33.350","Text":"the xz plane and it will cut the y-axis where y is 2 and this is the parabola."},{"Start":"14:33.350 ","End":"14:35.435","Text":"These are the traces."},{"Start":"14:35.435 ","End":"14:43.610","Text":"This is the trace and this is a trace and for the first one, z equals 0."},{"Start":"14:43.610 ","End":"14:47.690","Text":"Like I said, if we just saw where it cuts,"},{"Start":"14:47.690 ","End":"14:56.150","Text":"I could maybe just outline that ellipse by just going along here and if we continued it,"},{"Start":"14:56.150 ","End":"14:58.279","Text":"if it was transparent,"},{"Start":"14:58.279 ","End":"15:02.330","Text":"we would get the ellipse where z equals 0,."},{"Start":"15:02.330 ","End":"15:04.560","Text":"That\u0027s the x, y plane."},{"Start":"15:04.560 ","End":"15:08.780","Text":"This topic is now covered, we\u0027re done."}],"ID":9782},{"Watched":false,"Name":"Exercise 1","Duration":"2m 23s","ChapterTopicVideoID":9820,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9820.jpeg","UploadDate":"2017-08-01T05:12:41.4200000","DurationForVideoObject":"PT2M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise, we have a function of 2 variables,"},{"Start":"00:03.390 ","End":"00:05.745","Text":"f of x, y as follows."},{"Start":"00:05.745 ","End":"00:09.090","Text":"We want to find the domain of the function,"},{"Start":"00:09.090 ","End":"00:12.750","Text":"some region in the xy-plane, and,"},{"Start":"00:12.750 ","End":"00:14.460","Text":"if possible, to give a rough sketch,"},{"Start":"00:14.460 ","End":"00:18.190","Text":"although that\u0027s less important."},{"Start":"00:18.290 ","End":"00:21.465","Text":"What could go wrong here?"},{"Start":"00:21.465 ","End":"00:25.290","Text":"We have a square root and under the square root sign,"},{"Start":"00:25.290 ","End":"00:28.260","Text":"we need something that\u0027s not negative,"},{"Start":"00:28.260 ","End":"00:29.640","Text":"could be 0 or positive."},{"Start":"00:29.640 ","End":"00:31.680","Text":"Other than that, there\u0027s no restriction."},{"Start":"00:31.680 ","End":"00:36.900","Text":"All we need to write is x"},{"Start":"00:36.900 ","End":"00:44.270","Text":"squared minus 2y is bigger or equal to 0,"},{"Start":"00:44.270 ","End":"00:52.790","Text":"which means that 2y is less than or equal to x squared."},{"Start":"00:52.790 ","End":"00:56.780","Text":"I brought this over and I switched sides because it\u0027s easier for me than to"},{"Start":"00:56.780 ","End":"01:02.370","Text":"see that y is less than or equal to 1 half x squared."},{"Start":"01:02.370 ","End":"01:05.150","Text":"Now, 1 half x squared,"},{"Start":"01:05.150 ","End":"01:09.480","Text":"if it was equal, would be a parabola upward facing."},{"Start":"01:09.700 ","End":"01:12.140","Text":"If we sketch this parabola,"},{"Start":"01:12.140 ","End":"01:17.330","Text":"what we would want is from the parabola and downwards."},{"Start":"01:17.360 ","End":"01:21.435","Text":"Here is a sketch I brought with me."},{"Start":"01:21.435 ","End":"01:24.005","Text":"I found it on the Internet."},{"Start":"01:24.005 ","End":"01:29.645","Text":"This red line is the line y equals 1 half x squared."},{"Start":"01:29.645 ","End":"01:32.390","Text":"You would make a table of a few values."},{"Start":"01:32.390 ","End":"01:34.580","Text":"For example, when x is naught,"},{"Start":"01:34.580 ","End":"01:37.700","Text":"then y is also naught."},{"Start":"01:37.700 ","End":"01:40.100","Text":"When x is 1, y is a half."},{"Start":"01:40.100 ","End":"01:44.525","Text":"So I take 1 half here and 1 here."},{"Start":"01:44.525 ","End":"01:47.970","Text":"When x is 2, 2 squared over 2 is 4,"},{"Start":"01:47.970 ","End":"01:50.450","Text":"over 2 is 2, and so on."},{"Start":"01:50.450 ","End":"01:52.220","Text":"Also, it\u0027s an even function,"},{"Start":"01:52.220 ","End":"01:59.635","Text":"so I could get the same points on the other side, here and here."},{"Start":"01:59.635 ","End":"02:02.190","Text":"Then we get y equals 1 half x squared."},{"Start":"02:02.190 ","End":"02:09.510","Text":"Now the less than or equal to means it includes this curve and anything below the curve,"},{"Start":"02:09.510 ","End":"02:11.500","Text":"downwards from the curve,"},{"Start":"02:11.500 ","End":"02:13.010","Text":"less than or equal to."},{"Start":"02:13.010 ","End":"02:16.790","Text":"That\u0027s the part that\u0027s shaded in green."},{"Start":"02:16.790 ","End":"02:23.550","Text":"But the algebraic solution is just fine. That\u0027s it."}],"ID":9776},{"Watched":false,"Name":"Exercise 2","Duration":"2m 33s","ChapterTopicVideoID":9821,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9821.jpeg","UploadDate":"2017-08-01T05:12:47.7000000","DurationForVideoObject":"PT2M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.120","Text":"In this exercise, we have to find the domain of this function of 2 variables, f of x,"},{"Start":"00:06.120 ","End":"00:10.170","Text":"y as follows and if possible,"},{"Start":"00:10.170 ","End":"00:12.135","Text":"to give a rough sketch,"},{"Start":"00:12.135 ","End":"00:14.580","Text":"although that\u0027s less important."},{"Start":"00:14.580 ","End":"00:21.105","Text":"We want to find it more algebraically as an equation or inequality."},{"Start":"00:21.105 ","End":"00:24.495","Text":"Let\u0027s see. What could be the problem here?"},{"Start":"00:24.495 ","End":"00:26.610","Text":"The only snag here is"},{"Start":"00:26.610 ","End":"00:29.940","Text":"the natural logarithm that brings"},{"Start":"00:29.940 ","End":"00:34.650","Text":"the restriction that its argument has to be strictly positive."},{"Start":"00:34.650 ","End":"00:38.535","Text":"Really, we have the following,"},{"Start":"00:38.535 ","End":"00:47.980","Text":"2x minus 3y plus 1 has got to be bigger than 0."},{"Start":"00:47.980 ","End":"00:55.940","Text":"I\u0027d like to get it in terms of y is bigger or smaller or whatever,"},{"Start":"00:55.940 ","End":"00:57.185","Text":"than the rest of it,"},{"Start":"00:57.185 ","End":"00:58.550","Text":"bring y on 1 side."},{"Start":"00:58.550 ","End":"01:00.890","Text":"If I bring the 3y over there,"},{"Start":"01:00.890 ","End":"01:05.705","Text":"I\u0027ve got 2x plus 1 is bigger than 3y."},{"Start":"01:05.705 ","End":"01:10.400","Text":"Switch sides, 3y less than 2x plus 1,"},{"Start":"01:10.400 ","End":"01:12.455","Text":"now divide by 3."},{"Start":"01:12.455 ","End":"01:18.785","Text":"Y is less than 2/3x plus 1/3."},{"Start":"01:18.785 ","End":"01:26.265","Text":"For the sketch, what I would do is draw the straight line y equals this,"},{"Start":"01:26.265 ","End":"01:29.749","Text":"just plot a few points or use the intercept,"},{"Start":"01:29.749 ","End":"01:35.000","Text":"or whatever, and then take everything below that line,"},{"Start":"01:35.000 ","End":"01:37.025","Text":"because the less than means downwards."},{"Start":"01:37.025 ","End":"01:39.514","Text":"Let me just give you my sketch."},{"Start":"01:39.514 ","End":"01:42.935","Text":"Here\u0027s the sketch which I got by first of all,"},{"Start":"01:42.935 ","End":"01:50.655","Text":"drawing the line y equals 2/3x plus 1/3."},{"Start":"01:50.655 ","End":"01:55.015","Text":"This you can do say with a table of values like when x is 1,"},{"Start":"01:55.015 ","End":"01:57.600","Text":"y is 1, when x is 0,"},{"Start":"01:57.600 ","End":"02:00.735","Text":"y is 1/3, a few points line through that."},{"Start":"02:00.735 ","End":"02:05.885","Text":"Now this, I drew dotted or dashed because this is not included."},{"Start":"02:05.885 ","End":"02:10.555","Text":"What we want is strictly less than, it\u0027s below the line,"},{"Start":"02:10.555 ","End":"02:13.440","Text":"but not including the line."},{"Start":"02:13.440 ","End":"02:16.235","Text":"If it\u0027s included, it\u0027s a solid line and if not,"},{"Start":"02:16.235 ","End":"02:19.410","Text":"we draw it dotted or dashed."},{"Start":"02:19.510 ","End":"02:28.160","Text":"This gives you an idea that this is the domain of this function in the x,"},{"Start":"02:28.160 ","End":"02:32.670","Text":"y plane. That\u0027s all."}],"ID":9777},{"Watched":false,"Name":"Exercise 3","Duration":"3m 10s","ChapterTopicVideoID":9822,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9822.jpeg","UploadDate":"2017-08-01T05:12:55.6030000","DurationForVideoObject":"PT3M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"In this exercise, we have a function f of 2 variables, x and y, defined as follows."},{"Start":"00:07.050 ","End":"00:13.365","Text":"We\u0027re interested in the domain which will be a subset of the x, y plane."},{"Start":"00:13.365 ","End":"00:19.335","Text":"Let\u0027s see what are the legal values for x and y."},{"Start":"00:19.335 ","End":"00:22.185","Text":"There are actually 3 restrictions."},{"Start":"00:22.185 ","End":"00:24.855","Text":"Each of these terms provide the restriction."},{"Start":"00:24.855 ","End":"00:31.140","Text":"This 1 tells us that x cannot be equal to 0,"},{"Start":"00:31.140 ","End":"00:34.230","Text":"because we can\u0027t have a 0 on the denominator."},{"Start":"00:34.230 ","End":"00:40.920","Text":"This here tells us that what\u0027s under the square root sign has to be bigger or equal to 0."},{"Start":"00:40.920 ","End":"00:44.155","Text":"In other words, I\u0027ll write the word and,"},{"Start":"00:44.155 ","End":"00:46.040","Text":"all these conditions have to hold,"},{"Start":"00:46.040 ","End":"00:51.860","Text":"and y plus 4 has to be non-negative."},{"Start":"00:51.860 ","End":"00:58.170","Text":"Similarly what\u0027s under this square root sign also has to be non-negative,"},{"Start":"00:58.170 ","End":"01:03.705","Text":"so x plus 1 bigger or equal to 0."},{"Start":"01:03.705 ","End":"01:06.660","Text":"Now this I\u0027ll leave as is."},{"Start":"01:06.660 ","End":"01:08.550","Text":"These 2 could be rewritten."},{"Start":"01:08.550 ","End":"01:15.410","Text":"This 1 would tell us that y is bigger or equal to negative 4."},{"Start":"01:15.410 ","End":"01:21.775","Text":"This 1 we could rewrite as x bigger or equal to negative 1."},{"Start":"01:21.775 ","End":"01:24.925","Text":"Now if I take these 2,"},{"Start":"01:24.925 ","End":"01:28.570","Text":"that would give me a 1/4 of the plane,"},{"Start":"01:28.570 ","End":"01:31.730","Text":"because x is to the right of something,"},{"Start":"01:31.730 ","End":"01:36.715","Text":"and y is above something so we have an upper right 1/4 plane."},{"Start":"01:36.715 ","End":"01:39.980","Text":"Well I\u0027ll just bring in the sketch and then we\u0027ll explain it."},{"Start":"01:39.980 ","End":"01:44.165","Text":"That would be easier. Here\u0027s the sketch."},{"Start":"01:44.165 ","End":"01:46.130","Text":"Need some explaining."},{"Start":"01:46.130 ","End":"01:47.644","Text":"This is the line"},{"Start":"01:47.644 ","End":"01:57.640","Text":"where y is equal to minus 4,"},{"Start":"01:57.640 ","End":"02:03.340","Text":"and this line here is where x equals negative 1."},{"Start":"02:03.340 ","End":"02:10.670","Text":"Now the bigger or equal to means that we take this line and anything to the right of it,"},{"Start":"02:10.670 ","End":"02:19.010","Text":"and the bigger or equal to minus 4 means this line and above including the lines."},{"Start":"02:19.010 ","End":"02:20.990","Text":"If we just had these 2,"},{"Start":"02:20.990 ","End":"02:24.880","Text":"we\u0027d have this 1/4 of a plane."},{"Start":"02:24.880 ","End":"02:29.440","Text":"This extends indefinitely, but we can\u0027t draw it indefinitely."},{"Start":"02:29.440 ","End":"02:31.680","Text":"We have to cut it off somewhere."},{"Start":"02:31.680 ","End":"02:37.534","Text":"The x not equal to 0 means that we rule out x equals 0,"},{"Start":"02:37.534 ","End":"02:40.415","Text":"x equals 0 is the y-axis,"},{"Start":"02:40.415 ","End":"02:46.865","Text":"so this is indicated with a dotted, dashed line."},{"Start":"02:46.865 ","End":"02:50.090","Text":"The y-axis is removed in short."},{"Start":"02:50.090 ","End":"02:57.430","Text":"It\u0027s the green infinite square, infinite 1/4 plane,"},{"Start":"02:58.280 ","End":"03:09.640","Text":"excluding the y-axis, the stuff that\u0027s shaded in green here. That\u0027s it."}],"ID":9778},{"Watched":false,"Name":"Exercise 4","Duration":"3m 1s","ChapterTopicVideoID":9823,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9823.jpeg","UploadDate":"2017-08-01T05:13:04.8300000","DurationForVideoObject":"PT3M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have a function of three variables,"},{"Start":"00:03.720 ","End":"00:05.550","Text":"x, y, and z."},{"Start":"00:05.550 ","End":"00:08.550","Text":"It\u0027s a function defined on three space"},{"Start":"00:08.550 ","End":"00:13.845","Text":"and it\u0027s given as follows with variables x, y, and z."},{"Start":"00:13.845 ","End":"00:15.885","Text":"We want to find its domain."},{"Start":"00:15.885 ","End":"00:18.120","Text":"We\u0027ll just do this algebraically,"},{"Start":"00:18.120 ","End":"00:22.380","Text":"a sketch is more difficult because we\u0027d have to sketch"},{"Start":"00:22.380 ","End":"00:27.029","Text":"a sub-region or a subset of the 3D space."},{"Start":"00:27.029 ","End":"00:30.645","Text":"Let\u0027s just do it algebraically as I say,"},{"Start":"00:30.645 ","End":"00:32.850","Text":"the only thing that can go wrong here,"},{"Start":"00:32.850 ","End":"00:35.670","Text":"everything behaves nicely except for the 1 over,"},{"Start":"00:35.670 ","End":"00:38.550","Text":"we have to make sure the denominator is not 0."},{"Start":"00:38.550 ","End":"00:44.390","Text":"So the domain is everywhere that x squared plus"},{"Start":"00:44.390 ","End":"00:51.515","Text":"y squared plus 4z is not equal to 0."},{"Start":"00:51.515 ","End":"01:00.455","Text":"I can write this as 4z is not equal to,"},{"Start":"01:00.455 ","End":"01:01.895","Text":"bring stuff to the other side,"},{"Start":"01:01.895 ","End":"01:06.905","Text":"minus x squared minus y squared,"},{"Start":"01:06.905 ","End":"01:12.120","Text":"and then divide by 4 and I"},{"Start":"01:12.120 ","End":"01:17.565","Text":"have the z is not equal to,"},{"Start":"01:17.565 ","End":"01:22.525","Text":"you know what?, I\u0027ll put the minus on the left and the 4 on the right."},{"Start":"01:22.525 ","End":"01:29.990","Text":"Z over minus 1 is not equal to x squared over 4,"},{"Start":"01:29.990 ","End":"01:32.780","Text":"let me write it like this."},{"Start":"01:32.780 ","End":"01:34.700","Text":"Y squared over 2 squared."},{"Start":"01:34.700 ","End":"01:43.010","Text":"The reason I\u0027m doing that is because the standard form of an elliptic paraboloid is z"},{"Start":"01:43.010 ","End":"01:51.920","Text":"over c is equal to x squared over a squared plus y squared over b squared."},{"Start":"01:51.920 ","End":"01:58.280","Text":"This is in the section on quadric surfaces and it\u0027s a circular or elliptic paraboloid."},{"Start":"01:58.280 ","End":"02:03.065","Text":"In this case it\u0027s a circular paraboloid because a equals b"},{"Start":"02:03.065 ","End":"02:10.500","Text":"and it\u0027s downward facing because c is negative, it\u0027s minus 1."},{"Start":"02:11.200 ","End":"02:18.755","Text":"I could show you the sketch of this but I won\u0027t."},{"Start":"02:18.755 ","End":"02:21.965","Text":"We\u0027ll just settle for it algebraically."},{"Start":"02:21.965 ","End":"02:31.999","Text":"This is the inequality that z has to satisfy or we could say that z is not on"},{"Start":"02:31.999 ","End":"02:37.520","Text":"the circular paraboloid given by z"},{"Start":"02:37.520 ","End":"02:44.210","Text":"over minus 1 equals x squared over 4 plus y squared over 4."},{"Start":"02:44.210 ","End":"02:47.969","Text":"I\u0027ll just emphasize, not,"},{"Start":"02:48.160 ","End":"02:55.230","Text":"z is going to be everywhere in space except for on this surface."},{"Start":"02:56.350 ","End":"03:00.750","Text":"That\u0027s it. We had a three-dimensional example."}],"ID":9779},{"Watched":false,"Name":"Exercise 5","Duration":"2m 30s","ChapterTopicVideoID":9824,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9824.jpeg","UploadDate":"2017-08-01T05:13:12.9400000","DurationForVideoObject":"PT2M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we have a function of 2 variables, f of x,"},{"Start":"00:03.630 ","End":"00:08.355","Text":"y equals this and we want to sketch a few contours."},{"Start":"00:08.355 ","End":"00:12.435","Text":"Contours are also called level curves, especially in economics."},{"Start":"00:12.435 ","End":"00:17.670","Text":"A level curve is when we take the function and assign it to some constant value."},{"Start":"00:17.670 ","End":"00:22.110","Text":"A level curve would be x squared plus y squared"},{"Start":"00:22.110 ","End":"00:29.250","Text":"equals K and different values of K would give different level curves."},{"Start":"00:29.450 ","End":"00:34.550","Text":"Now, obviously, K has to be bigger or equal"},{"Start":"00:34.550 ","End":"00:39.605","Text":"to 0 because this is non-negative and this is non-negative."},{"Start":"00:39.605 ","End":"00:46.340","Text":"We could try values of K that are convenient for us."},{"Start":"00:46.340 ","End":"00:48.710","Text":"For example, if K was something squared,"},{"Start":"00:48.710 ","End":"00:51.634","Text":"if I have that K was equal to c squared,"},{"Start":"00:51.634 ","End":"00:58.150","Text":"then this would be a circle of radius c. I\u0027ll bring in the sketch now."},{"Start":"00:58.150 ","End":"00:59.850","Text":"Here we have a few circles,"},{"Start":"00:59.850 ","End":"01:02.445","Text":"let\u0027s say this is our typical one."},{"Start":"01:02.445 ","End":"01:07.955","Text":"This one here is x squared plus y squared"},{"Start":"01:07.955 ","End":"01:14.000","Text":"equals K and we said that K is c squared or c is square root of K,"},{"Start":"01:14.000 ","End":"01:15.380","Text":"then that\u0027s the radius."},{"Start":"01:15.380 ","End":"01:18.410","Text":"Then this point here would be the point c,"},{"Start":"01:18.410 ","End":"01:24.050","Text":"which is the square root of K. For example,"},{"Start":"01:24.050 ","End":"01:27.545","Text":"this circle where c equals 3,"},{"Start":"01:27.545 ","End":"01:30.680","Text":"that would be where K is equal to 9,"},{"Start":"01:30.680 ","End":"01:33.095","Text":"x squared plus y squared equals 9."},{"Start":"01:33.095 ","End":"01:35.825","Text":"Here, K would equal 4,"},{"Start":"01:35.825 ","End":"01:39.860","Text":"x squared plus y squared equals 2 squared, which is 4."},{"Start":"01:39.860 ","End":"01:46.085","Text":"The radius 1 would correspond to K equals 1 and actually we could even take K equals 0,"},{"Start":"01:46.085 ","End":"01:48.545","Text":"which just gives us a single point,"},{"Start":"01:48.545 ","End":"01:52.970","Text":"that single point is the level curve for K equals 0,"},{"Start":"01:52.970 ","End":"01:56.660","Text":"x squared plus y squared is 0 and both of them are 0."},{"Start":"01:56.660 ","End":"01:58.850","Text":"Here we have K is 16,"},{"Start":"01:58.850 ","End":"02:01.790","Text":"and here we have K is 25."},{"Start":"02:01.790 ","End":"02:04.550","Text":"You could take any K you want."},{"Start":"02:04.550 ","End":"02:09.345","Text":"If you took, let\u0027s say, any value of K,"},{"Start":"02:09.345 ","End":"02:12.950","Text":"maybe this K is 12,"},{"Start":"02:12.950 ","End":"02:18.920","Text":"and then the point here would be the square root of 12 with a circle."},{"Start":"02:18.920 ","End":"02:25.940","Text":"Okay, that\u0027s just to give you an idea of level curves for a function."},{"Start":"02:25.940 ","End":"02:30.150","Text":"I\u0027m done for this exercise."}],"ID":9780},{"Watched":false,"Name":"Exercise 6","Duration":"3m 6s","ChapterTopicVideoID":9816,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9816.jpeg","UploadDate":"2017-08-01T05:11:52.3570000","DurationForVideoObject":"PT3M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.920","Text":"In this exercise we have an implicit function and actually I didn\u0027t say it here,"},{"Start":"00:07.920 ","End":"00:16.860","Text":"but I intended that z is the function of x and y implicitly in this relationship,"},{"Start":"00:16.860 ","End":"00:20.310","Text":"z is like a function of x and y."},{"Start":"00:20.310 ","End":"00:24.915","Text":"We want some level curves for this function."},{"Start":"00:24.915 ","End":"00:29.370","Text":"Level curves means that we set the value of z to some constant"},{"Start":"00:29.370 ","End":"00:35.850","Text":"k. If we set z is equal to k,"},{"Start":"00:35.850 ","End":"00:45.170","Text":"then we get that 2x minus 3y plus instead of z,"},{"Start":"00:45.170 ","End":"00:49.085","Text":"I put k, k squared equals 1."},{"Start":"00:49.085 ","End":"00:51.800","Text":"We want to rewrite this as the first step."},{"Start":"00:51.800 ","End":"00:55.000","Text":"I can put 3y on one side,"},{"Start":"00:55.000 ","End":"00:58.100","Text":"3y and everything else on the other side."},{"Start":"00:58.100 ","End":"00:59.930","Text":"Let\u0027s say 3y is on the right,"},{"Start":"00:59.930 ","End":"01:04.410","Text":"and then I\u0027d bring this over so I\u0027d get 2x,"},{"Start":"01:06.160 ","End":"01:13.300","Text":"that\u0027s right, 2x and then k squared minus 1."},{"Start":"01:15.410 ","End":"01:18.600","Text":"Then if I divide by 3,"},{"Start":"01:18.600 ","End":"01:20.730","Text":"have got y equals"},{"Start":"01:20.730 ","End":"01:31.085","Text":"2/3x plus k squared minus 1 over 3."},{"Start":"01:31.085 ","End":"01:36.460","Text":"This is the equation of a straight line where the slope is"},{"Start":"01:36.460 ","End":"01:45.285","Text":"the 2/3 and the y-intercept is this k squared minus 1 over 3."},{"Start":"01:45.285 ","End":"01:51.245","Text":"For example, if I took k equals 4,"},{"Start":"01:51.245 ","End":"01:56.035","Text":"then I would get that the slope of the line"},{"Start":"01:56.035 ","End":"01:59.140","Text":"is 2/3 but the slope is always going to be"},{"Start":"01:59.140 ","End":"02:02.080","Text":"2/3 so these are all going to be parallel lines,"},{"Start":"02:02.080 ","End":"02:03.880","Text":"whatever k is and"},{"Start":"02:03.880 ","End":"02:11.545","Text":"the y-intercept would be"},{"Start":"02:11.545 ","End":"02:15.610","Text":"4 squared minus 1/3 is 16,"},{"Start":"02:15.610 ","End":"02:19.475","Text":"minus 1/3 is 5."},{"Start":"02:19.475 ","End":"02:24.010","Text":"Actually that would work for both plus or minus"},{"Start":"02:24.010 ","End":"02:29.280","Text":"4 because I\u0027m squaring it so I\u0027ve got 2-level curves,"},{"Start":"02:29.280 ","End":"02:30.975","Text":"4 and minus 4."},{"Start":"02:30.975 ","End":"02:34.360","Text":"Now if I sketch this as well as a few others,"},{"Start":"02:34.360 ","End":"02:36.220","Text":"and here it is."},{"Start":"02:36.220 ","End":"02:40.850","Text":"The one that we illustrated was k plus or minus 4,"},{"Start":"02:40.850 ","End":"02:44.080","Text":"and indeed we saw that the y-intercept was"},{"Start":"02:44.080 ","End":"02:49.110","Text":"5 and the slope of all of these, just take it on trust,"},{"Start":"02:49.110 ","End":"02:55.200","Text":"this is 2/3 and different values of k gave different lines,"},{"Start":"02:55.200 ","End":"02:58.990","Text":"the plus or minus gives the same."},{"Start":"03:00.020 ","End":"03:03.360","Text":"These are more than a few level curves,"},{"Start":"03:03.360 ","End":"03:06.670","Text":"contours and so we\u0027re done."}],"ID":9772},{"Watched":false,"Name":"Exercise 7","Duration":"2m 11s","ChapterTopicVideoID":9817,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9817.jpeg","UploadDate":"2017-08-01T05:11:59.3630000","DurationForVideoObject":"PT2M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this exercise, we want to sketch some contours,"},{"Start":"00:02.820 ","End":"00:06.750","Text":"also known as level curves for the following implicit function."},{"Start":"00:06.750 ","End":"00:09.210","Text":"I guess I should have stated that z is the"},{"Start":"00:09.210 ","End":"00:12.960","Text":"variable and it\u0027s an implicit function of x and y."},{"Start":"00:12.960 ","End":"00:14.730","Text":"In fact, in this case,"},{"Start":"00:14.730 ","End":"00:18.975","Text":"I actually could isolate z. I could put everything to the right and divide by 4."},{"Start":"00:18.975 ","End":"00:24.285","Text":"The event z is the variable and it depends on x and y."},{"Start":"00:24.285 ","End":"00:32.880","Text":"Level curves are gotten by putting z equals k for some constant k. Let\u0027s write that,"},{"Start":"00:32.880 ","End":"00:36.190","Text":"z equals k are the level curves."},{"Start":"00:36.190 ","End":"00:39.155","Text":"If we set z equals k,"},{"Start":"00:39.155 ","End":"00:47.925","Text":"what we get is 4k plus 2y squared minus x equals 0."},{"Start":"00:47.925 ","End":"00:50.445","Text":"Now, in previous examples,"},{"Start":"00:50.445 ","End":"00:57.290","Text":"I got it so that y was a function of x and the parameter k. Here,"},{"Start":"00:57.290 ","End":"00:59.975","Text":"it\u0027s going to be easier if we isolate x."},{"Start":"00:59.975 ","End":"01:04.140","Text":"Let\u0027s write this as x equals,"},{"Start":"01:04.790 ","End":"01:10.780","Text":"let\u0027s say 2y squared plus 4k."},{"Start":"01:11.360 ","End":"01:16.070","Text":"We know that this is a sideways parabola,"},{"Start":"01:16.070 ","End":"01:23.050","Text":"x equals something positive y squared is going to be some parabola."},{"Start":"01:23.050 ","End":"01:26.600","Text":"It\u0027s going to start at this point here,"},{"Start":"01:26.600 ","End":"01:28.370","Text":"which will be 4k."},{"Start":"01:28.370 ","End":"01:30.845","Text":"Okay, this is the terrible little sketch."},{"Start":"01:30.845 ","End":"01:33.305","Text":"I\u0027ve got a decent 1 for you,"},{"Start":"01:33.305 ","End":"01:35.105","Text":"and here we are."},{"Start":"01:35.105 ","End":"01:37.010","Text":"For example, if k is 0,"},{"Start":"01:37.010 ","End":"01:39.595","Text":"we get x equals 2y squared."},{"Start":"01:39.595 ","End":"01:41.730","Text":"That\u0027s this 1."},{"Start":"01:41.730 ","End":"01:44.370","Text":"Just parabolas that open to the right,"},{"Start":"01:44.370 ","End":"01:45.390","Text":"x is a function of y,"},{"Start":"01:45.390 ","End":"01:47.010","Text":"instead of y being a function of x."},{"Start":"01:47.010 ","End":"01:50.810","Text":"Usually, it\u0027s y equals something x squared, and so on."},{"Start":"01:50.810 ","End":"01:54.754","Text":"If k was equal to, let\u0027s say 1,"},{"Start":"01:54.754 ","End":"01:59.020","Text":"then we have the same thing shifted 4 to the right,"},{"Start":"01:59.020 ","End":"02:01.470","Text":"because 4k would be 4 and here,"},{"Start":"02:01.470 ","End":"02:03.975","Text":"k would be minus 1 and so on."},{"Start":"02:03.975 ","End":"02:06.015","Text":"Here\u0027s 8, so k is 2,"},{"Start":"02:06.015 ","End":"02:07.695","Text":"and you get the idea."},{"Start":"02:07.695 ","End":"02:10.660","Text":"Okay, that\u0027s all."}],"ID":9773},{"Watched":false,"Name":"Exercise 8","Duration":"6m 53s","ChapterTopicVideoID":9818,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9818.jpeg","UploadDate":"2017-08-01T05:12:20.6830000","DurationForVideoObject":"PT6M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.755","Text":"In this exercise, we have an implicit function written here."},{"Start":"00:05.755 ","End":"00:09.460","Text":"We want to sketch some contours,"},{"Start":"00:09.460 ","End":"00:12.290","Text":"also known as level curves."},{"Start":"00:12.290 ","End":"00:19.150","Text":"It\u0027s implicit, but we understand that z is a function of x and y."},{"Start":"00:19.150 ","End":"00:20.740","Text":"Of course, in this case,"},{"Start":"00:20.740 ","End":"00:24.400","Text":"I could actually isolate z and say z equals y squared"},{"Start":"00:24.400 ","End":"00:25.960","Text":"minus 2x squared."},{"Start":"00:25.960 ","End":"00:28.855","Text":"But just for the practice with implicit functions,"},{"Start":"00:28.855 ","End":"00:30.775","Text":"I\u0027ll leave it like this."},{"Start":"00:30.775 ","End":"00:35.410","Text":"We get contours or level curves by assigning z"},{"Start":"00:35.410 ","End":"00:40.689","Text":"equaling some constant k, z equals k,"},{"Start":"00:40.689 ","End":"00:42.450","Text":"or in this case,"},{"Start":"00:42.450 ","End":"00:49.940","Text":"y squared equals 2x squared plus k."},{"Start":"00:49.940 ","End":"00:53.300","Text":"Now it turns out that we have to divide into 3 cases."},{"Start":"00:53.300 ","End":"00:55.970","Text":"The general shape changes,"},{"Start":"00:55.970 ","End":"00:58.880","Text":"k could be 0, positive or negative."},{"Start":"00:58.880 ","End":"01:01.070","Text":"Let\u0027s take case a,"},{"Start":"01:01.070 ","End":"01:03.450","Text":"where k equals 0,"},{"Start":"01:03.450 ","End":"01:06.470","Text":"and then, well, all right, I\u0027m ready now,"},{"Start":"01:06.470 ","End":"01:10.220","Text":"but later we\u0027ll have case b where k is positive"},{"Start":"01:10.220 ","End":"01:14.060","Text":"and case c will take k negative."},{"Start":"01:14.060 ","End":"01:16.730","Text":"Let\u0027s start off with case a."},{"Start":"01:16.730 ","End":"01:25.930","Text":"If k is 0, we get just that y squared equals 2x squared,"},{"Start":"01:25.930 ","End":"01:29.660","Text":"and what that gives us is,"},{"Start":"01:29.660 ","End":"01:31.940","Text":"if you take the square root of both sides,"},{"Start":"01:31.940 ","End":"01:35.030","Text":"the y is, well,"},{"Start":"01:35.030 ","End":"01:41.960","Text":"initially I would say the square root of 2 times x,"},{"Start":"01:41.960 ","End":"01:43.280","Text":"but it\u0027s not really x,"},{"Start":"01:43.280 ","End":"01:46.230","Text":"it\u0027s plus or minus x."},{"Start":"01:46.270 ","End":"01:50.690","Text":"Because the square root of x squared"},{"Start":"01:50.690 ","End":"01:52.700","Text":"could be plus or minus x,"},{"Start":"01:52.700 ","End":"01:56.120","Text":"or if you like, square this and you\u0027ll get,"},{"Start":"01:56.120 ","End":"01:58.895","Text":"in each case whether we take plus or minus,"},{"Start":"01:58.895 ","End":"02:00.785","Text":"this is what we get."},{"Start":"02:00.785 ","End":"02:04.595","Text":"Now each of these is a straight line through the origin,"},{"Start":"02:04.595 ","End":"02:08.585","Text":"so we\u0027ve got 2 straight lines through the origin."},{"Start":"02:08.585 ","End":"02:10.910","Text":"I\u0027ll draw a nice sketch later,"},{"Start":"02:10.910 ","End":"02:14.125","Text":"but meanwhile, just to give you a general idea,"},{"Start":"02:14.125 ","End":"02:18.080","Text":"1 of them has a positive slope of square root of 2,"},{"Start":"02:18.080 ","End":"02:21.980","Text":"y equals square root of 2x, and the other 1,"},{"Start":"02:21.980 ","End":"02:26.895","Text":"y equals minus square root of 2x, something like this."},{"Start":"02:26.895 ","End":"02:33.825","Text":"Here, k is equal to 0 and here k is 0,"},{"Start":"02:33.825 ","End":"02:39.375","Text":"but the equations are what\u0027s written here."},{"Start":"02:39.375 ","End":"02:44.560","Text":"Now let\u0027s do case b,"},{"Start":"02:44.710 ","End":"02:50.455","Text":"which is where k is positive if you don\u0027t need them here."},{"Start":"02:50.455 ","End":"02:57.080","Text":"In this case, let me bring the 2x squared to the other side,"},{"Start":"02:57.080 ","End":"02:59.225","Text":"and I\u0027ll also divide by k,"},{"Start":"02:59.225 ","End":"03:05.990","Text":"so I\u0027ve got y squared over k minus"},{"Start":"03:05.990 ","End":"03:15.185","Text":"2x squared over k is equal to 1."},{"Start":"03:15.185 ","End":"03:20.720","Text":"In general, when we have something of"},{"Start":"03:20.720 ","End":"03:26.710","Text":"the form y squared over something positive,"},{"Start":"03:26.710 ","End":"03:34.150","Text":"usually it\u0027s given as a squared minus x squared"},{"Start":"03:34.150 ","End":"03:35.680","Text":"over something positive,"},{"Start":"03:35.680 ","End":"03:36.950","Text":"in this case k over 2,"},{"Start":"03:36.950 ","End":"03:38.560","Text":"but let\u0027s call it b squared,"},{"Start":"03:38.560 ","End":"03:41.575","Text":"that will indicate positive equals 1."},{"Start":"03:41.575 ","End":"03:46.305","Text":"Then this is a hyperbola,"},{"Start":"03:46.305 ","End":"03:53.590","Text":"it looks something like this and this,"},{"Start":"03:53.590 ","End":"03:56.740","Text":"and there are actually asymptotes here and here."},{"Start":"03:56.740 ","End":"03:59.094","Text":"I\u0027ll give you a better picture in a moment,"},{"Start":"03:59.094 ","End":"04:02.400","Text":"just want to give you the general idea."},{"Start":"04:02.400 ","End":"04:06.840","Text":"The 3rd case, which will be,"},{"Start":"04:06.840 ","End":"04:09.135","Text":"let me see all the cases here,"},{"Start":"04:09.135 ","End":"04:13.155","Text":"k negative, and that\u0027s cool."},{"Start":"04:13.155 ","End":"04:16.715","Text":"Case c where k is negative,"},{"Start":"04:16.715 ","End":"04:19.670","Text":"then I\u0027m going to write the equation of,"},{"Start":"04:19.670 ","End":"04:21.980","Text":"just lost it, here it is,"},{"Start":"04:21.980 ","End":"04:33.220","Text":"we\u0027ll write it the other way that x squared, just a second,"},{"Start":"04:33.220 ","End":"04:37.935","Text":"write it as 2x squared over k"},{"Start":"04:37.935 ","End":"04:48.400","Text":"minus y squared over k. Now,"},{"Start":"04:48.400 ","End":"04:51.340","Text":"I\u0027m going to put a minus here also because I want"},{"Start":"04:51.340 ","End":"04:55.100","Text":"the denominator to be positive and k is negative,"},{"Start":"04:55.100 ","End":"05:02.350","Text":"then that will equal 1."},{"Start":"05:02.350 ","End":"05:05.545","Text":"You can check by multiplying out that this is correct."},{"Start":"05:05.545 ","End":"05:10.090","Text":"What I have is x squared and then"},{"Start":"05:10.090 ","End":"05:12.685","Text":"divided by something positive,"},{"Start":"05:12.685 ","End":"05:15.760","Text":"it\u0027s actually the denominator would be minus k over 2,"},{"Start":"05:15.760 ","End":"05:20.140","Text":"which is positive, I can write it as something like this,"},{"Start":"05:20.140 ","End":"05:22.270","Text":"and then minus, again,"},{"Start":"05:22.270 ","End":"05:23.380","Text":"minus k is positive,"},{"Start":"05:23.380 ","End":"05:25.060","Text":"I can call it b squared,"},{"Start":"05:25.060 ","End":"05:29.725","Text":"so y squared over b squared equals 1."},{"Start":"05:29.725 ","End":"05:32.690","Text":"When we have this equation,"},{"Start":"05:32.690 ","End":"05:35.030","Text":"this is also a hyperbola,"},{"Start":"05:35.030 ","End":"05:37.340","Text":"but in contrast to this 1,"},{"Start":"05:37.340 ","End":"05:38.930","Text":"it goes the other way."},{"Start":"05:38.930 ","End":"05:40.460","Text":"If these are the axes,"},{"Start":"05:40.460 ","End":"05:43.070","Text":"and I forgot arrows here,"},{"Start":"05:43.070 ","End":"05:45.590","Text":"y and x, of course,"},{"Start":"05:45.590 ","End":"05:51.170","Text":"it actually has asymptotes similar to this,"},{"Start":"05:51.170 ","End":"05:54.230","Text":"but the hyperbola has 2 branches"},{"Start":"05:54.230 ","End":"05:58.345","Text":"like before only this way round."},{"Start":"05:58.345 ","End":"06:02.720","Text":"This is when the values of k are negative,"},{"Start":"06:02.720 ","End":"06:07.010","Text":"values of k are positive and k equals 0, just 2 lines."},{"Start":"06:07.010 ","End":"06:09.890","Text":"Now I\u0027ll give you a nicer picture which I borrowed"},{"Start":"06:09.890 ","End":"06:11.725","Text":"from the Internet,"},{"Start":"06:11.725 ","End":"06:14.075","Text":"and here it is."},{"Start":"06:14.075 ","End":"06:17.060","Text":"We can see case a where k is 0,"},{"Start":"06:17.060 ","End":"06:19.580","Text":"that\u0027s these 2 lines there in black,"},{"Start":"06:19.580 ","End":"06:21.960","Text":"and there\u0027s something funny around here."},{"Start":"06:21.960 ","End":"06:26.450","Text":"These are supposed to be straight lines, k bigger than 0."},{"Start":"06:26.450 ","End":"06:29.015","Text":"I gave the general shape and that\u0027s these,"},{"Start":"06:29.015 ","End":"06:32.495","Text":"this is a purple 1 and this orange 1,"},{"Start":"06:32.495 ","End":"06:36.270","Text":"where k is 1 and k is 2."},{"Start":"06:36.620 ","End":"06:39.170","Text":"When k is negative,"},{"Start":"06:39.170 ","End":"06:40.360","Text":"we got the general shape,"},{"Start":"06:40.360 ","End":"06:42.050","Text":"and there\u0027s 2 examples here."},{"Start":"06:42.050 ","End":"06:44.810","Text":"This blue 1 where k is negative 2,"},{"Start":"06:44.810 ","End":"06:47.105","Text":"and this green 1 where k is negative 1."},{"Start":"06:47.105 ","End":"06:50.795","Text":"This all correspond, this is a bit nicer."},{"Start":"06:50.795 ","End":"06:54.000","Text":"That\u0027s it. We\u0027re done."}],"ID":9774},{"Watched":false,"Name":"Exercise 9","Duration":"4m 56s","ChapterTopicVideoID":9819,"CourseChapterTopicPlaylistID":57478,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9819.jpeg","UploadDate":"2017-08-01T05:12:34.9700000","DurationForVideoObject":"PT4M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this exercise, we\u0027re using the concept of a trace,"},{"Start":"00:05.580 ","End":"00:07.110","Text":"which I hope you remember."},{"Start":"00:07.110 ","End":"00:09.375","Text":"If not, you could go back to the tutorial."},{"Start":"00:09.375 ","End":"00:11.160","Text":"But I will say a few words."},{"Start":"00:11.160 ","End":"00:16.035","Text":"It\u0027s similar to a contour or a level curve."},{"Start":"00:16.035 ","End":"00:22.620","Text":"With contours, we had a function z of x, y could be implicit."},{"Start":"00:22.620 ","End":"00:28.050","Text":"We let z equals a constant and where it cut the curve,"},{"Start":"00:28.050 ","End":"00:31.920","Text":"then we got a contour or level curve."},{"Start":"00:31.920 ","End":"00:37.925","Text":"Z equals a constant is a plane parallel to the xy plane."},{"Start":"00:37.925 ","End":"00:40.175","Text":"Instead of the xy plane,"},{"Start":"00:40.175 ","End":"00:44.090","Text":"we could take the yz plane"},{"Start":"00:44.090 ","End":"00:48.960","Text":"and then get x as a constant or the xz plane"},{"Start":"00:48.960 ","End":"00:51.355","Text":"and get y equals a constant."},{"Start":"00:51.355 ","End":"00:55.700","Text":"Any plane parallel to any of the 3 coordinate planes"},{"Start":"00:55.700 ","End":"01:01.130","Text":"and some generalize it even further to take any kind of skew or diagonal plane"},{"Start":"01:01.130 ","End":"01:08.270","Text":"where it cuts the surface and get different curves."},{"Start":"01:08.270 ","End":"01:10.570","Text":"But we\u0027ll just take these 2 examples,"},{"Start":"01:10.570 ","End":"01:16.109","Text":"where x is some constant and y is some constant and this is the implicit function."},{"Start":"01:16.109 ","End":"01:19.400","Text":"Z is a function of x and y implicitly,"},{"Start":"01:19.400 ","End":"01:21.785","Text":"but it doesn\u0027t really matter which you take."},{"Start":"01:21.785 ","End":"01:27.260","Text":"In this case, we just intersect this surface with each of these separately"},{"Start":"01:27.260 ","End":"01:29.000","Text":"and see what we get."},{"Start":"01:29.000 ","End":"01:32.430","Text":"Let\u0027s start with a."},{"Start":"01:32.980 ","End":"01:38.990","Text":"In part a, we let our variable x equals a"},{"Start":"01:38.990 ","End":"01:47.850","Text":"and so we get 2a minus 3y plus z squared equals 1."},{"Start":"01:48.860 ","End":"01:53.345","Text":"It\u0027s easier for me to put y in terms of z,"},{"Start":"01:53.345 ","End":"01:55.925","Text":"though you could do it the other way around if it was easier."},{"Start":"01:55.925 ","End":"02:08.570","Text":"So y equals, 1/3z squared plus 2a minus 1 over 3."},{"Start":"02:08.570 ","End":"02:11.810","Text":"You can see this easiest if you put the 3y on 1 side,"},{"Start":"02:11.810 ","End":"02:13.280","Text":"everything else on the other side,"},{"Start":"02:13.280 ","End":"02:16.020","Text":"and then you just divide by 3."},{"Start":"02:16.180 ","End":"02:29.610","Text":"This will be a parabola y upwards and z horizontally."},{"Start":"02:29.610 ","End":"02:33.190","Text":"You can choose whichever way you want to draw this."},{"Start":"02:33.190 ","End":"02:38.960","Text":"Here I would take the vertical axis as being the y-axis"},{"Start":"02:38.960 ","End":"02:41.810","Text":"and the horizontal axis as being the z-axis,"},{"Start":"02:41.810 ","End":"02:43.175","Text":"but that\u0027s up to you."},{"Start":"02:43.175 ","End":"02:46.675","Text":"If it was just 1/3z squared,"},{"Start":"02:46.675 ","End":"02:51.065","Text":"then it would be something like this going through the origin."},{"Start":"02:51.065 ","End":"02:53.030","Text":"But when we add this constant"},{"Start":"02:53.030 ","End":"03:00.720","Text":"and we get different curves depending on the value of a,"},{"Start":"03:00.720 ","End":"03:03.160","Text":"they could go up or down."},{"Start":"03:03.160 ","End":"03:07.765","Text":"In part b, we get something similar."},{"Start":"03:07.765 ","End":"03:12.345","Text":"Here, we get that y is a constant b."},{"Start":"03:12.345 ","End":"03:16.600","Text":"Just coincidence that here b, here b, and a, and a, anyway."},{"Start":"03:17.090 ","End":"03:26.805","Text":"We get that 2x minus 3b plus z squared equals 1."},{"Start":"03:26.805 ","End":"03:31.114","Text":"Here, I\u0027d rather have x as a function of z."},{"Start":"03:31.114 ","End":"03:32.495","Text":"That would be easier,"},{"Start":"03:32.495 ","End":"03:35.310","Text":"then we\u0027ll get also a parabola."},{"Start":"03:35.450 ","End":"03:47.400","Text":"We get x equals minus a 1/2z squared plus 3b plus 1 over 2."},{"Start":"03:47.400 ","End":"03:57.130","Text":"This part here is the minus a 1/2z squared,"},{"Start":"03:57.350 ","End":"04:01.150","Text":"just get a little more space here,"},{"Start":"04:01.550 ","End":"04:08.445","Text":"is a facing down parabola because of the minus."},{"Start":"04:08.445 ","End":"04:12.245","Text":"Then it goes up or down depending on the constant."},{"Start":"04:12.245 ","End":"04:14.000","Text":"This is a general idea,"},{"Start":"04:14.000 ","End":"04:17.930","Text":"but I\u0027ll give you a nicer sketch."},{"Start":"04:17.930 ","End":"04:21.980","Text":"Here they are the nicer pictures in color."},{"Start":"04:21.980 ","End":"04:27.530","Text":"This is the family where x equals a"},{"Start":"04:27.530 ","End":"04:31.070","Text":"and the different values of a are indicated on each of the curves,"},{"Start":"04:31.070 ","End":"04:33.875","Text":"but they\u0027re all upward facing parabolas."},{"Start":"04:33.875 ","End":"04:37.640","Text":"This is the family for y equals b."},{"Start":"04:37.640 ","End":"04:40.060","Text":"The family of traces,"},{"Start":"04:40.060 ","End":"04:42.530","Text":"sketch more than a few,"},{"Start":"04:42.530 ","End":"04:46.574","Text":"we\u0027ve sketched about 5 of them and the different values of b"},{"Start":"04:46.574 ","End":"04:51.994","Text":"are marked on each of the parabolas and they\u0027re all downward facing."},{"Start":"04:51.994 ","End":"04:56.310","Text":"That\u0027s all and so we\u0027re done."}],"ID":9775}],"Thumbnail":null,"ID":57478},{"Name":"Vector Functions in 3D Coordinates System","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Vector Functions","Duration":"25m ","ChapterTopicVideoID":10481,"CourseChapterTopicPlaylistID":57479,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10481.jpeg","UploadDate":"2020-01-16T07:14:03.7970000","DurationForVideoObject":"PT25M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.309","Text":"Continuing with the topic of 3D coordinate system, 3D space."},{"Start":"00:05.309 ","End":"00:10.455","Text":"The next subtopic is vector functions."},{"Start":"00:10.455 ","End":"00:12.540","Text":"We\u0027ve seen this before,"},{"Start":"00:12.540 ","End":"00:15.150","Text":"but I\u0027m going to add a little bit more detail."},{"Start":"00:15.150 ","End":"00:24.429","Text":"Also, 3D really means 2D or any higher multi-dimensional but particularly 3D."},{"Start":"00:24.429 ","End":"00:28.815","Text":"Let me give an example of a vector function that you\u0027ve already seen."},{"Start":"00:28.815 ","End":"00:30.840","Text":"Well, not a specific example,"},{"Start":"00:30.840 ","End":"00:32.310","Text":"but the general form."},{"Start":"00:32.310 ","End":"00:40.110","Text":"In 2D, we had r of t is equal to 2 functions."},{"Start":"00:40.110 ","End":"00:43.140","Text":"1 for x, say f of t,"},{"Start":"00:43.140 ","End":"00:46.730","Text":"g of t. Or we could have written x of t,"},{"Start":"00:46.730 ","End":"00:49.415","Text":"y of t and that\u0027s in 2D."},{"Start":"00:49.415 ","End":"00:52.070","Text":"In 3D we would have something like"},{"Start":"00:52.070 ","End":"01:01.560","Text":"r position vector as a function of parameter t. Would be let\u0027s say f of t,"},{"Start":"01:01.560 ","End":"01:05.550","Text":"g of t, h of t,"},{"Start":"01:05.550 ","End":"01:08.435","Text":"and sometimes we would say x of t,"},{"Start":"01:08.435 ","End":"01:12.530","Text":"y of t, z of t. Similarly here sometimes x,"},{"Start":"01:12.530 ","End":"01:14.870","Text":"y, sometimes actual letters,"},{"Start":"01:14.870 ","End":"01:17.405","Text":"and in practice actual functions here."},{"Start":"01:17.405 ","End":"01:21.730","Text":"These would be vector functions of a parameter t of a variable"},{"Start":"01:21.730 ","End":"01:25.815","Text":"t. Promote t from parameter to variable,"},{"Start":"01:25.815 ","End":"01:27.515","Text":"so r is a function of t,"},{"Start":"01:27.515 ","End":"01:32.560","Text":"and later on we\u0027ll actually have vector functions of more than 1 variable."},{"Start":"01:32.560 ","End":"01:42.575","Text":"A word on notation like f of t and g of t are called the component functions."},{"Start":"01:42.575 ","End":"01:46.980","Text":"Likewise here these 3 are the component functions"},{"Start":"01:46.980 ","End":"01:52.410","Text":"separately and just like with the non-vector functions,"},{"Start":"01:52.410 ","End":"01:56.390","Text":"regular functions, the variable has a domain."},{"Start":"01:56.390 ","End":"01:59.850","Text":"We\u0027re going to talk about domain which is a set of"},{"Start":"01:59.850 ","End":"02:05.390","Text":"values which you can substitute t into for it to make sense."},{"Start":"02:05.390 ","End":"02:08.540","Text":"The domain applies in both the 2D,"},{"Start":"02:08.540 ","End":"02:11.165","Text":"3D, any dimensional case."},{"Start":"02:11.165 ","End":"02:13.650","Text":"Let\u0027s start with an example."},{"Start":"02:13.850 ","End":"02:21.530","Text":"In the example I\u0027m going to take a 3D case where r of t is equal to,"},{"Start":"02:21.530 ","End":"02:25.400","Text":"the first component function is going to be sine of t."},{"Start":"02:25.400 ","End":"02:34.445","Text":"The second component function will be natural log of 3 minus t,"},{"Start":"02:34.445 ","End":"02:42.970","Text":"and the third component function will be the square root of t plus 2."},{"Start":"02:43.610 ","End":"02:46.700","Text":"Let\u0027s see what the domain is."},{"Start":"02:46.700 ","End":"02:49.175","Text":"Just wrote that down, find the domain."},{"Start":"02:49.175 ","End":"02:52.370","Text":"We look at each of the 3 component functions,"},{"Start":"02:52.370 ","End":"02:58.850","Text":"for sine t the domain is all t. No restriction."},{"Start":"02:58.850 ","End":"03:02.030","Text":"For natural log of 3 minus t,"},{"Start":"03:02.030 ","End":"03:08.720","Text":"I need 3 minus t to be strictly positive so if 3 minus t is bigger than 0,"},{"Start":"03:08.720 ","End":"03:12.980","Text":"that makes it that t is less than 3."},{"Start":"03:12.980 ","End":"03:18.020","Text":"For the third component function,"},{"Start":"03:18.020 ","End":"03:22.985","Text":"we need what\u0027s under the square root sign to be bigger or equal to 0."},{"Start":"03:22.985 ","End":"03:25.895","Text":"If t plus 2 is bigger or equal to 0,"},{"Start":"03:25.895 ","End":"03:31.195","Text":"that gives me that t is bigger or equal to minus 2."},{"Start":"03:31.195 ","End":"03:33.980","Text":"Now if I put this and this, and this,"},{"Start":"03:33.980 ","End":"03:37.790","Text":"what I get from all of this is that this has no restrictions,"},{"Start":"03:37.790 ","End":"03:39.515","Text":"so we ignore that."},{"Start":"03:39.515 ","End":"03:42.500","Text":"Less than 3, bigger or equal to minus 2,"},{"Start":"03:42.500 ","End":"03:46.685","Text":"so the answer is minus 2 less than or equal to"},{"Start":"03:46.685 ","End":"03:52.025","Text":"t less than 3 and that would be the answer."},{"Start":"03:52.025 ","End":"03:55.234","Text":"Okay, example of domain."},{"Start":"03:55.234 ","End":"04:02.690","Text":"The next thing I want to do is look at the graph of vector functions."},{"Start":"04:02.690 ","End":"04:07.100","Text":"But it\u0027s going to be a lot easier if we do it in 2D,"},{"Start":"04:07.100 ","End":"04:08.930","Text":"because it\u0027s much easier to draw but"},{"Start":"04:08.930 ","End":"04:13.300","Text":"the same principle will hold and maybe later we\u0027ll do something in 3D."},{"Start":"04:13.300 ","End":"04:20.540","Text":"The example I\u0027m going to take to graph will be the following."},{"Start":"04:20.540 ","End":"04:23.510","Text":"Will take r of t,"},{"Start":"04:23.510 ","End":"04:26.735","Text":"and now we need 2 functions an x and a y function,"},{"Start":"04:26.735 ","End":"04:34.465","Text":"t minus 2 sine t, t squared."},{"Start":"04:34.465 ","End":"04:37.460","Text":"I deliberately chose a more difficult example."},{"Start":"04:37.460 ","End":"04:38.840","Text":"I\u0027m only going to give 1."},{"Start":"04:38.840 ","End":"04:41.780","Text":"They\u0027ll be more in the exercises."},{"Start":"04:41.780 ","End":"04:52.470","Text":"What you might do is draw a table just like we do with regular functions of 1 variable."},{"Start":"04:52.840 ","End":"04:56.705","Text":"Here we\u0027ll need a table with 3 columns,"},{"Start":"04:56.705 ","End":"05:02.309","Text":"we\u0027ll need a value of t,"},{"Start":"05:02.309 ","End":"05:07.100","Text":"but for r of t. Just to make it easier,"},{"Start":"05:07.100 ","End":"05:09.440","Text":"I\u0027ll call this function f of t,"},{"Start":"05:09.440 ","End":"05:11.540","Text":"and this function g of t,"},{"Start":"05:11.540 ","End":"05:15.470","Text":"and then I can do that here and say f of t,"},{"Start":"05:15.470 ","End":"05:22.030","Text":"g of t. But of course together these form r of t. I\u0027ll take"},{"Start":"05:22.030 ","End":"05:31.005","Text":"some values maybe I\u0027ll start 0 in the middle and work both ways."},{"Start":"05:31.005 ","End":"05:39.210","Text":"If t is 0, and I put an extra dividing line here just to make it neater."},{"Start":"05:39.440 ","End":"05:46.395","Text":"Let\u0027s see, f of t, is t minus 2 sine t. Sine of 0 is 0,"},{"Start":"05:46.395 ","End":"05:51.180","Text":"everything 0 and g of t, is also 0."},{"Start":"05:51.180 ","End":"05:53.865","Text":"It\u0027s actually 0 squared but it\u0027s 0."},{"Start":"05:53.865 ","End":"05:59.200","Text":"Now maybe I\u0027ll try t equals 1."},{"Start":"05:59.840 ","End":"06:02.370","Text":"This 1 is easier to compute,"},{"Start":"06:02.370 ","End":"06:08.630","Text":"t squared would be 1 and I actually want to take not a whole number,"},{"Start":"06:08.630 ","End":"06:16.015","Text":"say 2.5, so g of t would be 2.5 squared is 6.25."},{"Start":"06:16.015 ","End":"06:19.940","Text":"Now the f of t is a bit harder to compute and you need a calculator,"},{"Start":"06:19.940 ","End":"06:22.640","Text":"t is in radians so when t is 1,"},{"Start":"06:22.640 ","End":"06:26.485","Text":"we need 1 minus 2 sine 1."},{"Start":"06:26.485 ","End":"06:30.870","Text":"I\u0027ll look that up. I make it minus"},{"Start":"06:30.870 ","End":"06:39.900","Text":"0.683 rounded and for t equals 2.5,"},{"Start":"06:39.900 ","End":"06:44.830","Text":"1.303, I make it."},{"Start":"06:44.830 ","End":"06:47.630","Text":"Let\u0027s see. Let\u0027s take a couple of negative ones,"},{"Start":"06:47.630 ","End":"06:51.905","Text":"say minus 1 and minus 2.5."},{"Start":"06:51.905 ","End":"06:56.180","Text":"Well, the second component function is even,"},{"Start":"06:56.180 ","End":"06:57.875","Text":"so that will be the same."},{"Start":"06:57.875 ","End":"07:04.075","Text":"This will also be 1 and this will be 6.25 and the first component is odd,"},{"Start":"07:04.075 ","End":"07:06.360","Text":"t and sine t are odd functions."},{"Start":"07:06.360 ","End":"07:08.600","Text":"So just reverse the signs here,"},{"Start":"07:08.600 ","End":"07:16.770","Text":"0.683 and minus 1.303."},{"Start":"07:17.520 ","End":"07:24.055","Text":"Then if I plot these points,"},{"Start":"07:24.055 ","End":"07:26.635","Text":"these are the ones I want to plot,"},{"Start":"07:26.635 ","End":"07:33.040","Text":"like an x and a y, they\u0027ll get 5 points on the graph,"},{"Start":"07:33.040 ","End":"07:37.730","Text":"and I\u0027ll show you what the graph looks like."},{"Start":"07:38.250 ","End":"07:41.830","Text":"Here it is with 2 extra points."},{"Start":"07:41.830 ","End":"07:44.845","Text":"We took t equals 3 and minus 3 as well."},{"Start":"07:44.845 ","End":"07:48.925","Text":"I mean, this 0,0 would be that point."},{"Start":"07:48.925 ","End":"07:53.500","Text":"If I take minus 0.6831,"},{"Start":"07:53.500 ","End":"07:55.810","Text":"that would be this point."},{"Start":"07:55.810 ","End":"08:02.170","Text":"The height would be 1, this point 1.3 something,"},{"Start":"08:02.170 ","End":"08:05.215","Text":"and then 6.25 would be this."},{"Start":"08:05.215 ","End":"08:09.280","Text":"But the thing is that the graph is not the"},{"Start":"08:09.280 ","End":"08:14.185","Text":"same as the graph of a regular function because it\u0027s a vector function."},{"Start":"08:14.185 ","End":"08:15.940","Text":"Several ways of showing this,"},{"Start":"08:15.940 ","End":"08:20.605","Text":"but 1 way would be dotted lines with a vector to each point."},{"Start":"08:20.605 ","End":"08:22.840","Text":"As each of these points is not really a point,"},{"Start":"08:22.840 ","End":"08:27.955","Text":"it\u0027s a vector, and we show it as a position vector coming out of the origin."},{"Start":"08:27.955 ","End":"08:30.790","Text":"So that\u0027s the only difference really is that we just have to"},{"Start":"08:30.790 ","End":"08:34.390","Text":"remember they\u0027re not points, they\u0027re position vectors."},{"Start":"08:34.390 ","End":"08:39.970","Text":"This whole concept is very similar to parametric equations."},{"Start":"08:39.970 ","End":"08:42.310","Text":"In 2-dimensions, we would write,"},{"Start":"08:42.310 ","End":"08:46.000","Text":"for example, a set of parametric equations,"},{"Start":"08:46.000 ","End":"08:53.980","Text":"x equals 2 minus 2 sine t and y equals t squared,"},{"Start":"08:53.980 ","End":"08:56.800","Text":"and this would be a set of parametric equations."},{"Start":"08:56.800 ","End":"09:04.690","Text":"We\u0027d even call it maybe x of t and y of t. That would give the same graph,"},{"Start":"09:04.690 ","End":"09:06.460","Text":"but not as a vector,"},{"Start":"09:06.460 ","End":"09:08.290","Text":"just as a set of points."},{"Start":"09:08.290 ","End":"09:09.790","Text":"That\u0027s the only really difference."},{"Start":"09:09.790 ","End":"09:11.875","Text":"I mean, the actual curve is the same,"},{"Start":"09:11.875 ","End":"09:15.620","Text":"but we think about it as vectors rather than points."},{"Start":"09:15.990 ","End":"09:19.030","Text":"Perhaps I should emphasize that in general,"},{"Start":"09:19.030 ","End":"09:21.055","Text":"we don\u0027t draw the dotted lines."},{"Start":"09:21.055 ","End":"09:22.870","Text":"This was really done for"},{"Start":"09:22.870 ","End":"09:27.370","Text":"educational purposes to remind you that these are vectors rather than points."},{"Start":"09:27.370 ","End":"09:28.660","Text":"But from now on,"},{"Start":"09:28.660 ","End":"09:31.960","Text":"I\u0027m not going to be drawing the dotted lines."},{"Start":"09:31.960 ","End":"09:33.925","Text":"Let\u0027s move on from,"},{"Start":"09:33.925 ","End":"09:36.550","Text":"this was 2D, and now,"},{"Start":"09:36.550 ","End":"09:38.800","Text":"let\u0027s move on to 3D,"},{"Start":"09:38.800 ","End":"09:42.085","Text":"so let\u0027s take a vector function in 3D,"},{"Start":"09:42.085 ","End":"09:45.955","Text":"and I will take, let\u0027s say,"},{"Start":"09:45.955 ","End":"09:48.040","Text":"r of t. Now,"},{"Start":"09:48.040 ","End":"09:51.605","Text":"I need 3 component functions."},{"Start":"09:51.605 ","End":"09:55.140","Text":"Here, I\u0027ll put 2 minus 4t."},{"Start":"09:55.140 ","End":"10:00.930","Text":"The second component, minus 1 plus 5t,"},{"Start":"10:00.930 ","End":"10:09.890","Text":"and the third component make it 3 plus t. I forgot to write example."},{"Start":"10:12.960 ","End":"10:16.390","Text":"Now, in a way,"},{"Start":"10:16.390 ","End":"10:20.425","Text":"we\u0027ve studied this, and to see this,"},{"Start":"10:20.425 ","End":"10:27.760","Text":"I could rewrite this as 2 comma minus 1, 3 plus,"},{"Start":"10:27.760 ","End":"10:29.980","Text":"and then take the t out the brackets,"},{"Start":"10:29.980 ","End":"10:36.280","Text":"minus 4, 5, 1, this is 1t."},{"Start":"10:36.280 ","End":"10:41.230","Text":"Now, when we have this form of vector function,"},{"Start":"10:41.230 ","End":"10:44.275","Text":"we recognize immediately this is a straight line."},{"Start":"10:44.275 ","End":"10:45.820","Text":"If you\u0027re not sure,"},{"Start":"10:45.820 ","End":"10:51.834","Text":"go back to the straight line section in vectors."},{"Start":"10:51.834 ","End":"10:58.900","Text":"This is a point on the line and this is the direction vector of the line."},{"Start":"10:58.900 ","End":"11:01.120","Text":"Just to clarify it,"},{"Start":"11:01.120 ","End":"11:05.920","Text":"I\u0027ll bring a sketch for this too. Here\u0027s a diagram."},{"Start":"11:05.920 ","End":"11:07.960","Text":"Of course, you wouldn\u0027t be expected to do something"},{"Start":"11:07.960 ","End":"11:11.390","Text":"like this on a test or an exam or anything."},{"Start":"11:11.940 ","End":"11:15.580","Text":"This is the 0.2,"},{"Start":"11:15.580 ","End":"11:19.525","Text":"minus 13 from the same coordinates here,"},{"Start":"11:19.525 ","End":"11:25.105","Text":"and this minus 451 is what\u0027s in blue here."},{"Start":"11:25.105 ","End":"11:27.280","Text":"It\u0027s a direction vector for the line,"},{"Start":"11:27.280 ","End":"11:31.150","Text":"so we just take the line parallel to this vector and through this point,"},{"Start":"11:31.150 ","End":"11:32.395","Text":"that\u0027s the red line,"},{"Start":"11:32.395 ","End":"11:36.055","Text":"and that\u0027s the graph of the vector function."},{"Start":"11:36.055 ","End":"11:39.220","Text":"You have to remember these are vector function."},{"Start":"11:39.220 ","End":"11:41.679","Text":"Let\u0027s give another example,"},{"Start":"11:41.679 ","End":"11:44.020","Text":"not of a straight line."},{"Start":"11:44.020 ","End":"11:47.305","Text":"This time, I\u0027ll do the example here."},{"Start":"11:47.305 ","End":"11:51.130","Text":"Example also 3-dimensional."},{"Start":"11:51.130 ","End":"11:59.145","Text":"R as a function of t is going to be 2 cosine t,"},{"Start":"11:59.145 ","End":"12:03.135","Text":"2 sine t and 3."},{"Start":"12:03.135 ","End":"12:06.090","Text":"You could actually imagine this without me bringing you"},{"Start":"12:06.090 ","End":"12:09.495","Text":"a picture and let me leave the picture aside for the moment."},{"Start":"12:09.495 ","End":"12:13.840","Text":"If we just look at the first 2 parts,"},{"Start":"12:13.840 ","End":"12:16.870","Text":"the 2 cosine t and the 2 sine t,"},{"Start":"12:16.870 ","End":"12:25.075","Text":"you might recall that r cosine t and r sine t give us a circle with radius r,"},{"Start":"12:25.075 ","End":"12:26.875","Text":"in this case, 2."},{"Start":"12:26.875 ","End":"12:33.085","Text":"The first 2 in 2-dimensions as a circle of radius 2, that\u0027s the x and the y,"},{"Start":"12:33.085 ","End":"12:36.894","Text":"and now we\u0027ve bring in the z which is always 3,"},{"Start":"12:36.894 ","End":"12:44.140","Text":"so we raise that circle up to the plane where z equals 3 parallel to the x,"},{"Start":"12:44.140 ","End":"12:47.240","Text":"y plane, and now I\u0027ll bring the picture."},{"Start":"12:47.460 ","End":"12:52.510","Text":"Here we are as the picture."},{"Start":"12:52.510 ","End":"12:55.465","Text":"This would be the circle of radius 2."},{"Start":"12:55.465 ","End":"12:57.910","Text":"I mean, if I project it down onto the x,"},{"Start":"12:57.910 ","End":"13:00.790","Text":"y plane, it will be through these points,"},{"Start":"13:00.790 ","End":"13:06.760","Text":"2 minus 2, but its center is raised up 3 units."},{"Start":"13:06.760 ","End":"13:10.915","Text":"It\u0027s at this point on the z-axis where it\u0027s 3."},{"Start":"13:10.915 ","End":"13:15.410","Text":"That\u0027s another 3D example."},{"Start":"13:16.380 ","End":"13:21.760","Text":"You can easily see how you would generalize this via"},{"Start":"13:21.760 ","End":"13:26.515","Text":"circle parallel to any of the coordinate planes."},{"Start":"13:26.515 ","End":"13:32.260","Text":"If we put the cosine and the sine in the first and last place and a constant here,"},{"Start":"13:32.260 ","End":"13:36.430","Text":"then it would be parallel to the x,"},{"Start":"13:36.430 ","End":"13:37.810","Text":"z plane and so on."},{"Start":"13:37.810 ","End":"13:39.670","Text":"If you put the constant here,"},{"Start":"13:39.670 ","End":"13:44.200","Text":"you could get circles around any of"},{"Start":"13:44.200 ","End":"13:49.450","Text":"the axes and parallel to any of the coordinate planes."},{"Start":"13:49.450 ","End":"13:52.195","Text":"Let\u0027s take a variation of this."},{"Start":"13:52.195 ","End":"13:54.550","Text":"Suppose I didn\u0027t have a constant here."},{"Start":"13:54.550 ","End":"14:03.490","Text":"My next example will be then r of t is equal to and I won\u0027t reuse the 2 this time,"},{"Start":"14:03.490 ","End":"14:06.534","Text":"let\u0027s take 4 cosine t,"},{"Start":"14:06.534 ","End":"14:15.355","Text":"4 sine t. I haven\u0027t told you yet what\u0027s in the third component, the z component."},{"Start":"14:15.355 ","End":"14:21.370","Text":"But I do know already that if I look at it from above a projection,"},{"Start":"14:21.370 ","End":"14:22.870","Text":"it will be a circle on the x,"},{"Start":"14:22.870 ","End":"14:25.570","Text":"y plane with radius 4, but that\u0027s it."},{"Start":"14:25.570 ","End":"14:27.385","Text":"This time, I\u0027m going to put t,"},{"Start":"14:27.385 ","End":"14:33.460","Text":"not a constant, so that when t increases, the z increases."},{"Start":"14:33.460 ","End":"14:34.815","Text":"If you think about it,"},{"Start":"14:34.815 ","End":"14:38.705","Text":"I\u0027m going around in a circle when I look from above,"},{"Start":"14:38.705 ","End":"14:41.030","Text":"but the z is constantly increasing."},{"Start":"14:41.030 ","End":"14:43.295","Text":"What I\u0027m going to get is a helix,"},{"Start":"14:43.295 ","End":"14:45.780","Text":"and here is the picture."},{"Start":"14:45.780 ","End":"14:49.580","Text":"Here is the helix."},{"Start":"14:49.580 ","End":"14:51.890","Text":"It\u0027s like from above,"},{"Start":"14:51.890 ","End":"14:56.615","Text":"would look just like a circle of radius 4."},{"Start":"14:56.615 ","End":"15:00.995","Text":"But from the side, we can see that it\u0027s increasing or decreasing"},{"Start":"15:00.995 ","End":"15:06.290","Text":"depending which way you\u0027re taking t. This is not called a spiral,"},{"Start":"15:06.290 ","End":"15:10.860","Text":"it\u0027s called the helix, actually. Okay."},{"Start":"15:11.190 ","End":"15:14.965","Text":"On the same comment I made for circles applies here."},{"Start":"15:14.965 ","End":"15:20.875","Text":"If you want the helix not to spiral up around the z-axis but around another axis,"},{"Start":"15:20.875 ","End":"15:25.900","Text":"then you would put the t or a constant times t instead"},{"Start":"15:25.900 ","End":"15:31.255","Text":"of the z position and the y position or the x position and so on."},{"Start":"15:31.255 ","End":"15:35.740","Text":"I\u0027m going to move on to the next example and we\u0027ll start a new page."},{"Start":"15:35.740 ","End":"15:38.290","Text":"This will be a reverse example."},{"Start":"15:38.290 ","End":"15:46.690","Text":"I\u0027ll give you a specification and you\u0027ve got to find the vector function or equation."},{"Start":"15:46.690 ","End":"15:51.775","Text":"We\u0027ve already done in the section on vectors"},{"Start":"15:51.775 ","End":"15:59.350","Text":"the equation of a line given 2 points on the line."},{"Start":"15:59.350 ","End":"16:04.670","Text":"Let\u0027s do this in general and we want it as a vector function."},{"Start":"16:06.540 ","End":"16:12.040","Text":"I\u0027ll take the point P to equal,"},{"Start":"16:12.040 ","End":"16:14.080","Text":"and I\u0027m not doing it with specific numbers,"},{"Start":"16:14.080 ","End":"16:18.205","Text":"we\u0027ll do it in general as I said, x_1, y_1, z_1."},{"Start":"16:18.205 ","End":"16:26.965","Text":"The other point Q is going to be the point x_2, y_2, z_2."},{"Start":"16:26.965 ","End":"16:30.475","Text":"Not the same point, assume that."},{"Start":"16:30.475 ","End":"16:38.380","Text":"If you remember, what we do is to find a point in a position vector."},{"Start":"16:38.380 ","End":"16:39.940","Text":"Well, we\u0027ve already got a point."},{"Start":"16:39.940 ","End":"16:41.710","Text":"You can choose any one of these."},{"Start":"16:41.710 ","End":"16:43.660","Text":"I\u0027ll choose this one."},{"Start":"16:43.660 ","End":"16:45.790","Text":"Did I say position vector?"},{"Start":"16:45.790 ","End":"16:47.200","Text":"I meant direction vector."},{"Start":"16:47.200 ","End":"16:52.150","Text":"Direction vector for the line would be say, PQ,"},{"Start":"16:52.150 ","End":"16:54.505","Text":"which I will call V,"},{"Start":"16:54.505 ","End":"17:00.550","Text":"which will be just subtract the coordinates of this from the coordinates of this."},{"Start":"17:00.550 ","End":"17:05.080","Text":"What we have is x_2 minus x_1,"},{"Start":"17:05.080 ","End":"17:11.440","Text":"y_2 minus y_1, z_2 minus z_1."},{"Start":"17:11.440 ","End":"17:14.455","Text":"Direction vector point."},{"Start":"17:14.455 ","End":"17:19.735","Text":"The standard formula is that r, as a function of t,"},{"Start":"17:19.735 ","End":"17:28.134","Text":"is equal to the point but as a position vector from the origin."},{"Start":"17:28.134 ","End":"17:31.240","Text":"So it\u0027s x_1, y_1,"},{"Start":"17:31.240 ","End":"17:36.490","Text":"z_1 plus t times the direction vector,"},{"Start":"17:36.490 ","End":"17:41.155","Text":"which is copy-paste from here."},{"Start":"17:41.155 ","End":"17:48.530","Text":"It\u0027s actually possible to write this in a slightly neater or different form anyway."},{"Start":"17:48.540 ","End":"17:51.280","Text":"Suppose we just look at x_1."},{"Start":"17:51.280 ","End":"17:57.310","Text":"We have one from here and minus t x_1 from here,"},{"Start":"17:57.310 ","End":"18:01.780","Text":"so we\u0027ve got 1 minus t x_1."},{"Start":"18:01.780 ","End":"18:04.510","Text":"Before the x_1, look at y_1."},{"Start":"18:04.510 ","End":"18:09.370","Text":"Same thing, y_1, and then from here minus t y_1,"},{"Start":"18:09.370 ","End":"18:11.830","Text":"so it\u0027s also 1 minus t times y_1."},{"Start":"18:11.830 ","End":"18:17.635","Text":"In short, what we get is 1 minus t x_1, y_1, z_1."},{"Start":"18:17.635 ","End":"18:20.065","Text":"If we look at the x_2,"},{"Start":"18:20.065 ","End":"18:23.200","Text":"y_2, z_2, we just have t of each of these,"},{"Start":"18:23.200 ","End":"18:31.790","Text":"so plus t x_2, y_2, z_2."},{"Start":"18:32.040 ","End":"18:34.180","Text":"This is the way we do it."},{"Start":"18:34.180 ","End":"18:40.840","Text":"You take the vectors corresponding to these 2 points and you put a 1 minus t and a t,"},{"Start":"18:40.840 ","End":"18:43.660","Text":"and that\u0027s not so hard to remember."},{"Start":"18:43.660 ","End":"18:51.444","Text":"That gives us the equation of the line through PQ as t travels over all numbers."},{"Start":"18:51.444 ","End":"18:53.830","Text":"Now, if I said to you,"},{"Start":"18:53.830 ","End":"18:58.645","Text":"not the line PQ but the line segment PQ,"},{"Start":"18:58.645 ","End":"19:06.685","Text":"then we can get that by restricting t. Notice that if I put t equals 0,"},{"Start":"19:06.685 ","End":"19:16.975","Text":"then this is 0 and I just get the first point P. This corresponds to when t equals 0,"},{"Start":"19:16.975 ","End":"19:22.585","Text":"we get P or actually the vector OP."},{"Start":"19:22.585 ","End":"19:26.395","Text":"Sometimes we confuse points with their position vectors."},{"Start":"19:26.395 ","End":"19:29.989","Text":"If I set t equals 1,"},{"Start":"19:30.000 ","End":"19:35.380","Text":"then what I get is 1 minus 1 is 0,"},{"Start":"19:35.380 ","End":"19:36.400","Text":"so this part is 0."},{"Start":"19:36.400 ","End":"19:41.875","Text":"T is 1, so I get the other point x_2, y_2,"},{"Start":"19:41.875 ","End":"19:46.720","Text":"z_2, which is practically the same as Q,"},{"Start":"19:46.720 ","End":"19:48.370","Text":"so I get Q."},{"Start":"19:48.370 ","End":"19:51.085","Text":"It turns out that as we go from 0 to 1,"},{"Start":"19:51.085 ","End":"19:53.335","Text":"we go from P to Q."},{"Start":"19:53.335 ","End":"19:57.535","Text":"This is the equation of the line,"},{"Start":"19:57.535 ","End":"20:07.120","Text":"but if we restrict 0 less than or equal to t less than or equal to 1,"},{"Start":"20:07.120 ","End":"20:10.375","Text":"then we get the segment."},{"Start":"20:10.375 ","End":"20:13.570","Text":"I should have mentioned this is the line PQ,"},{"Start":"20:13.570 ","End":"20:15.925","Text":"and this is the segment PQ."},{"Start":"20:15.925 ","End":"20:21.370","Text":"If I restrict, k between 0 and 1 inclusive."},{"Start":"20:21.370 ","End":"20:26.980","Text":"This equation for the line between 2 points is useful,"},{"Start":"20:26.980 ","End":"20:30.290","Text":"so I\u0027m going to highlight it."},{"Start":"20:32.730 ","End":"20:37.090","Text":"Also, the restriction if we just want the segment and not the line,"},{"Start":"20:37.090 ","End":"20:44.215","Text":"then we\u0027d restrict t. Let\u0027s move on to the next topic."},{"Start":"20:44.215 ","End":"20:48.715","Text":"We can take vector functions of more than 1 variable."},{"Start":"20:48.715 ","End":"20:51.025","Text":"Let\u0027s start with 2 variables."},{"Start":"20:51.025 ","End":"20:54.145","Text":"Again, the vectors can be any dimension,"},{"Start":"20:54.145 ","End":"20:59.120","Text":"but we\u0027ll usually be using 2D or 3D."},{"Start":"21:00.090 ","End":"21:04.165","Text":"As an example, we\u0027ll take a vector function."},{"Start":"21:04.165 ","End":"21:06.370","Text":"I could use x and y,"},{"Start":"21:06.370 ","End":"21:09.550","Text":"or I could use s and t, any 2 letters."},{"Start":"21:09.550 ","End":"21:12.440","Text":"I\u0027ll use s and t this time."},{"Start":"21:15.930 ","End":"21:21.490","Text":"It\u0027s going to be a 3-dimensional vector, s,"},{"Start":"21:21.490 ","End":"21:27.939","Text":"t, s squared plus t squared."},{"Start":"21:27.939 ","End":"21:32.065","Text":"I don\u0027t think it\u0027s time for a reminder for the other notation,"},{"Start":"21:32.065 ","End":"21:40.300","Text":"but I would say it\u0027s s times i plus t times j."},{"Start":"21:40.300 ","End":"21:42.700","Text":"These are the standard-basis vectors i, j,"},{"Start":"21:42.700 ","End":"21:46.615","Text":"and k plus s squared plus t squared"},{"Start":"21:46.615 ","End":"21:51.340","Text":"k. Just reminding you that\u0027s another way of writing the same thing."},{"Start":"21:51.340 ","End":"21:56.275","Text":"I want to somehow describe the surface, what is this."},{"Start":"21:56.275 ","End":"21:59.560","Text":"Well, first of all, I gave it away that it\u0027s a surface."},{"Start":"21:59.560 ","End":"22:02.499","Text":"In general, when you have 2 variables,"},{"Start":"22:02.499 ","End":"22:03.685","Text":"it will be a surface,"},{"Start":"22:03.685 ","End":"22:06.925","Text":"1 variable is a line or curve."},{"Start":"22:06.925 ","End":"22:10.900","Text":"The easiest way to see this would be to let say,"},{"Start":"22:10.900 ","End":"22:14.440","Text":"write it in parametric form."},{"Start":"22:14.440 ","End":"22:17.305","Text":"We could say that x equals s,"},{"Start":"22:17.305 ","End":"22:23.650","Text":"y equals t, z equals s squared plus t squared."},{"Start":"22:23.650 ","End":"22:25.960","Text":"From this we can see that, well,"},{"Start":"22:25.960 ","End":"22:30.190","Text":"x and y are not restricted just because s and t are not restricted,"},{"Start":"22:30.190 ","End":"22:39.205","Text":"but that z is equal to x squared plus y squared and that really says all that this says."},{"Start":"22:39.205 ","End":"22:44.320","Text":"From the chapter on Qualtrics sections,"},{"Start":"22:44.320 ","End":"22:47.170","Text":"this turns out to be elliptical."},{"Start":"22:47.170 ","End":"22:53.035","Text":"Actually, a circular paraboloid."},{"Start":"22:53.035 ","End":"22:59.410","Text":"Actually, this example leads me to a certain generalization that in general,"},{"Start":"22:59.410 ","End":"23:04.015","Text":"if I have z as a function of x and y,"},{"Start":"23:04.015 ","End":"23:07.105","Text":"let\u0027s call it g of x and y,"},{"Start":"23:07.105 ","End":"23:14.005","Text":"I can always write this as a vector function of 2 variables."},{"Start":"23:14.005 ","End":"23:17.815","Text":"In this case, a 3-dimensional vector."},{"Start":"23:17.815 ","End":"23:21.835","Text":"I can just do a reverse of what I did here."},{"Start":"23:21.835 ","End":"23:28.360","Text":"I can write the vector r as a function of,"},{"Start":"23:28.360 ","End":"23:36.950","Text":"I could keep x and y or I could switch it to s and t that this equals s,"},{"Start":"23:37.800 ","End":"23:46.960","Text":"t, and instead of x squared plus y squared they are generalized function g of s"},{"Start":"23:46.960 ","End":"23:57.210","Text":"and t. All this is in 3 dimensions and the 2 variables."},{"Start":"23:57.210 ","End":"24:01.770","Text":"The same concept actually works also if I have,"},{"Start":"24:01.770 ","End":"24:06.400","Text":"let\u0027s say, y is some function of x."},{"Start":"24:06.400 ","End":"24:08.110","Text":"In a simpler case,"},{"Start":"24:08.110 ","End":"24:15.475","Text":"I can write this as a 2-dimensional vector function this time of 1 variable."},{"Start":"24:15.475 ","End":"24:24.160","Text":"I can write that as vector function r of t is equal to t,"},{"Start":"24:24.160 ","End":"24:29.485","Text":"f of t. Because if we do what we did before,"},{"Start":"24:29.485 ","End":"24:36.130","Text":"this really just says that x equals t and y equals f of t,"},{"Start":"24:36.130 ","End":"24:38.380","Text":"and because x and t could be anything,"},{"Start":"24:38.380 ","End":"24:40.270","Text":"we could just says y equals f of x."},{"Start":"24:40.270 ","End":"24:42.610","Text":"What I\u0027m saying is that a function of"},{"Start":"24:42.610 ","End":"24:46.630","Text":"2 variables or a function of 1 variable can be always"},{"Start":"24:46.630 ","End":"24:54.085","Text":"written as a vector function of dimension 1 more than the number of variables."},{"Start":"24:54.085 ","End":"24:56.980","Text":"I think that\u0027s about it."},{"Start":"24:56.980 ","End":"25:00.920","Text":"That conclude this clip."}],"ID":10843},{"Watched":false,"Name":"Exercise 1","Duration":"3m 22s","ChapterTopicVideoID":9826,"CourseChapterTopicPlaylistID":57479,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9826.jpeg","UploadDate":"2017-08-06T04:29:22.8100000","DurationForVideoObject":"PT3M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"In this exercise,"},{"Start":"00:02.010 ","End":"00:03.840","Text":"which is really 2 exercises,"},{"Start":"00:03.840 ","End":"00:05.970","Text":"we have to find the domain"},{"Start":"00:05.970 ","End":"00:08.190","Text":"of the following vector functions."},{"Start":"00:08.190 ","End":"00:10.560","Text":"I\u0027ve given 1 example in 3D"},{"Start":"00:10.560 ","End":"00:12.360","Text":"and 1 example in 2D"},{"Start":"00:12.360 ","End":"00:15.210","Text":"using the angular brackets notation."},{"Start":"00:15.210 ","End":"00:17.070","Text":"Remember, we can either use angular"},{"Start":"00:17.070 ","End":"00:21.120","Text":"brackets for vectors or the IJK notation."},{"Start":"00:21.120 ","End":"00:23.220","Text":"Let\u0027s start with part a,"},{"Start":"00:23.220 ","End":"00:25.920","Text":"and let\u0027s see what the domain is."},{"Start":"00:25.920 ","End":"00:28.050","Text":"Well, it has 3 components,"},{"Start":"00:28.050 ","End":"00:29.520","Text":"that is 3D and all 3 of them"},{"Start":"00:29.520 ","End":"00:30.854","Text":"have to be defined."},{"Start":"00:30.854 ","End":"00:32.970","Text":"The first part, first component"},{"Start":"00:32.970 ","End":"00:35.265","Text":"is t^3 minus 3t is a polynomial,"},{"Start":"00:35.265 ","End":"00:36.300","Text":"defined everywhere,"},{"Start":"00:36.300 ","End":"00:38.325","Text":"so no problem, no restriction."},{"Start":"00:38.325 ","End":"00:40.390","Text":"Here, we have a square root."},{"Start":"00:40.390 ","End":"00:41.810","Text":"What\u0027s under the square root"},{"Start":"00:41.810 ","End":"00:43.655","Text":"has to be non-negative."},{"Start":"00:43.655 ","End":"00:45.770","Text":"We get that t minus 1"},{"Start":"00:45.770 ","End":"00:47.825","Text":"has to be bigger or equal to 0,"},{"Start":"00:47.825 ","End":"00:49.130","Text":"which just means that"},{"Start":"00:49.130 ","End":"00:51.565","Text":"t is bigger or equal to 1."},{"Start":"00:51.565 ","End":"00:53.790","Text":"As for the last part,"},{"Start":"00:53.790 ","End":"00:55.700","Text":"we see that the only problem"},{"Start":"00:55.700 ","End":"00:58.640","Text":"could be a 0 in the denominator"},{"Start":"00:58.640 ","End":"01:02.210","Text":"and that means that t minus 3"},{"Start":"01:02.210 ","End":"01:04.130","Text":"must not be 0."},{"Start":"01:04.130 ","End":"01:07.880","Text":"Therefore, t must not be 3."},{"Start":"01:07.880 ","End":"01:10.310","Text":"If I take this condition"},{"Start":"01:10.310 ","End":"01:12.020","Text":"and this condition together,"},{"Start":"01:12.020 ","End":"01:15.170","Text":"so the answer will be t"},{"Start":"01:15.170 ","End":"01:17.360","Text":"bigger or equal to 1,"},{"Start":"01:17.360 ","End":"01:23.145","Text":"and t not equal to 3."},{"Start":"01:23.145 ","End":"01:27.540","Text":"You could say from 1 to 3,"},{"Start":"01:27.540 ","End":"01:29.480","Text":"and from 3 to infinity."},{"Start":"01:29.480 ","End":"01:31.950","Text":"Now, we\u0027ll just leave it like that."},{"Start":"01:32.480 ","End":"01:36.440","Text":"In part b, we only have 2 things to check."},{"Start":"01:36.440 ","End":"01:38.780","Text":"The first component has a square root,"},{"Start":"01:38.780 ","End":"01:39.830","Text":"so we have to make sure"},{"Start":"01:39.830 ","End":"01:41.780","Text":"that\u0027s bigger or equal to 0."},{"Start":"01:41.780 ","End":"01:45.485","Text":"So t plus 2 is bigger or equal to 0,"},{"Start":"01:45.485 ","End":"01:48.410","Text":"which gives us that t is bigger"},{"Start":"01:48.410 ","End":"01:51.215","Text":"or equal to minus 2."},{"Start":"01:51.215 ","End":"01:52.820","Text":"In the second part,"},{"Start":"01:52.820 ","End":"01:54.740","Text":"we have a natural logarithm"},{"Start":"01:54.740 ","End":"01:57.890","Text":"and natural logarithm can only"},{"Start":"01:57.890 ","End":"02:00.290","Text":"be taken off a positive quantity."},{"Start":"02:00.290 ","End":"02:05.180","Text":"We also have that 9 minus t squared"},{"Start":"02:05.180 ","End":"02:08.815","Text":"has to be strictly bigger than 0."},{"Start":"02:08.815 ","End":"02:11.480","Text":"Now, if we solve where this is 0,"},{"Start":"02:11.480 ","End":"02:13.915","Text":"we see it\u0027s plus or minus 3,"},{"Start":"02:13.915 ","End":"02:16.595","Text":"if we are to sketch it for you."},{"Start":"02:16.595 ","End":"02:19.430","Text":"If we do a check with sample values,"},{"Start":"02:19.430 ","End":"02:20.720","Text":"I\u0027m not going to do the whole thing."},{"Start":"02:20.720 ","End":"02:24.590","Text":"If we check this is 3 and this is minus 3,"},{"Start":"02:24.590 ","End":"02:26.645","Text":"it\u0027s an upside down parabola."},{"Start":"02:26.645 ","End":"02:29.810","Text":"We can see that bigger than 0"},{"Start":"02:29.810 ","End":"02:32.180","Text":"is going to be in the middle part,"},{"Start":"02:32.180 ","End":"02:37.730","Text":"but not including the 3 and the minus 3."},{"Start":"02:37.730 ","End":"02:39.992","Text":"his gives us minus 3"},{"Start":"02:39.992 ","End":"02:47.355","Text":"less than x less than 3."},{"Start":"02:47.355 ","End":"02:51.130","Text":"Now, to take these 2 conditions together,"},{"Start":"02:54.080 ","End":"02:57.635","Text":"if it\u0027s bigger or equal to minus 2,"},{"Start":"02:57.635 ","End":"03:01.200","Text":"it\u0027s automatically bigger than minus 3."},{"Start":"03:01.200 ","End":"03:02.870","Text":"But I still have this restriction."},{"Start":"03:02.870 ","End":"03:05.810","Text":"If I take this together with this,"},{"Start":"03:05.810 ","End":"03:07.910","Text":"you could do it with a sketch on the number line,"},{"Start":"03:07.910 ","End":"03:10.400","Text":"or you think we can see that this gives us"},{"Start":"03:10.400 ","End":"03:15.095","Text":"that t has to be bigger or equal to minus 2,"},{"Start":"03:15.095 ","End":"03:17.690","Text":"but has to be less than 3,"},{"Start":"03:17.690 ","End":"03:20.675","Text":"and then it will satisfy both this and this."},{"Start":"03:20.675 ","End":"03:23.070","Text":"That\u0027s it."}],"ID":9695},{"Watched":false,"Name":"Exercise 2","Duration":"3m 53s","ChapterTopicVideoID":9827,"CourseChapterTopicPlaylistID":57479,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9827.jpeg","UploadDate":"2017-08-06T04:29:36.6070000","DurationForVideoObject":"PT3M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"In this exercise well,"},{"Start":"00:02.250 ","End":"00:03.725","Text":"its 2 parts, each of them,"},{"Start":"00:03.725 ","End":"00:09.600","Text":"we have a 2-dimensional vector function in parametric form and we"},{"Start":"00:09.600 ","End":"00:12.600","Text":"want to sketch both of them but with different methods in Part A"},{"Start":"00:12.600 ","End":"00:16.140","Text":"by making a table of values and in Part B,"},{"Start":"00:16.140 ","End":"00:19.960","Text":"by eliminating t and getting an equation in x and y."},{"Start":"00:19.960 ","End":"00:25.574","Text":"Let\u0027s start with Part A. I drew a grid for the table,"},{"Start":"00:25.574 ","End":"00:28.680","Text":"and we\u0027re going to put here t and here x,"},{"Start":"00:28.680 ","End":"00:30.180","Text":"which is the first component,"},{"Start":"00:30.180 ","End":"00:32.385","Text":"and here y, the second component."},{"Start":"00:32.385 ","End":"00:34.995","Text":"Let\u0027s take 0, 1,"},{"Start":"00:34.995 ","End":"00:37.895","Text":"2, 3 for a start."},{"Start":"00:37.895 ","End":"00:42.170","Text":"When t is 0, well, x is 40."},{"Start":"00:42.170 ","End":"00:47.345","Text":"We can just go down this way and say 4 times 0 is 0,"},{"Start":"00:47.345 ","End":"00:51.660","Text":"4 times 1, 4 times 2, 4 times 3."},{"Start":"00:51.660 ","End":"00:54.060","Text":"Here we have 10 minus 2t,"},{"Start":"00:54.060 ","End":"00:59.250","Text":"so 10 minus 0, 10 minus 2."},{"Start":"00:59.250 ","End":"01:01.110","Text":"Then here we\u0027ll get 6,"},{"Start":"01:01.110 ","End":"01:03.370","Text":"here we\u0027ll get 4."},{"Start":"01:03.370 ","End":"01:06.880","Text":"Maybe take some negative values also."},{"Start":"01:06.880 ","End":"01:11.745","Text":"Maybe minus 1, minus 2, minus 3."},{"Start":"01:11.745 ","End":"01:15.710","Text":"4t will be minus 4, minus 8,"},{"Start":"01:15.710 ","End":"01:21.945","Text":"minus 12, 10 minus 2t so it\u0027ll be 10 plus 2."},{"Start":"01:21.945 ","End":"01:24.915","Text":"Anyway, we\u0027ll get 12 and see the differences are 2,"},{"Start":"01:24.915 ","End":"01:27.360","Text":"here 14 and 16."},{"Start":"01:27.360 ","End":"01:35.240","Text":"Then finally, we get a graph paper and just sketch these points,"},{"Start":"01:35.240 ","End":"01:36.725","Text":"these pairs, x, y,"},{"Start":"01:36.725 ","End":"01:40.110","Text":"I\u0027ll just cut to the chase show you what it looks like."},{"Start":"01:40.160 ","End":"01:48.180","Text":"Here it is. Actually, I took t from minus 4 and up to 4."},{"Start":"01:48.180 ","End":"01:53.570","Text":"For each t, we plot the point and this is the idea."},{"Start":"01:53.570 ","End":"01:57.520","Text":"Now, let\u0027s move on to Part B."},{"Start":"01:57.520 ","End":"02:01.220","Text":"In Part B, we\u0027re going to use a different technique."},{"Start":"02:01.220 ","End":"02:09.270","Text":"In Part B, I can write the equation parametrically as x equals"},{"Start":"02:09.270 ","End":"02:18.330","Text":"t plus 1 and the y is equal to 1/4 t squared plus 3."},{"Start":"02:18.330 ","End":"02:23.975","Text":"Now what I want to do is eliminate t. What I can do is from the first equation,"},{"Start":"02:23.975 ","End":"02:27.539","Text":"I could get t equals x minus 1,"},{"Start":"02:27.539 ","End":"02:31.190","Text":"and then substitute that in the second equation."},{"Start":"02:31.190 ","End":"02:33.815","Text":"This, I\u0027ll substitute in here."},{"Start":"02:33.815 ","End":"02:37.385","Text":"We\u0027ll get that y equals 1/4,"},{"Start":"02:37.385 ","End":"02:42.855","Text":"t is x minus 1 squared plus 3."},{"Start":"02:42.855 ","End":"02:45.720","Text":"If we open the brackets, lets see,"},{"Start":"02:45.720 ","End":"02:48.030","Text":"this is x squared minus 2x plus 1,"},{"Start":"02:48.030 ","End":"02:49.635","Text":"but if we divide by 4,"},{"Start":"02:49.635 ","End":"02:53.040","Text":"it\u0027s 1/4x squared minus 2x,"},{"Start":"02:53.040 ","End":"02:55.860","Text":"so it\u0027s minus 1/2x and plus 1,"},{"Start":"02:55.860 ","End":"02:59.280","Text":"so it\u0027s plus 1/4 plus 3."},{"Start":"02:59.280 ","End":"03:02.800","Text":"I\u0027ll change this 1/4 to 3 and 1/4."},{"Start":"03:05.240 ","End":"03:09.270","Text":"This is a parabola facing up."},{"Start":"03:09.270 ","End":"03:11.450","Text":"Here\u0027s what it looks like."},{"Start":"03:11.450 ","End":"03:14.420","Text":"You could sketch it using standard investigation of"},{"Start":"03:14.420 ","End":"03:17.825","Text":"function techniques or properties of the parabola."},{"Start":"03:17.825 ","End":"03:20.615","Text":"We could find the intersection with the xs."},{"Start":"03:20.615 ","End":"03:23.345","Text":"It never intersects the x-axis,"},{"Start":"03:23.345 ","End":"03:26.135","Text":"it intersects the y-axis at 3 and 1/4."},{"Start":"03:26.135 ","End":"03:28.325","Text":"You can see this is 3 and 1/4."},{"Start":"03:28.325 ","End":"03:31.520","Text":"The minimum, you can differentiate and set to 0,"},{"Start":"03:31.520 ","End":"03:33.785","Text":"or remember minus b over 2a,"},{"Start":"03:33.785 ","End":"03:36.560","Text":"which is 1/2 over twice 1/4,"},{"Start":"03:36.560 ","End":"03:41.370","Text":"which is 1, and the minimum does occur at 1, it faces up."},{"Start":"03:41.510 ","End":"03:46.655","Text":"Here also in case you want to do it with a table method, some values."},{"Start":"03:46.655 ","End":"03:51.425","Text":"Anyway, I\u0027m not going to spend too much time on that, just the idea."},{"Start":"03:51.425 ","End":"03:54.630","Text":"We\u0027re done with this exercise."}],"ID":9696},{"Watched":false,"Name":"Exercise 3","Duration":"3m 20s","ChapterTopicVideoID":9828,"CourseChapterTopicPlaylistID":57479,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9828.jpeg","UploadDate":"2017-08-06T04:29:46.9970000","DurationForVideoObject":"PT3M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"This exercise is 2-in-1."},{"Start":"00:03.360 ","End":"00:10.380","Text":"In each case, we have a parametric vector function with parameter t."},{"Start":"00:10.380 ","End":"00:13.320","Text":"Typically, when you have 1 parameter t,"},{"Start":"00:13.320 ","End":"00:14.850","Text":"it traces out a curve,"},{"Start":"00:14.850 ","End":"00:18.960","Text":"and we have to describe what the curve looks like just to identify the graph,"},{"Start":"00:18.960 ","End":"00:21.885","Text":"but not to actually sketch it."},{"Start":"00:21.885 ","End":"00:25.660","Text":"I\u0027ll start with part a."},{"Start":"00:25.910 ","End":"00:29.745","Text":"In part a, first of all,"},{"Start":"00:29.745 ","End":"00:42.060","Text":"I could write it with x, y, z as follows: x equals 2 cosine 3t, y equals sine 3t."},{"Start":"00:42.060 ","End":"00:44.550","Text":"Perhaps best to write the brackets,"},{"Start":"00:44.550 ","End":"00:48.115","Text":"and z is a constant, 4."},{"Start":"00:48.115 ","End":"00:53.870","Text":"Now, what I\u0027m going to do is eliminate t from these 2."},{"Start":"00:53.870 ","End":"00:57.340","Text":"I see there\u0027s a cosine and sine of the same angle."},{"Start":"00:57.340 ","End":"01:03.770","Text":"So I\u0027m going to use the famous identity that cosine squared plus sine squared is 1."},{"Start":"01:03.770 ","End":"01:06.500","Text":"So If I take x squared,"},{"Start":"01:06.500 ","End":"01:09.865","Text":"I\u0027ll get 4 cosine squared 3t."},{"Start":"01:09.865 ","End":"01:13.269","Text":"If I take x squared over 4,"},{"Start":"01:13.269 ","End":"01:14.835","Text":"I\u0027ll write it underneath,"},{"Start":"01:14.835 ","End":"01:18.000","Text":"this will be cosine squared 3t."},{"Start":"01:18.000 ","End":"01:20.550","Text":"If I take y squared,"},{"Start":"01:20.550 ","End":"01:24.450","Text":"that will be sine squared 3t."},{"Start":"01:24.450 ","End":"01:27.570","Text":"This plus this will equal 1,"},{"Start":"01:27.570 ","End":"01:33.225","Text":"and I can actually write this as this over 1."},{"Start":"01:33.225 ","End":"01:42.395","Text":"In general, when we have x squared over a squared plus y squared over b squared equals 1,"},{"Start":"01:42.395 ","End":"01:45.890","Text":"then this is the equation of an ellipse with"},{"Start":"01:45.890 ","End":"01:51.845","Text":"the horizontal axis of length a and the vertical axis of length b."},{"Start":"01:51.845 ","End":"01:54.620","Text":"In this case, a is 2 and b is 1."},{"Start":"01:54.620 ","End":"01:57.030","Text":"This is 2 squared, this is 1 squared."},{"Start":"01:57.050 ","End":"02:00.200","Text":"It\u0027s not an ellipse, though,"},{"Start":"02:00.200 ","End":"02:04.820","Text":"because it\u0027s in 3D and z is equal to 4."},{"Start":"02:04.820 ","End":"02:06.080","Text":"Well, it is an ellipse,"},{"Start":"02:06.080 ","End":"02:08.270","Text":"but it\u0027s hovering at height 4."},{"Start":"02:08.270 ","End":"02:11.390","Text":"So I can say that it is"},{"Start":"02:11.390 ","End":"02:20.700","Text":"an ellipse suspended 4 units above the xy-plane."},{"Start":"02:20.700 ","End":"02:23.205","Text":"I won\u0027t write that down, just said it."},{"Start":"02:23.205 ","End":"02:26.070","Text":"Now let\u0027s do Part B."},{"Start":"02:26.070 ","End":"02:29.195","Text":"Part B should look very familiar."},{"Start":"02:29.195 ","End":"02:32.540","Text":"If you think back to the equation of lines,"},{"Start":"02:32.540 ","End":"02:34.310","Text":"I can break it down."},{"Start":"02:34.310 ","End":"02:38.700","Text":"We can write this as r of t equals,"},{"Start":"02:38.700 ","End":"02:43.500","Text":"I can write it as 5, 3 minus 4,"},{"Start":"02:43.500 ","End":"02:46.565","Text":"I\u0027m getting the coefficients from here,"},{"Start":"02:46.565 ","End":"02:51.740","Text":"plus t times, and then I\u0027ve got 2,"},{"Start":"02:51.740 ","End":"02:55.500","Text":"minus 6, minus 1."},{"Start":"02:57.320 ","End":"03:01.550","Text":"We know that this is a line that passes through,"},{"Start":"03:01.550 ","End":"03:03.245","Text":"this is the point,"},{"Start":"03:03.245 ","End":"03:06.784","Text":"this is the position vector of the point anyway,"},{"Start":"03:06.784 ","End":"03:12.540","Text":"through this point p and with this direction vector v,"},{"Start":"03:13.600 ","End":"03:19.470","Text":"line passing through this point and parallel to this. That\u0027s it."}],"ID":9697},{"Watched":false,"Name":"Exercise 4","Duration":"2m 51s","ChapterTopicVideoID":9829,"CourseChapterTopicPlaylistID":57479,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9829.jpeg","UploadDate":"2017-08-06T04:29:56.4530000","DurationForVideoObject":"PT2M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"In this exercise, we\u0027re given a pair of points."},{"Start":"00:03.585 ","End":"00:06.120","Text":"In part a, we\u0027re given points in 2D,"},{"Start":"00:06.120 ","End":"00:09.165","Text":"and in part b we\u0027re given a 3D example."},{"Start":"00:09.165 ","End":"00:16.130","Text":"We want the vector equation of the line segment between the two points,"},{"Start":"00:16.130 ","End":"00:17.180","Text":"not the whole line,"},{"Start":"00:17.180 ","End":"00:19.565","Text":"just the segment from A to B here,"},{"Start":"00:19.565 ","End":"00:21.290","Text":"from P to Q here."},{"Start":"00:21.290 ","End":"00:26.180","Text":"Now this we\u0027ve already covered really in the section on lines,"},{"Start":"00:26.180 ","End":"00:29.570","Text":"equations of lines, especially in 3D parametric."},{"Start":"00:29.570 ","End":"00:31.760","Text":"Well, I\u0027ll go over it again."},{"Start":"00:31.760 ","End":"00:37.310","Text":"What we can do is take these points as position vectors."},{"Start":"00:37.310 ","End":"00:47.525","Text":"It\u0027s always, we take 1 minus t times the first position vector plus t times the second."},{"Start":"00:47.525 ","End":"00:51.815","Text":"In other words, r of t, is equal to,"},{"Start":"00:51.815 ","End":"00:56.780","Text":"it\u0027s 1 minus t always times the first position vector,"},{"Start":"00:56.780 ","End":"01:00.060","Text":"which is 2, 4 as a vector,"},{"Start":"01:00.060 ","End":"01:04.970","Text":"plus t times the second one, minus 3, 5"},{"Start":"01:04.970 ","End":"01:09.605","Text":"and t goes between 0 and 1."},{"Start":"01:09.605 ","End":"01:12.710","Text":"When t is 0, it gives the first point,"},{"Start":"01:12.710 ","End":"01:15.935","Text":"and t is one, which is the second point."},{"Start":"01:15.935 ","End":"01:18.680","Text":"That\u0027s part a, but we want to simplify it."},{"Start":"01:18.680 ","End":"01:21.380","Text":"This is equal to, let\u0027s see."},{"Start":"01:21.380 ","End":"01:23.600","Text":"The first component is 2,"},{"Start":"01:23.600 ","End":"01:27.615","Text":"minus 2t, minus 3t."},{"Start":"01:27.615 ","End":"01:31.785","Text":"I make it 2 minus 5t."},{"Start":"01:31.785 ","End":"01:43.710","Text":"The second component, 4 minus 4t, plus 5t, 4 plus t."},{"Start":"01:43.710 ","End":"01:44.775","Text":"That\u0027s it."},{"Start":"01:44.775 ","End":"01:47.265","Text":"Same thing in part b."},{"Start":"01:47.265 ","End":"01:50.090","Text":"Part b is just in 3D."},{"Start":"01:50.090 ","End":"01:55.055","Text":"Also we have the equation is 1 minus t,"},{"Start":"01:55.055 ","End":"02:01.700","Text":"times the first point position vector 3, 2, 0,"},{"Start":"02:01.700 ","End":"02:05.585","Text":"and t times the end point,"},{"Start":"02:05.585 ","End":"02:12.295","Text":"its position vector, which is vector 8 minus 5, 1."},{"Start":"02:12.295 ","End":"02:15.810","Text":"Again, we\u0027ll simplify, but we have a 3 computations here."},{"Start":"02:15.810 ","End":"02:24.990","Text":"First component, 3 minus 3t, plus 8t, is 3 plus 5t."},{"Start":"02:25.460 ","End":"02:27.600","Text":"That\u0027s the second one,"},{"Start":"02:27.600 ","End":"02:34.500","Text":"2 minus 2t, minus 5t, 2 minus 7t."},{"Start":"02:34.500 ","End":"02:37.530","Text":"Last one."},{"Start":"02:37.530 ","End":"02:43.360","Text":"The 0 here, just t from the second."},{"Start":"02:44.180 ","End":"02:46.470","Text":"I was going to say that\u0027s it, no."},{"Start":"02:46.470 ","End":"02:49.740","Text":"I have to also add that t goes from 0 to 1."},{"Start":"02:49.740 ","End":"02:51.730","Text":"Now we\u0027re done."}],"ID":9698},{"Watched":false,"Name":"Exercise 5","Duration":"6m 22s","ChapterTopicVideoID":9825,"CourseChapterTopicPlaylistID":57479,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9825.jpeg","UploadDate":"2017-08-06T04:29:12.0730000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:05.315","Text":"In this exercise, we have a straight line through 2 points,"},{"Start":"00:05.315 ","End":"00:13.425","Text":"actually I meant the line segment just from 1 point to the other,"},{"Start":"00:13.425 ","End":"00:15.570","Text":"not the whole line."},{"Start":"00:15.570 ","End":"00:17.670","Text":"We have these 2 points,"},{"Start":"00:17.670 ","End":"00:19.530","Text":"we have a line segment through them."},{"Start":"00:19.530 ","End":"00:24.570","Text":"We want to parametrize that with the parameter t,"},{"Start":"00:24.570 ","End":"00:26.370","Text":"going from 0 to 1,"},{"Start":"00:26.370 ","End":"00:29.565","Text":"but in polar coordinates."},{"Start":"00:29.565 ","End":"00:34.530","Text":"Of course we\u0027re in 2D. Cartesian is x,"},{"Start":"00:34.530 ","End":"00:38.265","Text":"y and polar is r theta."},{"Start":"00:38.265 ","End":"00:40.470","Text":"What I suggest is,"},{"Start":"00:40.470 ","End":"00:44.270","Text":"let\u0027s parametrize it using"},{"Start":"00:44.270 ","End":"00:51.850","Text":"Cartesian coordinates and then use the conversion formulas from Cartesian to polar."},{"Start":"00:51.850 ","End":"00:55.740","Text":"Now, in Cartesian, I\u0027ll call it x,"},{"Start":"00:55.740 ","End":"01:04.090","Text":"y. I could have just used vector r. I would like to have x, y."},{"Start":"01:04.640 ","End":"01:07.080","Text":"They\u0027re both functions of t,"},{"Start":"01:07.080 ","End":"01:09.530","Text":"I just didn\u0027t write the t. The formula for"},{"Start":"01:09.530 ","End":"01:15.474","Text":"the line segment between 2 points is we take the first point,"},{"Start":"01:15.474 ","End":"01:24.545","Text":"and then we add parameter t times the vector from 1 point to the other."},{"Start":"01:24.545 ","End":"01:26.890","Text":"I\u0027m going to subtract,"},{"Start":"01:26.890 ","End":"01:30.570","Text":"well the same things, but considered as vectors."},{"Start":"01:30.570 ","End":"01:35.790","Text":"Sometimes I write angular brackets to distinguish points from vectors,"},{"Start":"01:35.790 ","End":"01:37.395","Text":"there\u0027s no confusion here."},{"Start":"01:37.395 ","End":"01:41.010","Text":"We\u0027ll take the head minus"},{"Start":"01:41.010 ","End":"01:49.595","Text":"the tail and then we also make t go from 0 to 1."},{"Start":"01:49.595 ","End":"01:52.100","Text":"Notice that when t is 0, this bit 0,"},{"Start":"01:52.100 ","End":"01:56.570","Text":"so we\u0027re at the first part."},{"Start":"01:56.570 ","End":"02:02.495","Text":"When t is 1, then this cancels with this and we\u0027re at the second point."},{"Start":"02:02.495 ","End":"02:07.460","Text":"This is equal to, if I just write it out, let\u0027s see."},{"Start":"02:07.460 ","End":"02:10.085","Text":"1, 4 minus 3,"},{"Start":"02:10.085 ","End":"02:17.790","Text":"2 is minus 2,2."},{"Start":"02:17.790 ","End":"02:22.890","Text":"If I do the computation, 3 minus 2t,"},{"Start":"02:22.890 ","End":"02:25.755","Text":"that\u0027s this 3 and then the minus 2t,"},{"Start":"02:25.755 ","End":"02:31.720","Text":"and then y is 2 plus 2t."},{"Start":"02:32.690 ","End":"02:35.870","Text":"Now I have it in Cartesian,"},{"Start":"02:35.870 ","End":"02:41.049","Text":"really I could write it as x or even x of t,"},{"Start":"02:41.049 ","End":"02:44.565","Text":"the full answer, 3 minus 2t."},{"Start":"02:44.565 ","End":"02:49.440","Text":"This answer is okay, I just wanted to use the curly-brace form,"},{"Start":"02:49.640 ","End":"02:52.904","Text":"parametric rather than vector,"},{"Start":"02:52.904 ","End":"02:57.975","Text":"y of t is equal to 2 plus 2t."},{"Start":"02:57.975 ","End":"03:00.260","Text":"Now I want to convert from x,"},{"Start":"03:00.260 ","End":"03:02.785","Text":"y to r theta."},{"Start":"03:02.785 ","End":"03:07.775","Text":"There are formulas, 1 of the formulas is that r"},{"Start":"03:07.775 ","End":"03:12.775","Text":"is the square root of x squared plus y squared,"},{"Start":"03:12.775 ","End":"03:23.420","Text":"and the other formula is that theta is the arc tangent of y over x,"},{"Start":"03:23.420 ","End":"03:25.460","Text":"provided x is not 0."},{"Start":"03:25.460 ","End":"03:30.760","Text":"This also has to be adjusted sometimes if we\u0027re not in the first quadrant,"},{"Start":"03:30.760 ","End":"03:33.590","Text":"actually first and fourth quadrant."},{"Start":"03:33.590 ","End":"03:35.720","Text":"Anyway, we\u0027re always in the first quadrant."},{"Start":"03:35.720 ","End":"03:38.105","Text":"If 2 points are in the first quadrant,"},{"Start":"03:38.105 ","End":"03:46.410","Text":"then the line joining them will also be in the first quadrant."},{"Start":"03:46.990 ","End":"03:51.125","Text":"This formula will work just fine."},{"Start":"03:51.125 ","End":"03:58.595","Text":"What we get is 2r of t first,"},{"Start":"03:58.595 ","End":"04:08.475","Text":"r of t will be the square root of 3 minus 2t squared plus 2,"},{"Start":"04:08.475 ","End":"04:16.850","Text":"plus 2t squared, and it\u0027s just going to be messy algebra."},{"Start":"04:16.850 ","End":"04:19.225","Text":"Okay. Let\u0027s do it."},{"Start":"04:19.225 ","End":"04:23.310","Text":"Square root of this here."},{"Start":"04:23.310 ","End":"04:24.915","Text":"I\u0027ll write it in order,"},{"Start":"04:24.915 ","End":"04:33.065","Text":"2t squared is 4t squared minus twice this times this is minus 12t,"},{"Start":"04:33.065 ","End":"04:35.555","Text":"and then plus 9."},{"Start":"04:35.555 ","End":"04:37.595","Text":"That\u0027s the first bit."},{"Start":"04:37.595 ","End":"04:42.110","Text":"The second bit is 4t squared twice this"},{"Start":"04:42.110 ","End":"04:48.750","Text":"times this 8t plus 4."},{"Start":"04:48.750 ","End":"04:54.149","Text":"Let\u0027s see, 4t squared and 4t squared is 8t squared"},{"Start":"04:54.149 ","End":"05:01.800","Text":"minus 12t plus 8t is minus 4t,"},{"Start":"05:01.800 ","End":"05:05.970","Text":"9 plus 4 is 13,"},{"Start":"05:05.970 ","End":"05:09.220","Text":"and the square root."},{"Start":"05:09.580 ","End":"05:12.020","Text":"I thought I had cooked it up so,"},{"Start":"05:12.020 ","End":"05:13.490","Text":"it\u0027d come out nice and neat,"},{"Start":"05:13.490 ","End":"05:17.975","Text":"it doesn\u0027t come out nice and neat. Never mind."},{"Start":"05:17.975 ","End":"05:27.290","Text":"That\u0027s r of t and theta of t is just the arc tangent of y over x,"},{"Start":"05:27.290 ","End":"05:33.790","Text":"which is 2 plus 2t over 3 minus 2t."},{"Start":"05:33.790 ","End":"05:39.580","Text":"Notice that the denominator is not 0 when t is between 0 and 1."},{"Start":"05:39.580 ","End":"05:43.005","Text":"When t is 0, it\u0027s 3, when t is 1,"},{"Start":"05:43.005 ","End":"05:47.980","Text":"3 minus 2 is 1, it goes down from 3 to 1, never 0."},{"Start":"05:49.100 ","End":"05:54.660","Text":"All we have to do really is to"},{"Start":"05:54.660 ","End":"06:01.890","Text":"just copy this over here,"},{"Start":"06:01.890 ","End":"06:05.810","Text":"and r of t, I\u0027ll just copy like so,"},{"Start":"06:05.810 ","End":"06:08.000","Text":"it was supposed to come out neater but never mind,"},{"Start":"06:08.000 ","End":"06:09.080","Text":"this is the answer."},{"Start":"06:09.080 ","End":"06:14.910","Text":"Then finally we add that t goes from 0"},{"Start":"06:14.910 ","End":"06:22.230","Text":"to 1, and that\u0027s it."}],"ID":9694}],"Thumbnail":null,"ID":57479},{"Name":"Vector Calculus in 3D Coordinates System","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Vector Calculus (1)","Duration":"18m 28s","ChapterTopicVideoID":9864,"CourseChapterTopicPlaylistID":57480,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9864.jpeg","UploadDate":"2020-01-16T07:13:32.9030000","DurationForVideoObject":"PT18M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.060","Text":"We\u0027re continuing with 3D and higher dimensions."},{"Start":"00:04.060 ","End":"00:06.310","Text":"We just finished vector functions."},{"Start":"00:06.310 ","End":"00:10.390","Text":"Next, we\u0027ll do some calculus with vector functions and I\u0027ll just"},{"Start":"00:10.390 ","End":"00:15.710","Text":"write that as vector calculus."},{"Start":"00:17.390 ","End":"00:20.950","Text":"The first thing one talks about in calculus,"},{"Start":"00:20.950 ","End":"00:24.115","Text":"the most basic concept is the concept of a limit."},{"Start":"00:24.115 ","End":"00:27.890","Text":"Let\u0027s talk about limits with vectors."},{"Start":"00:27.900 ","End":"00:37.470","Text":"I just hope you remember what is a limit of a regular function in 1 variable."},{"Start":"00:37.470 ","End":"00:42.900","Text":"Let\u0027s say the function is f. I\u0027m going to use t instead of x."},{"Start":"00:42.900 ","End":"00:48.650","Text":"If I say something like the limit as x goes to a f of x or t goes to a f of t,"},{"Start":"00:48.650 ","End":"00:51.830","Text":"I\u0027m assuming that you know what this means and if"},{"Start":"00:51.830 ","End":"00:55.540","Text":"not you should go back and review it because I\u0027m not going to give you a summary now."},{"Start":"00:55.540 ","End":"01:00.230","Text":"What I do want to do is extend the concept and we\u0027ll stick with 1 variable and we\u0027ll call"},{"Start":"01:00.230 ","End":"01:04.670","Text":"it t from a regular function or scalar function,"},{"Start":"01:04.670 ","End":"01:07.495","Text":"if you want, to a vector function."},{"Start":"01:07.495 ","End":"01:10.160","Text":"Now, here\u0027s the definition and I\u0027m taking"},{"Start":"01:10.160 ","End":"01:15.080","Text":"a 3D example though it works for 2D or any higher dimensions,"},{"Start":"01:15.080 ","End":"01:17.300","Text":"3D is just typical."},{"Start":"01:17.300 ","End":"01:22.760","Text":"If I have a vector function of t in 3D which means it\u0027s 3 separate functions,"},{"Start":"01:22.760 ","End":"01:27.040","Text":"component functions, then the limit of the vector function,"},{"Start":"01:27.040 ","End":"01:30.320","Text":"i.e., the limit of this is what you"},{"Start":"01:30.320 ","End":"01:33.560","Text":"get when you take the separate limit at each component,"},{"Start":"01:33.560 ","End":"01:35.750","Text":"we take the limit of f, the limit of g,"},{"Start":"01:35.750 ","End":"01:41.550","Text":"the limit of h at the same place t goes to a."},{"Start":"01:41.550 ","End":"01:49.235","Text":"Here it is in the other notation using the standard basis vector, so everything\u0027s here."},{"Start":"01:49.235 ","End":"01:55.534","Text":"The point I want to make is that all these 3 limits must exist"},{"Start":"01:55.534 ","End":"02:02.045","Text":"in order for us to say that this vector function has a limit."},{"Start":"02:02.045 ","End":"02:03.980","Text":"If 1 of them doesn\u0027t,"},{"Start":"02:03.980 ","End":"02:06.400","Text":"that\u0027s no good already."},{"Start":"02:06.400 ","End":"02:11.355","Text":"Let\u0027s take an example."},{"Start":"02:11.355 ","End":"02:19.410","Text":"I want you to compute a limit as t goes to 1 of"},{"Start":"02:19.410 ","End":"02:26.990","Text":"t cubed then sine of"},{"Start":"02:26.990 ","End":"02:34.310","Text":"3t minus 1 over t minus 1,"},{"Start":"02:34.310 ","End":"02:44.250","Text":"sorry that\u0027s a 3, and the third component function is t minus 1 over t squared minus 1."},{"Start":"02:45.140 ","End":"02:47.960","Text":"Let\u0027s take them 1 at a time."},{"Start":"02:47.960 ","End":"02:50.820","Text":"We take the limit of each."},{"Start":"02:51.580 ","End":"03:00.525","Text":"We can write this as limit as t goes to 1 of t cubed,"},{"Start":"03:00.525 ","End":"03:11.535","Text":"the limit as t goes to 1 of sine 3t minus 3 over t minus 1,"},{"Start":"03:11.535 ","End":"03:21.335","Text":"and lastly, the limit as t goes to 1 of t minus 1 over t squared minus 1."},{"Start":"03:21.335 ","End":"03:25.530","Text":"Now all you have to do is remember how to do limits."},{"Start":"03:25.550 ","End":"03:30.300","Text":"The first one is easy,"},{"Start":"03:30.300 ","End":"03:33.560","Text":"there\u0027s nothing problematic about polynomial,"},{"Start":"03:33.560 ","End":"03:36.335","Text":"say t cubed, we just substitute."},{"Start":"03:36.335 ","End":"03:40.390","Text":"Substitute 1, 1 cubed is 1."},{"Start":"03:40.390 ","End":"03:47.285","Text":"Next one won\u0027t work with substitution because we have a 0 in the denominator."},{"Start":"03:47.285 ","End":"03:50.335","Text":"There\u0027s 2 tricks we could use."},{"Start":"03:50.335 ","End":"03:56.460","Text":"One of them would be the L\u0027Hopital rule and there is another way of doing it."},{"Start":"03:56.470 ","End":"03:59.900","Text":"In fact, I\u0027ll quickly mention both ways."},{"Start":"03:59.900 ","End":"04:03.275","Text":"Maybe I\u0027ll do that as a side exercise."},{"Start":"04:03.275 ","End":"04:07.175","Text":"For this one, if I use L\u0027Hopital,"},{"Start":"04:07.175 ","End":"04:08.735","Text":"let\u0027s write his name,"},{"Start":"04:08.735 ","End":"04:10.735","Text":"give him some respect."},{"Start":"04:10.735 ","End":"04:13.295","Text":"If we use L\u0027Hopital\u0027s rule,"},{"Start":"04:13.295 ","End":"04:16.760","Text":"then we can differentiate top and bottom"},{"Start":"04:16.760 ","End":"04:20.420","Text":"because they\u0027re both 0 when I substitute t goes to 1."},{"Start":"04:20.420 ","End":"04:22.520","Text":"Notice that the denominator, we said."},{"Start":"04:22.520 ","End":"04:27.140","Text":"In the numerator, 3t minus 3 is 0 and sine 0 is 0."},{"Start":"04:27.140 ","End":"04:28.760","Text":"In a 0 over 0 case,"},{"Start":"04:28.760 ","End":"04:29.960","Text":"we can differentiate top,"},{"Start":"04:29.960 ","End":"04:32.420","Text":"differentiate bottom, and evaluate that limit."},{"Start":"04:32.420 ","End":"04:37.495","Text":"By L\u0027Hopital, we get the lim as t goes to 1."},{"Start":"04:37.495 ","End":"04:38.750","Text":"If I differentiate this,"},{"Start":"04:38.750 ","End":"04:43.590","Text":"the sine becomes cosine of 3t minus 3,"},{"Start":"04:43.590 ","End":"04:46.530","Text":"but that\u0027s not all, it\u0027s an inner derivative of 3,"},{"Start":"04:46.530 ","End":"04:48.705","Text":"so I\u0027ll put the 3 in front."},{"Start":"04:48.705 ","End":"04:52.970","Text":"Here, the derivative of t minus 1 is just 1."},{"Start":"04:52.970 ","End":"04:55.835","Text":"At this point, there\u0027s no problem in substituting,"},{"Start":"04:55.835 ","End":"04:57.485","Text":"if I let t equals 1."},{"Start":"04:57.485 ","End":"05:01.405","Text":"I\u0027ve got cosine 0 times 3,"},{"Start":"05:01.405 ","End":"05:07.035","Text":"it\u0027s 3 times 1 over 1 which is 3."},{"Start":"05:07.035 ","End":"05:09.960","Text":"Here, I can write 3."},{"Start":"05:09.960 ","End":"05:12.150","Text":"2 down, 1 to go."},{"Start":"05:12.150 ","End":"05:14.510","Text":"I said there\u0027s another way of doing this."},{"Start":"05:14.510 ","End":"05:17.930","Text":"The other way of doing this is by a substitution."},{"Start":"05:17.930 ","End":"05:19.775","Text":"I\u0027ll just show you just for practice."},{"Start":"05:19.775 ","End":"05:27.720","Text":"If I let u equal 3t minus 3,"},{"Start":"05:27.720 ","End":"05:38.050","Text":"then notice that u over 3 is equal to t minus 1."},{"Start":"05:38.050 ","End":"05:42.390","Text":"You can see that this is 3 times this, it stand out."},{"Start":"05:43.370 ","End":"05:52.890","Text":"What I can do is to replace this limit by a new limit."},{"Start":"05:52.890 ","End":"05:56.415","Text":"What I get is the limit."},{"Start":"05:56.415 ","End":"05:58.410","Text":"I also have to replace this."},{"Start":"05:58.410 ","End":"06:00.245","Text":"If t goes to 1,"},{"Start":"06:00.245 ","End":"06:03.730","Text":"notice that u goes to 0."},{"Start":"06:03.730 ","End":"06:06.950","Text":"It\u0027s u goes to 0."},{"Start":"06:06.950 ","End":"06:10.110","Text":"Here I have sine of u,"},{"Start":"06:10.120 ","End":"06:13.655","Text":"and t minus 1 is u over 3,"},{"Start":"06:13.655 ","End":"06:17.250","Text":"so I have to divide by u over 3."},{"Start":"06:18.470 ","End":"06:21.724","Text":"If I put the 3 in front,"},{"Start":"06:21.724 ","End":"06:24.650","Text":"I mean, I can put the 3 on top and it comes out of the limit."},{"Start":"06:24.650 ","End":"06:33.570","Text":"I get 3 times limit u goes to 0 of sine u over u,"},{"Start":"06:33.570 ","End":"06:36.450","Text":"and it\u0027s a famous limit although usually with a different letter,"},{"Start":"06:36.450 ","End":"06:38.130","Text":"Theta or Alpha or x,"},{"Start":"06:38.130 ","End":"06:40.830","Text":"whatever, this is equal to 1."},{"Start":"06:40.830 ","End":"06:46.560","Text":"Altogether, I\u0027ve got that this is equal to 3."},{"Start":"06:46.560 ","End":"06:50.475","Text":"That justifies the result we get."},{"Start":"06:50.475 ","End":"06:53.460","Text":"I did 2 different ways."},{"Start":"06:53.460 ","End":"06:56.535","Text":"Just practicing limits with u,"},{"Start":"06:56.535 ","End":"06:59.610","Text":"there\u0027s nothing really special."},{"Start":"06:59.610 ","End":"07:03.685","Text":"I did it 2 ways just to get into the swing of things."},{"Start":"07:03.685 ","End":"07:05.260","Text":"Let\u0027s do the last one."},{"Start":"07:05.260 ","End":"07:07.975","Text":"Again, there\u0027s 2 ways of doing this."},{"Start":"07:07.975 ","End":"07:13.050","Text":"1 way of doing it is the L\u0027Hopital way."},{"Start":"07:13.050 ","End":"07:15.775","Text":"I\u0027ll just use a different color."},{"Start":"07:15.775 ","End":"07:22.000","Text":"The reason I say L\u0027Hopital is because it\u0027s a 0 over 0 case if you substitute."},{"Start":"07:22.000 ","End":"07:26.199","Text":"Using L\u0027Hopital method on this one,"},{"Start":"07:26.199 ","End":"07:29.170","Text":"differentiate the top you get 1,"},{"Start":"07:29.170 ","End":"07:32.190","Text":"differentiate the bottom you get 2t."},{"Start":"07:32.190 ","End":"07:36.120","Text":"You take the limit as t goes to 1,"},{"Start":"07:36.120 ","End":"07:39.755","Text":"straightforward substitution we get 1/2."},{"Start":"07:39.755 ","End":"07:41.990","Text":"But I\u0027m practicing limits,"},{"Start":"07:41.990 ","End":"07:43.760","Text":"so let me show you another way."},{"Start":"07:43.760 ","End":"07:48.050","Text":"The other way to do it would be to say what we have here is"},{"Start":"07:48.050 ","End":"07:54.195","Text":"the limit as t goes to 1 and use factorization."},{"Start":"07:54.195 ","End":"07:56.040","Text":"The numerator, nothing to be done."},{"Start":"07:56.040 ","End":"08:06.090","Text":"The denominator, difference of squares t squared minus 1 squared is t minus 1, t plus 1."},{"Start":"08:06.890 ","End":"08:10.095","Text":"If t tends to 1,"},{"Start":"08:10.095 ","End":"08:13.560","Text":"it\u0027s not equal to 1 and so we"},{"Start":"08:13.560 ","End":"08:18.025","Text":"can cancel the t minus 1 here and here what we\u0027re left with is 1,"},{"Start":"08:18.025 ","End":"08:22.070","Text":"and then we substitute t equals 1 in this which is okay,"},{"Start":"08:22.070 ","End":"08:23.930","Text":"it\u0027s 1 over 1 plus 1,"},{"Start":"08:23.930 ","End":"08:25.760","Text":"and this is equal to 1/2,"},{"Start":"08:25.760 ","End":"08:27.620","Text":"and this is equal to this."},{"Start":"08:27.620 ","End":"08:36.665","Text":"Here we have 1/2, and this is the answer to the example question."},{"Start":"08:36.665 ","End":"08:41.075","Text":"Just a quick summary, the first one we did by straightforward substitution."},{"Start":"08:41.075 ","End":"08:43.070","Text":"Second one, 2 different ways,"},{"Start":"08:43.070 ","End":"08:46.860","Text":"L\u0027Hopital and using a famous limit."},{"Start":"08:46.860 ","End":"08:50.010","Text":"The second one also we did 2 ways or double-checked,"},{"Start":"08:50.010 ","End":"08:54.805","Text":"1 using L\u0027Hopital and 1 using factorization."},{"Start":"08:54.805 ","End":"08:57.590","Text":"That\u0027s the only example I\u0027m going to give on limits."},{"Start":"08:57.590 ","End":"09:03.020","Text":"We\u0027re going to go on to the next topic which will be derivatives."},{"Start":"09:08.630 ","End":"09:16.250","Text":"Once again, we just generalize the case of scalar functions, regular functions."},{"Start":"09:16.250 ","End":"09:19.030","Text":"We generalize them to vector functions,"},{"Start":"09:19.030 ","End":"09:21.335","Text":"and here is the formula."},{"Start":"09:21.335 ","End":"09:25.250","Text":"It\u0027s a bit hard to see but this is an r prime."},{"Start":"09:25.250 ","End":"09:27.740","Text":"r is the same as above,"},{"Start":"09:27.740 ","End":"09:30.920","Text":"r is fgh component-wise."},{"Start":"09:30.920 ","End":"09:35.180","Text":"The derivative, our vector function is what you"},{"Start":"09:35.180 ","End":"09:39.620","Text":"get if you just take the derivatives of each of the component functions separately."},{"Start":"09:39.620 ","End":"09:43.160","Text":"Here it is using the other notation with standard basis vectors,"},{"Start":"09:43.160 ","End":"09:46.075","Text":"although we tend to use this one more I see."},{"Start":"09:46.075 ","End":"09:49.400","Text":"I\u0027m assuming you remember what regular derivatives are."},{"Start":"09:49.400 ","End":"09:54.485","Text":"If not, go and review. An example."},{"Start":"09:54.485 ","End":"09:56.825","Text":"Just for a change,"},{"Start":"09:56.825 ","End":"10:02.945","Text":"I\u0027ll take an example using the other notation with standard basis vectors."},{"Start":"10:02.945 ","End":"10:09.354","Text":"I\u0027ll take natural log of t squared plus 1,"},{"Start":"10:09.354 ","End":"10:16.095","Text":"all this times i plus, let say,"},{"Start":"10:16.095 ","End":"10:23.430","Text":"te to the minus t j plus,"},{"Start":"10:23.430 ","End":"10:29.205","Text":"let\u0027s say, cosine t times"},{"Start":"10:29.205 ","End":"10:35.030","Text":"k. I\u0027m really just practicing derivatives here."},{"Start":"10:35.030 ","End":"10:39.710","Text":"Because what we do to get the derivative of the vector function is to"},{"Start":"10:39.710 ","End":"10:45.210","Text":"take the derivative of each piece separately to the components."},{"Start":"10:45.210 ","End":"10:47.510","Text":"The derivative of this,"},{"Start":"10:47.510 ","End":"10:52.750","Text":"the derivative of natural log is 1 over,"},{"Start":"10:52.750 ","End":"10:56.075","Text":"we start off with 1 over t squared plus 1,"},{"Start":"10:56.075 ","End":"10:58.475","Text":"but then we need the inner derivative."},{"Start":"10:58.475 ","End":"11:05.560","Text":"We need to multiply by 2t which goes on the numerator and the i we just copy."},{"Start":"11:05.560 ","End":"11:10.550","Text":"For the next one, we\u0027re going to need the product rule."},{"Start":"11:10.550 ","End":"11:12.170","Text":"I\u0027m not going to write it out."},{"Start":"11:12.170 ","End":"11:16.130","Text":"Each time you take a derivative of 1 and the other unchanged."},{"Start":"11:16.130 ","End":"11:18.380","Text":"The derivative of t is 1,"},{"Start":"11:18.380 ","End":"11:22.525","Text":"that gives us e to the minus t. The other way round,"},{"Start":"11:22.525 ","End":"11:25.355","Text":"t times the derivative of this,"},{"Start":"11:25.355 ","End":"11:35.950","Text":"so it\u0027s t times e to the minus t times minus 1 brings a minus here, j,"},{"Start":"11:35.950 ","End":"11:39.465","Text":"and the derivative of cosine t,"},{"Start":"11:39.465 ","End":"11:43.490","Text":"I was going to write plus but the derivative of cosine of t is minus sine"},{"Start":"11:43.490 ","End":"11:50.350","Text":"t k. That\u0027s all there is to that."},{"Start":"11:50.860 ","End":"11:56.400","Text":"Next thing is going to be some rules for derivatives."},{"Start":"11:56.930 ","End":"12:01.715","Text":"We had product rules and quotient rules and all that."},{"Start":"12:01.715 ","End":"12:04.940","Text":"With vectors we have dot products and cross-products."},{"Start":"12:04.940 ","End":"12:07.445","Text":"Let\u0027s see how those behave with derivatives."},{"Start":"12:07.445 ","End":"12:12.140","Text":"I\u0027ll just bring you the table all at once."},{"Start":"12:12.140 ","End":"12:17.290","Text":"Before I go over them let me just remind you that we have 2 notations for derivative."},{"Start":"12:17.290 ","End":"12:20.905","Text":"If I have a function f of t,"},{"Start":"12:20.905 ","End":"12:28.170","Text":"then the derivative in the Newton style is f prime of t,"},{"Start":"12:28.170 ","End":"12:34.350","Text":"but the Leibniz style would say df of t by dt."},{"Start":"12:34.350 ","End":"12:43.810","Text":"The prime and the d over dt are equivalent and like I said, are both important."},{"Start":"12:43.810 ","End":"12:48.654","Text":"But historically Newton used this notation and Leibniz used this notation,"},{"Start":"12:48.654 ","End":"12:50.870","Text":"and here they\u0027re mixed."},{"Start":"12:50.930 ","End":"12:54.585","Text":"I\u0027m not going to give examples just briefly go over them."},{"Start":"12:54.585 ","End":"12:57.500","Text":"This just says that if we take the derivative of"},{"Start":"12:57.500 ","End":"13:00.994","Text":"a sum and these are both vector functions,"},{"Start":"13:00.994 ","End":"13:03.800","Text":"we just take the derivative of each one separately and add,"},{"Start":"13:03.800 ","End":"13:05.880","Text":"just like we\u0027re used to."},{"Start":"13:05.880 ","End":"13:08.180","Text":"Also multiplication by a constant."},{"Start":"13:08.180 ","End":"13:11.210","Text":"The constant just stays, it just sticks."},{"Start":"13:11.210 ","End":"13:13.955","Text":"That was a product rule."},{"Start":"13:13.955 ","End":"13:18.985","Text":"This time it\u0027s a product of a scalar function by a vector function."},{"Start":"13:18.985 ","End":"13:20.740","Text":"If it\u0027s a regular function,"},{"Start":"13:20.740 ","End":"13:23.120","Text":"that gives you real numbers and then there\u0027s a vector function."},{"Start":"13:23.120 ","End":"13:24.335","Text":"If I multiply them,"},{"Start":"13:24.335 ","End":"13:27.680","Text":"we get the product rule that we differentiate"},{"Start":"13:27.680 ","End":"13:34.155","Text":"this and multiply by the other one as is plus this one as is,"},{"Start":"13:34.155 ","End":"13:36.945","Text":"and this one differentiated as a prime here."},{"Start":"13:36.945 ","End":"13:39.280","Text":"It would be hard to see."},{"Start":"13:40.100 ","End":"13:43.360","Text":"Then there is-"}],"ID":9702},{"Watched":false,"Name":"Exercise 6","Duration":"5m 35s","ChapterTopicVideoID":9830,"CourseChapterTopicPlaylistID":57480,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9830.jpeg","UploadDate":"2017-08-06T04:32:30.0200000","DurationForVideoObject":"PT5M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we have to compute the following limits."},{"Start":"00:03.510 ","End":"00:07.725","Text":"There are 3 of them and each of them is a 3D vector."},{"Start":"00:07.725 ","End":"00:10.725","Text":"What we do is we take the limit component-wise."},{"Start":"00:10.725 ","End":"00:11.910","Text":"Let\u0027s start with a."},{"Start":"00:11.910 ","End":"00:18.705","Text":"What I\u0027ve done is taking the limit and put it inside in each component."},{"Start":"00:18.705 ","End":"00:21.135","Text":"Now we have 3 limits to compute."},{"Start":"00:21.135 ","End":"00:23.445","Text":"Let me just get some more space here."},{"Start":"00:23.445 ","End":"00:26.580","Text":"First component is a cosine,"},{"Start":"00:26.580 ","End":"00:28.680","Text":"not of t, of Pi t,"},{"Start":"00:28.680 ","End":"00:31.590","Text":"but everything goes continuous here,"},{"Start":"00:31.590 ","End":"00:36.095","Text":"so there\u0027s no problem in just substituting t equals 2."},{"Start":"00:36.095 ","End":"00:42.770","Text":"What we get for the first one is plugin 2 instead of t,"},{"Start":"00:42.770 ","End":"00:47.220","Text":"so we get cosine of 2 Pi."},{"Start":"00:50.450 ","End":"00:58.550","Text":"I\u0027ll continue with the first one and cosine 2 Pi is the same as cosine of 0, it\u0027s just 1."},{"Start":"00:58.550 ","End":"01:07.310","Text":"Here also, t minus 2 continuous function exponent is continuous, everything\u0027s continuous,"},{"Start":"01:07.310 ","End":"01:09.620","Text":"so we just substitute t equals 2,"},{"Start":"01:09.620 ","End":"01:14.435","Text":"so we get e^2 minus 2,"},{"Start":"01:14.435 ","End":"01:18.540","Text":"and that is equal to e^0, which is 1."},{"Start":"01:18.540 ","End":"01:22.535","Text":"Now we come to the third component here."},{"Start":"01:22.535 ","End":"01:26.615","Text":"We can\u0027t just substitute because when t equals 2,"},{"Start":"01:26.615 ","End":"01:28.975","Text":"the denominator is 0,"},{"Start":"01:28.975 ","End":"01:30.390","Text":"and as a matter of fact,"},{"Start":"01:30.390 ","End":"01:32.115","Text":"so is the numerator,"},{"Start":"01:32.115 ","End":"01:35.990","Text":"which is good because now we can use L\u0027Hopital\u0027s rule."},{"Start":"01:35.990 ","End":"01:40.700","Text":"What we do is replace this with a different limit that\u0027s going to be equal to it,"},{"Start":"01:40.700 ","End":"01:47.460","Text":"or we differentiate the numerator and the denominator so we get 1 over 2 t."},{"Start":"01:47.460 ","End":"01:52.220","Text":"Now, when t goes to 2,"},{"Start":"01:52.220 ","End":"01:56.205","Text":"2t goes to 4."},{"Start":"01:56.205 ","End":"01:57.510","Text":"There\u0027s no problems with this,"},{"Start":"01:57.510 ","End":"01:59.430","Text":"this is continuous at t equals 2,"},{"Start":"01:59.430 ","End":"02:00.840","Text":"it only have problems at 0."},{"Start":"02:00.840 ","End":"02:05.710","Text":"So it\u0027s 1/4 and that\u0027s the answer."},{"Start":"02:05.720 ","End":"02:09.190","Text":"This time, instead of the angular bracket notation,"},{"Start":"02:09.190 ","End":"02:12.205","Text":"we have the i, j, k notation."},{"Start":"02:12.205 ","End":"02:15.040","Text":"But still, we take each component separately,"},{"Start":"02:15.040 ","End":"02:18.400","Text":"its limit, and lets work on each one."},{"Start":"02:18.400 ","End":"02:21.705","Text":"The first one, no problems just substitute."},{"Start":"02:21.705 ","End":"02:24.720","Text":"If you substitute t equals 0,"},{"Start":"02:24.720 ","End":"02:27.645","Text":"0 cube plus 3 is 3,"},{"Start":"02:27.645 ","End":"02:31.380","Text":"so this becomes 3i."},{"Start":"02:32.300 ","End":"02:35.684","Text":"Next one, the constant,"},{"Start":"02:35.684 ","End":"02:37.110","Text":"2 is a constant function,"},{"Start":"02:37.110 ","End":"02:39.510","Text":"so the limit when t goes to 0,"},{"Start":"02:39.510 ","End":"02:40.800","Text":"it doesn\u0027t matter t goes to what."},{"Start":"02:40.800 ","End":"02:45.230","Text":"This is just 2, so minus 2j."},{"Start":"02:45.230 ","End":"02:50.195","Text":"The last one\u0027s a bit trickier because when we plug in t equals 0,"},{"Start":"02:50.195 ","End":"02:51.930","Text":"but it\u0027s not defined at 0,"},{"Start":"02:51.930 ","End":"02:55.190","Text":"you see the denominator\u0027s 0, and as a matter of fact,"},{"Start":"02:55.190 ","End":"02:57.995","Text":"so is the numerator because e^0 is 1,"},{"Start":"02:57.995 ","End":"03:01.250","Text":"1 minus 1 is 0, so we have a 0 over 0."},{"Start":"03:01.250 ","End":"03:05.675","Text":"Once again, we\u0027re going to use L\u0027Hopital\u0027s rule and"},{"Start":"03:05.675 ","End":"03:10.345","Text":"differentiate both numerator and denominator."},{"Start":"03:10.345 ","End":"03:13.370","Text":"We\u0027ve got limit t goes to 0."},{"Start":"03:13.370 ","End":"03:23.030","Text":"The derivative of this is minus e^t and the derivative of the denominator is 2t minus 1."},{"Start":"03:23.030 ","End":"03:26.735","Text":"There\u0027s no problem in substituting t equals 0 here."},{"Start":"03:26.735 ","End":"03:29.029","Text":"What we get is minus e^0,"},{"Start":"03:29.029 ","End":"03:30.875","Text":"which is minus 1."},{"Start":"03:30.875 ","End":"03:32.510","Text":"Let me just write that at the side,"},{"Start":"03:32.510 ","End":"03:37.790","Text":"minus e^0 over 2, 0 minus 1."},{"Start":"03:37.790 ","End":"03:39.770","Text":"This is minus 1."},{"Start":"03:39.770 ","End":"03:42.125","Text":"The denominator is minus 1,"},{"Start":"03:42.125 ","End":"03:44.345","Text":"so this is equal to 1."},{"Start":"03:44.345 ","End":"03:53.630","Text":"The answer is 3i minus 2j plus k."},{"Start":"03:53.630 ","End":"03:56.725","Text":"I forgot the k here."},{"Start":"03:56.725 ","End":"04:00.270","Text":"That\u0027s the answer. Now on to part C."},{"Start":"04:00.270 ","End":"04:03.340","Text":"In this part also,"},{"Start":"04:03.340 ","End":"04:06.670","Text":"we just take the limit and put it in front of each component."},{"Start":"04:06.670 ","End":"04:10.420","Text":"This time the limit is t goes to infinity."},{"Start":"04:10.420 ","End":"04:13.180","Text":"We have to remember our stuff about infinity."},{"Start":"04:13.180 ","End":"04:15.295","Text":"Let\u0027s do it component-wise."},{"Start":"04:15.295 ","End":"04:18.850","Text":"The first one is a polynomial over a polynomial."},{"Start":"04:18.850 ","End":"04:26.460","Text":"What we do is we look at the degrees of the polynomial and they\u0027re equal degrees."},{"Start":"04:26.460 ","End":"04:29.215","Text":"When the polynomials have equal degrees,"},{"Start":"04:29.215 ","End":"04:31.480","Text":"then we take the leading coefficients,"},{"Start":"04:31.480 ","End":"04:32.890","Text":"in this case, 3t squared."},{"Start":"04:32.890 ","End":"04:35.680","Text":"In this case, I\u0027ll write it as 1t squared."},{"Start":"04:35.680 ","End":"04:39.440","Text":"This is just 3 over 1."},{"Start":"04:39.440 ","End":"04:44.610","Text":"I\u0027ll just write that. The first bit is 3 over 1."},{"Start":"04:44.610 ","End":"04:47.560","Text":"Now, the second bit,"},{"Start":"04:47.560 ","End":"04:51.140","Text":"we have to remember when t goes to infinity,"},{"Start":"04:51.140 ","End":"04:53.810","Text":"this exponent goes to minus infinity,"},{"Start":"04:53.810 ","End":"04:56.270","Text":"and we know this limit and"},{"Start":"04:56.270 ","End":"05:00.275","Text":"symbolically we write it as e to the minus infinity, we know is 0,"},{"Start":"05:00.275 ","End":"05:03.260","Text":"which means that when the exponent goes to minus infinity,"},{"Start":"05:03.260 ","End":"05:08.470","Text":"the e to the power of it goes to 0."},{"Start":"05:08.470 ","End":"05:11.055","Text":"Just write 0 here."},{"Start":"05:11.055 ","End":"05:13.125","Text":"As for the last one,"},{"Start":"05:13.125 ","End":"05:17.890","Text":"we have 2 over infinity."},{"Start":"05:18.320 ","End":"05:23.140","Text":"Any finite number over infinity is also 0."},{"Start":"05:24.340 ","End":"05:29.795","Text":"That\u0027s all there is, and then I just have to rewrite this instead of 3 over 1,"},{"Start":"05:29.795 ","End":"05:32.360","Text":"I\u0027ll write it as just 3."},{"Start":"05:32.360 ","End":"05:34.980","Text":"That\u0027s all, so we\u0027re done."}],"ID":9700},{"Watched":false,"Name":"Exercise 7","Duration":"4m 49s","ChapterTopicVideoID":9831,"CourseChapterTopicPlaylistID":57480,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9831.jpeg","UploadDate":"2017-08-06T04:33:24.4170000","DurationForVideoObject":"PT4M49S","Description":null,"VideoComments":[],"Subtitles":[],"ID":9701}],"Thumbnail":null,"ID":57480},{"Name":"Tangent, Normal and Binormal Vectors","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Integrals","Duration":"12m 9s","ChapterTopicVideoID":9867,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9867.jpeg","UploadDate":"2017-08-06T05:31:24.9770000","DurationForVideoObject":"PT12M9S","Description":null,"VideoComments":[],"Subtitles":[],"ID":9703},{"Watched":false,"Name":"3D Space - Tangent , Normal and Binormal Vectors","Duration":"10m 48s","ChapterTopicVideoID":9865,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9865.jpeg","UploadDate":"2020-03-26T10:15:46.4830000","DurationForVideoObject":"PT10M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"This topic follows the previous topic on vector calculus."},{"Start":"00:04.320 ","End":"00:07.260","Text":"We\u0027re going to learn about 3 special kinds of vectors;"},{"Start":"00:07.260 ","End":"00:10.875","Text":"tangent, normal, and binormal."},{"Start":"00:10.875 ","End":"00:14.325","Text":"Let\u0027s get started with the tangent."},{"Start":"00:14.325 ","End":"00:22.170","Text":"Here\u0027s the setup. Suppose we have a vector function r of t. If I don\u0027t say otherwise,"},{"Start":"00:22.170 ","End":"00:24.075","Text":"we\u0027re working in 3D space."},{"Start":"00:24.075 ","End":"00:27.405","Text":"We have r of t and let\u0027s assume that it\u0027s smooth."},{"Start":"00:27.405 ","End":"00:30.270","Text":"In case you don\u0027t remember what smooth is,"},{"Start":"00:30.270 ","End":"00:36.130","Text":"it means that the derivative is continuous."},{"Start":"00:36.130 ","End":"00:38.620","Text":"I\u0027ll just write it again."},{"Start":"00:39.380 ","End":"00:43.445","Text":"What\u0027s important to remember is that"},{"Start":"00:43.445 ","End":"00:53.265","Text":"r prime of t is never 0, the 0 vector."},{"Start":"00:53.265 ","End":"00:57.910","Text":"Turns out this is important and we\u0027ll see."},{"Start":"00:57.920 ","End":"01:04.340","Text":"In actual fact, this r prime of t is"},{"Start":"01:04.340 ","End":"01:08.090","Text":"a tangent vector in the sense that"},{"Start":"01:08.090 ","End":"01:12.665","Text":"if we drew the tangent line to the curve at that particular point,"},{"Start":"01:12.665 ","End":"01:20.795","Text":"then this would be parallel or included in that tangent line."},{"Start":"01:20.795 ","End":"01:24.410","Text":"We can always compute the tangent we\u0027re"},{"Start":"01:24.410 ","End":"01:28.190","Text":"given a t. Let me give an example to explain this."},{"Start":"01:28.190 ","End":"01:31.760","Text":"Let me take the vector function in 3D, r of t,"},{"Start":"01:31.760 ","End":"01:38.380","Text":"which is given by 7e to the power of 2 minus t,"},{"Start":"01:38.420 ","End":"01:42.645","Text":"16 over t cubed,"},{"Start":"01:42.645 ","End":"01:47.760","Text":"and 5 minus t. Now,"},{"Start":"01:47.760 ","End":"01:50.550","Text":"if I differentiate this,"},{"Start":"01:50.550 ","End":"01:53.925","Text":"I didn\u0027t give the question."},{"Start":"01:53.925 ","End":"02:00.540","Text":"What I want to do is I want to find the tangent,"},{"Start":"02:01.650 ","End":"02:06.260","Text":"the equation of the tangent line that is,"},{"Start":"02:06.300 ","End":"02:10.240","Text":"we just have to give the value of the parameter,"},{"Start":"02:10.240 ","End":"02:14.990","Text":"parameter of the variable at t equals 2."},{"Start":"02:15.560 ","End":"02:22.569","Text":"In general, r prime of t is given by,"},{"Start":"02:22.569 ","End":"02:26.830","Text":"we just differentiate each component separately."},{"Start":"02:26.830 ","End":"02:33.000","Text":"Here we\u0027ll get, the inner derivative of 2 minus t is minus 1,"},{"Start":"02:33.000 ","End":"02:36.490","Text":"so we\u0027re going to get minus 7e to the power of"},{"Start":"02:36.490 ","End":"02:43.395","Text":"2 minus t. Here we have 16 t to the minus 4,"},{"Start":"02:43.395 ","End":"02:53.490","Text":"and if we differentiate it we\u0027ll get 16 times minus 4 is minus 64 and t to the minus 4,"},{"Start":"02:53.490 ","End":"02:55.815","Text":"so I\u0027ll put it over t to the 4th,"},{"Start":"02:55.815 ","End":"03:00.065","Text":"and here the derivative will be minus 1 always."},{"Start":"03:00.065 ","End":"03:05.580","Text":"Now at any point, this is going to be a tangent vector."},{"Start":"03:05.580 ","End":"03:11.400","Text":"It could be more than 1, multiples of this and even if it\u0027s not 0."},{"Start":"03:11.450 ","End":"03:16.850","Text":"What I want to say is that I can find both of these when t is 2."},{"Start":"03:16.850 ","End":"03:18.830","Text":"When t equals 2,"},{"Start":"03:18.830 ","End":"03:20.810","Text":"what I get is r of t,"},{"Start":"03:20.810 ","End":"03:27.084","Text":"which will give me the point or its position vector will equal,"},{"Start":"03:27.084 ","End":"03:30.600","Text":"if t is 2 then 2 minus 2 is 0,"},{"Start":"03:30.600 ","End":"03:32.775","Text":"e to the 0 is 1 that\u0027s 7."},{"Start":"03:32.775 ","End":"03:35.790","Text":"If t is 2, 16 over 2 cubed is 8,"},{"Start":"03:35.790 ","End":"03:37.860","Text":"that\u0027s 2, if t is 2,"},{"Start":"03:37.860 ","End":"03:41.205","Text":"5 minus t is 3."},{"Start":"03:41.205 ","End":"03:45.420","Text":"The tangent, a tangent,"},{"Start":"03:45.420 ","End":"03:49.080","Text":"I keep saying the, just 1."},{"Start":"03:49.080 ","End":"03:55.660","Text":"A tangent is the derivative vector or any multiple of it in fact,"},{"Start":"03:55.660 ","End":"04:03.495","Text":"and that is equal to just plugging 2 into here and we\u0027ve got minus 7,"},{"Start":"04:03.495 ","End":"04:07.545","Text":"and then 64 over 16 is 4,"},{"Start":"04:07.545 ","End":"04:12.120","Text":"minus 4, and minus 1 is a constant."},{"Start":"04:12.120 ","End":"04:15.680","Text":"That\u0027s that. Essentially what we have to"},{"Start":"04:15.680 ","End":"04:19.625","Text":"do is write the equation that we know already how to do."},{"Start":"04:19.625 ","End":"04:23.390","Text":"This is the point or its position vector."},{"Start":"04:23.390 ","End":"04:26.490","Text":"We have a point and we have a direction,"},{"Start":"04:27.890 ","End":"04:30.915","Text":"point and a direction vector,"},{"Start":"04:30.915 ","End":"04:35.789","Text":"so we apply the standard formula for the tangent line,"},{"Start":"04:40.640 ","End":"04:43.860","Text":"would be r of t,"},{"Start":"04:43.860 ","End":"04:45.660","Text":"I guess I\u0027m reusing the letter r,"},{"Start":"04:45.660 ","End":"04:47.655","Text":"but in this context it\u0027s understood,"},{"Start":"04:47.655 ","End":"04:52.830","Text":"is equal to 7,2,3"},{"Start":"04:52.830 ","End":"04:57.989","Text":"plus t times direction vector,"},{"Start":"04:57.989 ","End":"05:03.910","Text":"minus 7, minus 4, minus 1."},{"Start":"05:04.190 ","End":"05:10.130","Text":"That\u0027s the answer, although I could rewrite it by combining and saying,"},{"Start":"05:10.130 ","End":"05:14.230","Text":"this is 7 minus 7t,"},{"Start":"05:14.230 ","End":"05:23.760","Text":"2 minus 4t, and 3 minus t. Another way."},{"Start":"05:23.760 ","End":"05:28.260","Text":"Of course I could write it with the vectors i, j,"},{"Start":"05:28.260 ","End":"05:33.155","Text":"k. What we\u0027ve learned"},{"Start":"05:33.155 ","End":"05:39.500","Text":"basically is that the derivative at any given t is a direction vector of the tangent."},{"Start":"05:39.500 ","End":"05:41.910","Text":"It\u0027s a tangent vector."},{"Start":"05:42.140 ","End":"05:45.140","Text":"Besides the concept of tangent vector,"},{"Start":"05:45.140 ","End":"05:47.585","Text":"there is something called a unit"},{"Start":"05:47.585 ","End":"05:53.075","Text":"tangent vector and I\u0027ll just write the name unit tangent vector."},{"Start":"05:53.075 ","End":"05:58.040","Text":"What it is is a unit vector in the same direction as the tangent,"},{"Start":"05:58.040 ","End":"06:04.140","Text":"so what it is, is we take this r prime of t,"},{"Start":"06:07.550 ","End":"06:09.690","Text":"but I divide it,"},{"Start":"06:09.690 ","End":"06:11.900","Text":"if you divide any vector by its magnitude,"},{"Start":"06:11.900 ","End":"06:13.745","Text":"we get a unit vector,"},{"Start":"06:13.745 ","End":"06:21.120","Text":"by the magnitude of the same thing of r prime of t. For each given t,"},{"Start":"06:21.120 ","End":"06:27.360","Text":"this is going to give us the unit tangent vector and it has a name or rather a letter."},{"Start":"06:27.360 ","End":"06:30.665","Text":"I\u0027ll use T since t is used."},{"Start":"06:30.665 ","End":"06:36.950","Text":"This will be the tangent vector at the value t will equal this,"},{"Start":"06:36.950 ","End":"06:40.200","Text":"and that\u0027s the unit tangent vector."},{"Start":"06:40.510 ","End":"06:44.360","Text":"Note that the denominator is not 0 because we already"},{"Start":"06:44.360 ","End":"06:49.860","Text":"assumed that the derivative is never 0."},{"Start":"06:49.880 ","End":"06:52.960","Text":"Let\u0027s do an example."},{"Start":"06:52.960 ","End":"06:58.020","Text":"I\u0027ll take the function r of t,"},{"Start":"06:58.020 ","End":"07:00.270","Text":"vector function in 3D."},{"Start":"07:00.270 ","End":"07:03.665","Text":"This time we\u0027ll use the i, j, k notation."},{"Start":"07:03.665 ","End":"07:12.260","Text":"Let\u0027s take it as t times i plus e to the t times"},{"Start":"07:12.260 ","End":"07:17.655","Text":"j plus 3t squared"},{"Start":"07:17.655 ","End":"07:23.490","Text":"times vector k. What I want to know is,"},{"Start":"07:23.490 ","End":"07:30.065","Text":"what is the unit tangent vector equal to in general,"},{"Start":"07:30.065 ","End":"07:37.035","Text":"and specifically, what is it equal when t is 0?"},{"Start":"07:37.035 ","End":"07:43.950","Text":"The first thing we do is we differentiate and this gives us,"},{"Start":"07:43.950 ","End":"07:46.760","Text":"for each t it gives us a tangent,"},{"Start":"07:46.760 ","End":"07:48.365","Text":"but not the unit tangent,"},{"Start":"07:48.365 ","End":"07:53.065","Text":"so derivative of t is 1, that\u0027s just i,"},{"Start":"07:53.065 ","End":"07:59.675","Text":"and the derivative of e to the t is just e to the t j,"},{"Start":"07:59.675 ","End":"08:10.010","Text":"and derivative of 3t squared is 6t k. What I get is that"},{"Start":"08:10.010 ","End":"08:15.635","Text":"T of t unit tangent vector"},{"Start":"08:15.635 ","End":"08:20.810","Text":"is just this thing and often we just leave it as is,"},{"Start":"08:20.810 ","End":"08:26.820","Text":"so it\u0027s i plus e to the t j plus 6t k,"},{"Start":"08:26.820 ","End":"08:29.759","Text":"and then a big dividing line,"},{"Start":"08:29.759 ","End":"08:33.210","Text":"a normal denominator, a square root,"},{"Start":"08:33.210 ","End":"08:36.590","Text":"because the magnitude is the square root."},{"Start":"08:36.590 ","End":"08:42.800","Text":"Let\u0027s see, 1 squared plus e to"},{"Start":"08:42.800 ","End":"08:51.605","Text":"the t squared plus 6t squared and this is equal to,"},{"Start":"08:51.605 ","End":"08:54.650","Text":"of course, I could compute this and divide each 1"},{"Start":"08:54.650 ","End":"08:57.320","Text":"separately and say 1 over the square root,"},{"Start":"08:57.320 ","End":"08:59.090","Text":"e to the t over the square root,"},{"Start":"08:59.090 ","End":"09:00.730","Text":"and 6t over the square root,"},{"Start":"09:00.730 ","End":"09:03.440","Text":"but it\u0027s tedious to write the square root 3 times,"},{"Start":"09:03.440 ","End":"09:04.780","Text":"so we just write it once."},{"Start":"09:04.780 ","End":"09:08.270","Text":"All I need to do is just simplify the denominator,"},{"Start":"09:08.270 ","End":"09:14.970","Text":"took to the quick copy paste and now I can erase what\u0027s here and instead"},{"Start":"09:14.970 ","End":"09:21.970","Text":"write 1 plus e to the 2t plus 36t squared."},{"Start":"09:21.970 ","End":"09:25.080","Text":"That answers the first part."},{"Start":"09:25.080 ","End":"09:33.440","Text":"Now the second part as to what is the value of the unit tangent vector when t is 0,"},{"Start":"09:33.650 ","End":"09:36.990","Text":"well, when t is 0,"},{"Start":"09:36.990 ","End":"09:42.965","Text":"that\u0027s going to be 0, this is going to be 1 and this is going to be 1."},{"Start":"09:42.965 ","End":"09:47.795","Text":"I\u0027m going to get i plus"},{"Start":"09:47.795 ","End":"09:58.350","Text":"1j plus 0k over,"},{"Start":"09:58.350 ","End":"10:02.275","Text":"let\u0027s see, here t is 0,"},{"Start":"10:02.275 ","End":"10:06.245","Text":"e to the 0 is 1, that\u0027s 1 plus 1,"},{"Start":"10:06.245 ","End":"10:10.025","Text":"and this is 0, so I\u0027m going to get over the square root of"},{"Start":"10:10.025 ","End":"10:15.125","Text":"2 and this time why don\u0027t I just write each 1 over the square root of 2."},{"Start":"10:15.125 ","End":"10:20.270","Text":"I get 1 over the square root of 2 i plus 1"},{"Start":"10:20.270 ","End":"10:26.585","Text":"over the square root of 2 j and I don\u0027t even have to write that because it\u0027s 0."},{"Start":"10:26.585 ","End":"10:28.770","Text":"This is 0, it\u0027s not ok,"},{"Start":"10:28.770 ","End":"10:34.965","Text":"it\u0027s 0k. That\u0027s it."},{"Start":"10:34.965 ","End":"10:36.540","Text":"We didn\u0027t highlight them,"},{"Start":"10:36.540 ","End":"10:39.600","Text":"but the answer to this, the answer to that."},{"Start":"10:39.600 ","End":"10:42.000","Text":"So much for the tangent,"},{"Start":"10:42.000 ","End":"10:49.380","Text":"I\u0027m going to scroll back to the top and next we\u0027re going to discuss the normal."}],"ID":9704},{"Watched":false,"Name":"3D Space - Tangent , Normal and Binormal Vectors (continued)","Duration":"10m 14s","ChapterTopicVideoID":9866,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9866.jpeg","UploadDate":"2020-03-26T10:13:11.7870000","DurationForVideoObject":"PT10M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.230 ","End":"00:05.305","Text":"I just kept the definition of the unit tangent vector."},{"Start":"00:05.305 ","End":"00:09.820","Text":"Now let\u0027s go and define the unit normal vector."},{"Start":"00:09.820 ","End":"00:13.150","Text":"There is an actual normal vector,"},{"Start":"00:13.150 ","End":"00:16.870","Text":"but we\u0027re not interested so much in just any normal vector."},{"Start":"00:16.870 ","End":"00:19.614","Text":"We want the unit normal vector,"},{"Start":"00:19.614 ","End":"00:23.305","Text":"which is the derivative of this."},{"Start":"00:23.305 ","End":"00:29.390","Text":"We take the unit tangent vector."},{"Start":"00:29.750 ","End":"00:32.500","Text":"This would be a normal vector,"},{"Start":"00:32.500 ","End":"00:33.730","Text":"but we want the unit,"},{"Start":"00:33.730 ","End":"00:41.755","Text":"so we divide by the magnitude of the derivative of the tangent vector."},{"Start":"00:41.755 ","End":"00:49.585","Text":"This, we will use the N of"},{"Start":"00:49.585 ","End":"00:58.050","Text":"t. This will not work if the derivative,"},{"Start":"00:58.050 ","End":"01:00.090","Text":"if this is 0, it won\u0027t work."},{"Start":"01:00.090 ","End":"01:02.245","Text":"So I\u0027m going to assume non 0."},{"Start":"01:02.245 ","End":"01:03.680","Text":"It could happen."},{"Start":"01:03.680 ","End":"01:07.445","Text":"For example, if r was the equation of a straight line,"},{"Start":"01:07.445 ","End":"01:13.910","Text":"then the tangent comes out to be a constant and its derivative is always 0."},{"Start":"01:13.910 ","End":"01:18.855","Text":"In fact, there\u0027s no normal vector defined."},{"Start":"01:18.855 ","End":"01:20.420","Text":"But in the examples,"},{"Start":"01:20.420 ","End":"01:22.715","Text":"we\u0027ll see this will not be 0."},{"Start":"01:22.715 ","End":"01:27.635","Text":"Or if it is, we\u0027ll deal with it when the time comes."},{"Start":"01:27.635 ","End":"01:31.380","Text":"I\u0027m going to assume that this is not 0."},{"Start":"01:31.790 ","End":"01:34.650","Text":"Now I\u0027m not going to prove it,"},{"Start":"01:34.650 ","End":"01:42.400","Text":"but in fact the normal vector and the tangent vector are orthogonal."},{"Start":"01:42.400 ","End":"01:44.915","Text":"Remember it\u0027s another word for perpendicular."},{"Start":"01:44.915 ","End":"01:47.370","Text":"These are orthogonal."},{"Start":"01:49.010 ","End":"01:54.140","Text":"Another way of saying that is the dot product of these is 0,"},{"Start":"01:54.140 ","End":"01:56.945","Text":"but then also say that these 2 are"},{"Start":"01:56.945 ","End":"02:00.875","Text":"orthogonal to each other or perpendicular to each other."},{"Start":"02:00.875 ","End":"02:02.510","Text":"This is not hard to prove,"},{"Start":"02:02.510 ","End":"02:05.370","Text":"but I don\u0027t see the point of doing that."},{"Start":"02:05.420 ","End":"02:11.940","Text":"These are 2 unit vectors,"},{"Start":"02:11.940 ","End":"02:14.260","Text":"tangent and the normal."},{"Start":"02:14.630 ","End":"02:17.670","Text":"Even before I do an example,"},{"Start":"02:17.670 ","End":"02:22.180","Text":"let\u0027s get the third one out of the way, the binormal vector."},{"Start":"02:22.180 ","End":"02:32.090","Text":"We\u0027ll define the binormal at the point t to equal in this order."},{"Start":"02:32.090 ","End":"02:34.355","Text":"First the tangent vector,"},{"Start":"02:34.355 ","End":"02:40.825","Text":"then cross product with the normal vector."},{"Start":"02:40.825 ","End":"02:44.330","Text":"Now, being a cross product it\u0027s going to be"},{"Start":"02:44.330 ","End":"02:48.050","Text":"perpendicular to both of them and these are perpendicular to each other,"},{"Start":"02:48.050 ","End":"02:49.790","Text":"so that B and T,"},{"Start":"02:49.790 ","End":"02:54.385","Text":"and N are all 3 perpendicular vectors."},{"Start":"02:54.385 ","End":"02:59.810","Text":"In fact, this is also a unit vector because if I take 2 perpendicular unit vectors,"},{"Start":"02:59.810 ","End":"03:01.940","Text":"the cross product is also a unit vector."},{"Start":"03:01.940 ","End":"03:04.170","Text":"Again, I won\u0027t prove that."},{"Start":"03:04.820 ","End":"03:09.945","Text":"Like I said, these 3, this, and this,"},{"Start":"03:09.945 ","End":"03:16.390","Text":"and this are all orthogonal to each other."},{"Start":"03:16.910 ","End":"03:26.490","Text":"They are all unit vectors at any given point t. Like I said,"},{"Start":"03:26.490 ","End":"03:32.915","Text":"the only thing that I\u0027m supposing is that this thing is not 0."},{"Start":"03:32.915 ","End":"03:35.720","Text":"Of course, we originally suppose that this is not 0 either."},{"Start":"03:35.720 ","End":"03:45.810","Text":"Let me just remind us of that r prime of t is also not got to be 0."},{"Start":"03:46.850 ","End":"03:50.040","Text":"All I\u0027m missing now is an example."},{"Start":"03:50.040 ","End":"03:53.690","Text":"We\u0027re not going to say anymore about how to use N, t, and B,"},{"Start":"03:53.690 ","End":"03:58.490","Text":"they\u0027re here for reference for future that you will have covered these."},{"Start":"03:58.490 ","End":"04:06.560","Text":"I\u0027ll just do an example where I\u0027ll give you r of t and then we will find t and N and B."},{"Start":"04:06.560 ","End":"04:09.425","Text":"Let\u0027s take an example we\u0027ve seen already."},{"Start":"04:09.425 ","End":"04:13.460","Text":"Let\u0027s take the function r of t,"},{"Start":"04:13.460 ","End":"04:18.974","Text":"which is 4 cosine t,"},{"Start":"04:18.974 ","End":"04:24.000","Text":"4 sine t, and"},{"Start":"04:24.000 ","End":"04:29.360","Text":"t. This was in the section on vector functions."},{"Start":"04:29.360 ","End":"04:30.440","Text":"Take a look back."},{"Start":"04:30.440 ","End":"04:34.085","Text":"I\u0027ll just put the picture and maybe that will ring a bell."},{"Start":"04:34.085 ","End":"04:38.790","Text":"Here is the helix."},{"Start":"04:39.070 ","End":"04:43.835","Text":"Let\u0027s just start the computation."},{"Start":"04:43.835 ","End":"04:49.280","Text":"I\u0027ll keep some of the definitions as far as possible to start off with."},{"Start":"04:49.410 ","End":"04:54.175","Text":"First thing we want to do is compute r-prime of"},{"Start":"04:54.175 ","End":"04:58.610","Text":"t. That\u0027s just a straightforward differentiation."},{"Start":"04:58.610 ","End":"05:02.625","Text":"So minus 4 sine t,"},{"Start":"05:02.625 ","End":"05:07.930","Text":"4 cosine t, and 1."},{"Start":"05:07.930 ","End":"05:10.730","Text":"Then before we get to the unit tangent vector,"},{"Start":"05:10.730 ","End":"05:16.305","Text":"we need to find the magnitude of r-prime of"},{"Start":"05:16.305 ","End":"05:22.220","Text":"t. This is easy to compute."},{"Start":"05:22.220 ","End":"05:25.835","Text":"We need the square root of this squared plus this squared plus this squared."},{"Start":"05:25.835 ","End":"05:29.810","Text":"We get 16 sine squared plus 16 cosine squared."},{"Start":"05:29.810 ","End":"05:35.375","Text":"Well, that\u0027s just 16 plus 1 is 17."},{"Start":"05:35.375 ","End":"05:40.670","Text":"I of course used the property that sine squared plus cosine squared is 1."},{"Start":"05:40.670 ","End":"05:43.460","Text":"Yeah, square root of 17."},{"Start":"05:43.460 ","End":"05:54.380","Text":"That means that out unit tangent vector T of t is equal to"},{"Start":"05:54.380 ","End":"06:01.115","Text":"minus 4 over root 17 sine t. I\u0027m just dividing this"},{"Start":"06:01.115 ","End":"06:08.860","Text":"by the magnitude 4 over root 17 cosine t,"},{"Start":"06:08.860 ","End":"06:13.960","Text":"and 1 over root 17."},{"Start":"06:15.170 ","End":"06:17.925","Text":"That\u0027s the first one,"},{"Start":"06:17.925 ","End":"06:22.080","Text":"t. Now, to get N, we,"},{"Start":"06:22.080 ","End":"06:25.520","Text":"first of all, differentiate t. So we"},{"Start":"06:25.520 ","End":"06:30.710","Text":"need T prime of t. There\u0027s quite a few steps in this computation."},{"Start":"06:30.710 ","End":"06:41.535","Text":"What we get is minus 4 over root 17 cosine of t,"},{"Start":"06:41.535 ","End":"06:44.150","Text":"and derivative of cosine is minus sine."},{"Start":"06:44.150 ","End":"06:52.970","Text":"So minus 4 over root 17 sine t. But this is a constant,"},{"Start":"06:52.970 ","End":"06:56.880","Text":"so it is 0 here."},{"Start":"06:57.220 ","End":"07:05.685","Text":"Continuing, now we need to compute the magnitude of T-prime of"},{"Start":"07:05.685 ","End":"07:13.515","Text":"t. We can divide by it to get a unit vector which is going to be here,"},{"Start":"07:13.515 ","End":"07:16.495","Text":"the unit normal vector."},{"Start":"07:16.495 ","End":"07:18.570","Text":"This is equal to, again,"},{"Start":"07:18.570 ","End":"07:23.260","Text":"because cosine squared plus sine squared is 1,"},{"Start":"07:23.260 ","End":"07:30.465","Text":"we are just going to get 4 over root 17."},{"Start":"07:30.465 ","End":"07:36.490","Text":"You can just check that it comes out a plus because we\u0027re taking the squared,"},{"Start":"07:36.490 ","End":"07:37.690","Text":"now we\u0027re taking the square roots,"},{"Start":"07:37.690 ","End":"07:39.805","Text":"so it\u0027s going to come out plus."},{"Start":"07:39.805 ","End":"07:42.085","Text":"If we divide by this,"},{"Start":"07:42.085 ","End":"07:46.035","Text":"what we get is the unit normal vector,"},{"Start":"07:46.035 ","End":"07:51.930","Text":"is just minus cosine t,"},{"Start":"07:51.930 ","End":"07:57.520","Text":"minus sine t, and then 0."},{"Start":"07:57.650 ","End":"08:04.800","Text":"We\u0027ve got T of t. Yes,"},{"Start":"08:04.800 ","End":"08:06.255","Text":"we\u0027ve got N of t,."},{"Start":"08:06.255 ","End":"08:09.685","Text":"Yes, now we need the binormal,"},{"Start":"08:09.685 ","End":"08:14.755","Text":"which we just have to plug into this formula."},{"Start":"08:14.755 ","End":"08:18.975","Text":"We\u0027ve got that B of t is,"},{"Start":"08:18.975 ","End":"08:22.800","Text":"let\u0027s see now, we need T,"},{"Start":"08:22.800 ","End":"08:26.430","Text":"which is, I\u0027ll take"},{"Start":"08:26.430 ","End":"08:32.990","Text":"the square root of 17 outside the brackets,"},{"Start":"08:32.990 ","End":"08:36.855","Text":"and I\u0027ve got 1 over root 17."},{"Start":"08:36.855 ","End":"08:44.205","Text":"Then the tangent is minus 4 sine t,"},{"Start":"08:44.205 ","End":"08:53.070","Text":"4 cosine t, 1 cross with this 1,"},{"Start":"08:53.070 ","End":"08:56.925","Text":"with minus cosine t,"},{"Start":"08:56.925 ","End":"09:06.660","Text":"minus sine t, 0."},{"Start":"09:06.660 ","End":"09:11.705","Text":"What we get, I\u0027m not going to do the computation."},{"Start":"09:11.705 ","End":"09:13.550","Text":"I\u0027ll just give you the answer."},{"Start":"09:13.550 ","End":"09:18.860","Text":"What it comes out to be is 1 over root 17,"},{"Start":"09:18.860 ","End":"09:21.020","Text":"which I\u0027ll keep outside the bracket."},{"Start":"09:21.020 ","End":"09:26.245","Text":"Sine t in the first component,"},{"Start":"09:26.245 ","End":"09:30.410","Text":"minus cosine t in the second component,"},{"Start":"09:30.410 ","End":"09:33.845","Text":"and 4 in the third component."},{"Start":"09:33.845 ","End":"09:36.575","Text":"That\u0027s our binormal vector."},{"Start":"09:36.575 ","End":"09:39.370","Text":"I should have put the arrows."},{"Start":"09:39.370 ","End":"09:42.810","Text":"All along, I need to put the arrows,"},{"Start":"09:42.810 ","End":"09:47.770","Text":"and here, and wherever else I may have missed them."},{"Start":"09:47.960 ","End":"09:51.210","Text":"That\u0027s the binormal."},{"Start":"09:51.210 ","End":"09:53.460","Text":"I\u0027ll just highlight."},{"Start":"09:53.460 ","End":"09:57.470","Text":"The tangent vector is this one,"},{"Start":"09:57.470 ","End":"10:00.800","Text":"the normal vector is this one,"},{"Start":"10:00.800 ","End":"10:08.350","Text":"and the binormal vector is this one."},{"Start":"10:10.490 ","End":"10:14.680","Text":"We are done with this section."}],"ID":9705},{"Watched":false,"Name":"Exercise 8","Duration":"6m 34s","ChapterTopicVideoID":9838,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9838.jpeg","UploadDate":"2017-08-06T04:43:01.0100000","DurationForVideoObject":"PT6M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.040 ","End":"00:05.250","Text":"the following vector function in 3D,"},{"Start":"00:05.250 ","End":"00:07.560","Text":"and there\u0027s 2 parts."},{"Start":"00:07.560 ","End":"00:08.880","Text":"In part a, we have to find"},{"Start":"00:08.880 ","End":"00:11.000","Text":"the unit tangent vector,"},{"Start":"00:11.000 ","End":"00:12.930","Text":"and in part b we have to find"},{"Start":"00:12.930 ","End":"00:15.990","Text":"the tangent line at the given value of t."},{"Start":"00:15.990 ","End":"00:19.320","Text":"Let\u0027s start with part a."},{"Start":"00:19.320 ","End":"00:22.005","Text":"First of all, let\u0027s find a tangent vector."},{"Start":"00:22.005 ","End":"00:25.470","Text":"Now, tangent vector always is r prime,"},{"Start":"00:25.470 ","End":"00:26.670","Text":"so if we differentiate,"},{"Start":"00:26.670 ","End":"00:28.290","Text":"we\u0027ll get a tangent vector."},{"Start":"00:28.290 ","End":"00:30.420","Text":"What we get from here"},{"Start":"00:30.420 ","End":"00:32.730","Text":"is t squared gives us 2t,"},{"Start":"00:32.730 ","End":"00:37.350","Text":"sine 2t gives 2 cosine 2t,"},{"Start":"00:37.350 ","End":"00:44.110","Text":"and cosine 2t gives us minus 2 sine 2t."},{"Start":"00:44.360 ","End":"00:47.375","Text":"Now, how do I make this a unit vector?"},{"Start":"00:47.375 ","End":"00:49.670","Text":"Well, we divide by its magnitude."},{"Start":"00:49.670 ","End":"00:53.300","Text":"I\u0027ll need to compute the magnitude of this,"},{"Start":"00:53.300 ","End":"00:56.360","Text":"and that will be the square root"},{"Start":"00:56.360 ","End":"00:57.960","Text":"of this squared, and this squared,"},{"Start":"00:57.960 ","End":"00:58.890","Text":"and this squared,"},{"Start":"00:58.890 ","End":"01:05.780","Text":"so 2t squared plus 2 cosine 2t squared,"},{"Start":"01:05.780 ","End":"01:08.060","Text":"minus doesn\u0027t matter because it\u0027s squared,"},{"Start":"01:08.060 ","End":"01:14.325","Text":"so plus 2 sine 2t squared."},{"Start":"01:14.325 ","End":"01:16.530","Text":"Let\u0027s see if we can simplify this,"},{"Start":"01:16.530 ","End":"01:18.880","Text":"what we get."},{"Start":"01:18.890 ","End":"01:22.065","Text":"Well, I\u0027ll tell you what,"},{"Start":"01:22.065 ","End":"01:24.450","Text":"we can take the 2 out,"},{"Start":"01:24.450 ","End":"01:26.640","Text":"because here we have 2 squared,"},{"Start":"01:26.640 ","End":"01:27.570","Text":"here we have 2 squared,"},{"Start":"01:27.570 ","End":"01:28.890","Text":"and here we have 2 squared,"},{"Start":"01:28.890 ","End":"01:30.820","Text":"so I can take 2 squared out the brackets."},{"Start":"01:30.820 ","End":"01:32.260","Text":"But if I take it out of the root,"},{"Start":"01:32.260 ","End":"01:34.180","Text":"it just comes as 2."},{"Start":"01:34.180 ","End":"01:39.700","Text":"Then I get the root of t squared plus"},{"Start":"01:39.700 ","End":"01:46.520","Text":"cosine squared 2t plus sine squared 2t."},{"Start":"01:47.270 ","End":"01:50.230","Text":"The famous trigonometric identity"},{"Start":"01:50.230 ","End":"01:51.460","Text":"that the cosine squared plus"},{"Start":"01:51.460 ","End":"01:54.190","Text":"sine squared of anything is 1."},{"Start":"01:54.190 ","End":"01:59.650","Text":"What we get finally is 2 times"},{"Start":"01:59.650 ","End":"02:04.855","Text":"the square root of t squared plus 1."},{"Start":"02:04.855 ","End":"02:09.260","Text":"Now, we can write what the tangent is."},{"Start":"02:09.460 ","End":"02:15.320","Text":"The tangent vector at a point t is this"},{"Start":"02:15.320 ","End":"02:18.470","Text":"divided by this magnitude."},{"Start":"02:19.700 ","End":"02:23.270","Text":"We divide each of the components by this."},{"Start":"02:23.270 ","End":"02:25.415","Text":"The 2 is going to cancel everywhere."},{"Start":"02:25.415 ","End":"02:31.155","Text":"We\u0027ll get t over root t squared plus 1."},{"Start":"02:31.155 ","End":"02:36.965","Text":"Then we\u0027ll get cosine 2t over the same thing"},{"Start":"02:36.965 ","End":"02:44.540","Text":"and minus sine 2t over the same thing,"},{"Start":"02:44.540 ","End":"02:46.760","Text":"which is t squared plus 1,"},{"Start":"02:46.760 ","End":"02:49.615","Text":"t squared plus 1."},{"Start":"02:49.615 ","End":"02:53.310","Text":"That\u0027s a unit tangent vector."},{"Start":"02:53.310 ","End":"02:56.860","Text":"Should\u0027ve said part a."},{"Start":"02:56.930 ","End":"02:59.730","Text":"Now, let\u0027s go to part b where"},{"Start":"02:59.730 ","End":"03:02.400","Text":"we have to find the tangent line."},{"Start":"03:02.400 ","End":"03:07.565","Text":"In part b, I can use any tangent I want."},{"Start":"03:07.565 ","End":"03:10.955","Text":"I don\u0027t have to use the unit tangent."},{"Start":"03:10.955 ","End":"03:12.680","Text":"I could use this tangent"},{"Start":"03:12.680 ","End":"03:17.220","Text":"or any whole multiple of it."},{"Start":"03:17.220 ","End":"03:20.150","Text":"Scalar multiple, I could use this,"},{"Start":"03:20.150 ","End":"03:22.835","Text":"but I could ignore the 2 in front of each."},{"Start":"03:22.835 ","End":"03:26.580","Text":"I can take my tangent to be,"},{"Start":"03:26.580 ","End":"03:28.065","Text":"well I\u0027ll just write that."},{"Start":"03:28.065 ","End":"03:31.650","Text":"The vector t cosine 2t"},{"Start":"03:31.650 ","End":"03:37.960","Text":"minus sine 2t is a tangent vector."},{"Start":"03:40.040 ","End":"03:42.740","Text":"Usually, I just take this as-is."},{"Start":"03:42.740 ","End":"03:45.162","Text":"But when it\u0027s all divisible by 2,"},{"Start":"03:45.162 ","End":"03:47.465","Text":"it doesn\u0027t matter."},{"Start":"03:47.465 ","End":"03:49.580","Text":"A direction vector doesn\u0027t"},{"Start":"03:49.580 ","End":"03:52.860","Text":"depend on a scalar constant."},{"Start":"03:52.860 ","End":"03:55.455","Text":"This is a tangent vector."},{"Start":"03:55.455 ","End":"03:58.430","Text":"Now, I\u0027m going to write the equation"},{"Start":"03:58.430 ","End":"04:01.370","Text":"using a point and a direction vector."},{"Start":"04:01.370 ","End":"04:05.059","Text":"A point I have from substituting"},{"Start":"04:05.059 ","End":"04:06.995","Text":"t equals Pi over 2."},{"Start":"04:06.995 ","End":"04:13.290","Text":"What I need is r of Pi over 2"},{"Start":"04:13.290 ","End":"04:16.490","Text":"to see where the curve passes through,"},{"Start":"04:16.490 ","End":"04:19.250","Text":"and this is equal to,"},{"Start":"04:19.250 ","End":"04:22.940","Text":"well, we\u0027ve lost the original definition,"},{"Start":"04:22.940 ","End":"04:25.660","Text":"but it was t squared in the beginning."},{"Start":"04:25.660 ","End":"04:30.220","Text":"So it\u0027s Pi squared over 4."},{"Start":"04:30.950 ","End":"04:33.825","Text":"Then it was 2 sine 2t,"},{"Start":"04:33.825 ","End":"04:39.180","Text":"so it\u0027s 2 sine Pi because"},{"Start":"04:39.180 ","End":"04:42.270","Text":"2t is twice Pi over 2."},{"Start":"04:42.270 ","End":"04:46.200","Text":"Then we had 2 cosine 2t,"},{"Start":"04:46.200 ","End":"04:52.965","Text":"so 2 cosine of Pi and let\u0027s see,"},{"Start":"04:52.965 ","End":"04:58.830","Text":"this is equal to Pi squared over 4."},{"Start":"04:58.830 ","End":"05:02.805","Text":"Sine of Pi is 0."},{"Start":"05:02.805 ","End":"05:07.160","Text":"Cosine of Pi is minus 1,"},{"Start":"05:07.160 ","End":"05:09.545","Text":"so this is minus 2."},{"Start":"05:09.545 ","End":"05:13.915","Text":"We have a point or at least its position vector."},{"Start":"05:13.915 ","End":"05:16.310","Text":"We also need a direction vector,"},{"Start":"05:16.310 ","End":"05:18.475","Text":"which I can get from the tangent."},{"Start":"05:18.475 ","End":"05:20.540","Text":"I\u0027ll call the direction vector"},{"Start":"05:20.540 ","End":"05:23.060","Text":"at that point v is equal to,"},{"Start":"05:23.060 ","End":"05:25.600","Text":"I\u0027ll just substitute Pi over 2 in this,"},{"Start":"05:25.600 ","End":"05:30.990","Text":"and I get Pi over 2."},{"Start":"05:31.070 ","End":"05:38.795","Text":"Cosine of Pi is minus 1 and sine of Pi is 0,"},{"Start":"05:38.795 ","End":"05:41.350","Text":"so this is what I get."},{"Start":"05:41.350 ","End":"05:44.010","Text":"The equation of the line,"},{"Start":"05:44.010 ","End":"05:47.939","Text":"I combine these 2 in the usual formula."},{"Start":"05:47.939 ","End":"05:51.890","Text":"Let\u0027s make it l of also a parameter t"},{"Start":"05:51.890 ","End":"05:55.760","Text":"is equal to the position vector,"},{"Start":"05:55.760 ","End":"05:57.360","Text":"which is, just copy it,"},{"Start":"05:57.360 ","End":"05:59.210","Text":"Pi squared over 4,"},{"Start":"05:59.210 ","End":"06:05.435","Text":"0 minus 2 plus t times this 1,"},{"Start":"06:05.435 ","End":"06:09.390","Text":"Pi over 2 minus 1, 0."},{"Start":"06:09.390 ","End":"06:11.475","Text":"You could leave this as the answer."},{"Start":"06:11.475 ","End":"06:15.045","Text":"Optionally, some people like it combined."},{"Start":"06:15.045 ","End":"06:17.580","Text":"You would write Pi squared"},{"Start":"06:17.580 ","End":"06:26.310","Text":"over 4 plus Pi over 2t, minus t."},{"Start":"06:26.310 ","End":"06:31.290","Text":"The last 1 would be minus 2."},{"Start":"06:31.290 ","End":"06:35.380","Text":"That\u0027s it."}],"ID":9706},{"Watched":false,"Name":"Exercise 9","Duration":"5m 35s","ChapterTopicVideoID":9839,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9839.jpeg","UploadDate":"2017-08-06T04:43:15.0630000","DurationForVideoObject":"PT5M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.585","Text":"Here we\u0027re given a vector function in 3D and we have to find"},{"Start":"00:04.585 ","End":"00:12.185","Text":"the unit tangent vector and also the tangent line at the point where t is 0."},{"Start":"00:12.185 ","End":"00:16.510","Text":"Let\u0027s first of all find any tangent vector."},{"Start":"00:16.510 ","End":"00:22.690","Text":"The derivative is always a tangent and less than 0 of course."},{"Start":"00:22.690 ","End":"00:29.845","Text":"We can get it by differentiating each component separately."},{"Start":"00:29.845 ","End":"00:35.275","Text":"We get from the first component e^2t,"},{"Start":"00:35.275 ","End":"00:41.160","Text":"i, and from here minus 2e^t, j."},{"Start":"00:41.160 ","End":"00:47.260","Text":"The derivative is [inaudible] from here plus 2k."},{"Start":"00:49.130 ","End":"00:53.670","Text":"Now, let\u0027s say tangent vector."},{"Start":"00:53.670 ","End":"00:57.185","Text":"To make it a unit, we have to divide by its magnitude."},{"Start":"00:57.185 ","End":"01:03.065","Text":"Let\u0027s compute the magnitude of this and what we get is the square root"},{"Start":"01:03.065 ","End":"01:09.350","Text":"of this squared, which is e^4t."},{"Start":"01:09.350 ","End":"01:13.295","Text":"Then we need plus this thing squared."},{"Start":"01:13.295 ","End":"01:19.190","Text":"This thing squared is going to be plus 4e^t squared is e^2t."},{"Start":"01:19.190 ","End":"01:22.530","Text":"I\u0027m using rules of exponents."},{"Start":"01:23.060 ","End":"01:27.090","Text":"Finally 2 squared is 4."},{"Start":"01:27.090 ","End":"01:29.790","Text":"Now this looks messy."},{"Start":"01:29.790 ","End":"01:33.620","Text":"It is and we could leave it like that."},{"Start":"01:33.620 ","End":"01:37.940","Text":"But if you\u0027re sharped eye then you\u0027ll notice this should remind"},{"Start":"01:37.940 ","End":"01:46.440","Text":"you something like x squared plus 4x plus 4."},{"Start":"01:46.440 ","End":"01:49.000","Text":"If I take x as e^2t,"},{"Start":"01:49.000 ","End":"01:51.545","Text":"and this is a perfect square,"},{"Start":"01:51.545 ","End":"01:54.315","Text":"it\u0027s x plus 2 squared."},{"Start":"01:54.315 ","End":"01:57.990","Text":"I claim that this thing therefore is equal to"},{"Start":"01:57.990 ","End":"02:06.900","Text":"just e^2t plus 2."},{"Start":"02:06.900 ","End":"02:09.725","Text":"If we divide this by this,"},{"Start":"02:09.725 ","End":"02:13.330","Text":"we\u0027ll get a unit tangent vector,"},{"Start":"02:13.330 ","End":"02:17.565","Text":"capital T usually, which is equal."},{"Start":"02:17.565 ","End":"02:19.590","Text":"I\u0027ll just divide each component."},{"Start":"02:19.590 ","End":"02:27.889","Text":"I get e^2t over e^2t plus 2,"},{"Start":"02:27.889 ","End":"02:36.665","Text":"i minus 2e^t over e^2t"},{"Start":"02:36.665 ","End":"02:45.180","Text":"plus 2j plus 2 over e^2t plus 2k."},{"Start":"02:46.680 ","End":"02:49.510","Text":"To find the tangent line,"},{"Start":"02:49.510 ","End":"02:55.749","Text":"we need a point for its position vector and the direction vector."},{"Start":"02:55.749 ","End":"03:03.515","Text":"Now, the point we can get by just computing r of 0."},{"Start":"03:03.515 ","End":"03:07.345","Text":"Well, that will give us the direction vector of the point."},{"Start":"03:07.345 ","End":"03:09.565","Text":"Same thing almost."},{"Start":"03:09.565 ","End":"03:11.170","Text":"This is equal 2."},{"Start":"03:11.170 ","End":"03:12.670","Text":"If I let t equals 0,"},{"Start":"03:12.670 ","End":"03:18.670","Text":"I get 1/2 i because e^0 is 1"},{"Start":"03:18.670 ","End":"03:26.865","Text":"minus 2j and t is 0."},{"Start":"03:26.865 ","End":"03:30.430","Text":"I don\u0027t get anything in the k direction."},{"Start":"03:30.560 ","End":"03:34.235","Text":"For a direction vector,"},{"Start":"03:34.235 ","End":"03:38.660","Text":"what we can do is we don\u0027t have to use the unit tangent."},{"Start":"03:38.660 ","End":"03:40.070","Text":"We can use any tangent."},{"Start":"03:40.070 ","End":"03:42.125","Text":"The tangent is the direction vector."},{"Start":"03:42.125 ","End":"03:44.240","Text":"I\u0027m going to use it in this form,"},{"Start":"03:44.240 ","End":"03:53.180","Text":"so r prime of 0."},{"Start":"03:53.180 ","End":"04:00.610","Text":"This is rv direction vector is equal to,"},{"Start":"04:01.190 ","End":"04:08.920","Text":"plug in to r prime and we\u0027ll get just i"},{"Start":"04:08.920 ","End":"04:20.070","Text":"minus 2j adding t equals 0 plus 2k."},{"Start":"04:20.320 ","End":"04:23.420","Text":"Now from these two,"},{"Start":"04:23.420 ","End":"04:26.870","Text":"I can get the equation of the line."},{"Start":"04:26.870 ","End":"04:30.640","Text":"We want to call the line anything but r, say,"},{"Start":"04:30.640 ","End":"04:37.540","Text":"lt is going to equal this plus the parameter."},{"Start":"04:37.540 ","End":"04:41.150","Text":"We can use t again times this,"},{"Start":"04:41.150 ","End":"04:47.430","Text":"so we have 1/2i minus 2j"},{"Start":"04:47.430 ","End":"04:55.030","Text":"plus t times i minus 2j plus 2k."},{"Start":"04:55.760 ","End":"04:58.025","Text":"That\u0027s the answer."},{"Start":"04:58.025 ","End":"04:59.660","Text":"But optionally,"},{"Start":"04:59.660 ","End":"05:07.040","Text":"we could separate it into components and say that we have 1/2 plus ti,"},{"Start":"05:07.040 ","End":"05:09.290","Text":"combining this and this."},{"Start":"05:09.290 ","End":"05:18.230","Text":"Then here we have minus 2 and then minus 2t."},{"Start":"05:18.230 ","End":"05:23.040","Text":"It\u0027s 2 plus 2t,"},{"Start":"05:23.040 ","End":"05:26.535","Text":"j and no k here."},{"Start":"05:26.535 ","End":"05:29.290","Text":"Just from here 2tk,"},{"Start":"05:29.380 ","End":"05:35.130","Text":"either one of these forms will do. That\u0027s it."}],"ID":9707},{"Watched":false,"Name":"Exercise 10","Duration":"6m 41s","ChapterTopicVideoID":9836,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9836.jpeg","UploadDate":"2017-08-06T04:42:18.9630000","DurationForVideoObject":"PT6M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.825","Text":"Here we\u0027re given a vector function of a parameter t as follows."},{"Start":"00:06.825 ","End":"00:12.075","Text":"We want to find its unit normal and the unit binormal."},{"Start":"00:12.075 ","End":"00:17.520","Text":"Let\u0027s say this is part a and the binormal is part b,"},{"Start":"00:17.520 ","End":"00:20.470","Text":"so let\u0027s start with part a."},{"Start":"00:20.510 ","End":"00:25.270","Text":"We need to find the unit tangent first."},{"Start":"00:27.050 ","End":"00:35.460","Text":"We find a tangent by taking the derivative of r. This"},{"Start":"00:35.460 ","End":"00:43.695","Text":"will equal say 0, 3 cosine 3t."},{"Start":"00:43.695 ","End":"00:49.360","Text":"From here minus 3 sine 3t."},{"Start":"00:49.360 ","End":"00:53.810","Text":"Now, we want to divide this by its magnitude to get the unit tangent,"},{"Start":"00:53.810 ","End":"01:00.710","Text":"so the magnitude of this derivative is the magnitude of this,"},{"Start":"01:00.710 ","End":"01:06.379","Text":"which is the square root of 0 squared plus"},{"Start":"01:06.379 ","End":"01:16.740","Text":"9 cosine squared 3t plus 9 sine squared 3t."},{"Start":"01:17.030 ","End":"01:23.510","Text":"Now, we can take 9 out of the brackets and cosine squared plus sine squared is 1."},{"Start":"01:23.510 ","End":"01:29.680","Text":"This is just equal to 3, square root of 9."},{"Start":"01:29.680 ","End":"01:37.880","Text":"Now we can divide this by this and look how nicely it comes out."},{"Start":"01:37.880 ","End":"01:40.460","Text":"The unit tangent vector,"},{"Start":"01:40.460 ","End":"01:51.520","Text":"which is this divided by this is just 0 cosine 3t minus sine 3t."},{"Start":"01:52.790 ","End":"01:55.820","Text":"Now we found the tangent."},{"Start":"01:55.820 ","End":"01:58.640","Text":"What about the normal?"},{"Start":"01:58.640 ","End":"02:04.909","Text":"A normal is the derivative of the unit tangent,"},{"Start":"02:04.909 ","End":"02:12.350","Text":"so we\u0027ll need T prime of t and that is 0"},{"Start":"02:12.350 ","End":"02:18.690","Text":"minus 3 sine 3t"},{"Start":"02:18.700 ","End":"02:24.390","Text":"minus 3 cosine 3t."},{"Start":"02:25.970 ","End":"02:31.055","Text":"Once again, we need to find the magnitude."},{"Start":"02:31.055 ","End":"02:35.000","Text":"To get this to be a unit vector,"},{"Start":"02:35.000 ","End":"02:41.720","Text":"this will be the normal after I\u0027ve divided this by its magnitude."},{"Start":"02:41.720 ","End":"02:44.165","Text":"Anyway, this has got to equal."},{"Start":"02:44.165 ","End":"02:46.220","Text":"We get the same thing as before,"},{"Start":"02:46.220 ","End":"02:50.490","Text":"9 sine squared, 9 cosine squared."},{"Start":"02:50.490 ","End":"02:53.040","Text":"This is also going to be 3."},{"Start":"02:53.040 ","End":"02:54.900","Text":"I\u0027ll just write similarly,"},{"Start":"02:54.900 ","End":"02:59.165","Text":"just almost the same calculation as here."},{"Start":"02:59.165 ","End":"03:02.990","Text":"Square root of this squared plus this squared, we\u0027ll get 3."},{"Start":"03:02.990 ","End":"03:06.545","Text":"If we divide this by this,"},{"Start":"03:06.545 ","End":"03:15.270","Text":"that will give us the unit normal vector and this is equal to 0,"},{"Start":"03:15.270 ","End":"03:21.075","Text":"1/3, cosine,"},{"Start":"03:21.075 ","End":"03:27.095","Text":"sorry, I should be dividing this by this."},{"Start":"03:27.095 ","End":"03:30.170","Text":"So 0 over 3 is 0."},{"Start":"03:30.170 ","End":"03:35.535","Text":"This over 3 is minus sine 3t."},{"Start":"03:35.535 ","End":"03:41.550","Text":"This over 3 is minus cosine 3t."},{"Start":"03:41.550 ","End":"03:48.285","Text":"That\u0027s part a and I\u0027ll highlight it."},{"Start":"03:48.285 ","End":"03:54.920","Text":"I\u0027d also like to highlight the unit tangent because together the unit tangent and"},{"Start":"03:54.920 ","End":"04:01.745","Text":"the unit normal are going to help us define the unit binormal."},{"Start":"04:01.745 ","End":"04:07.820","Text":"The formula for the binormal is just the cross product"},{"Start":"04:07.820 ","End":"04:16.010","Text":"of the unit tangent with the unit normal cross-product."},{"Start":"04:16.010 ","End":"04:18.850","Text":"It\u0027s this cross with this."},{"Start":"04:18.850 ","End":"04:28.515","Text":"This gives us a 3 by 3 determinant where we take here i,"},{"Start":"04:28.515 ","End":"04:35.520","Text":"j, and k. Then we put the coordinates of the unit tangent."},{"Start":"04:35.520 ","End":"04:43.455","Text":"That\u0027s 0 cosine 3t minus sine 3t."},{"Start":"04:43.455 ","End":"04:50.865","Text":"Then this is the tangent,"},{"Start":"04:50.865 ","End":"05:01.840","Text":"this is the normal, 0 minus sine 3t minus cosine 3t."},{"Start":"05:02.590 ","End":"05:04.925","Text":"Now what I\u0027m going to do,"},{"Start":"05:04.925 ","End":"05:08.780","Text":"since I have 2 0s here,"},{"Start":"05:08.780 ","End":"05:14.460","Text":"best thing would be to expand along this column."},{"Start":"05:14.870 ","End":"05:18.735","Text":"What I can say is it\u0027s just i"},{"Start":"05:18.735 ","End":"05:26.735","Text":"times the determinant of what\u0027s left when I cross out its row and column,"},{"Start":"05:26.735 ","End":"05:28.880","Text":"which is this thing."},{"Start":"05:28.880 ","End":"05:35.375","Text":"What I get is now the determinant of this, maybe I\u0027ll write it."},{"Start":"05:35.375 ","End":"05:39.560","Text":"Let\u0027s write it again, the determinant of cosine 3t"},{"Start":"05:39.560 ","End":"05:52.490","Text":"minus sine 3t minus sine 3t minus cosine 3t i."},{"Start":"05:52.490 ","End":"05:55.770","Text":"Now, to compute this,"},{"Start":"05:56.020 ","End":"06:03.185","Text":"we take the product of this diagonal minus the product of this diagonal."},{"Start":"06:03.185 ","End":"06:11.600","Text":"I get this times this is minus cosine squared 3t minus this diagonal."},{"Start":"06:11.600 ","End":"06:13.565","Text":"It\u0027s minus, minus minus."},{"Start":"06:13.565 ","End":"06:22.320","Text":"So it\u0027s minus sine squared 3t i."},{"Start":"06:22.320 ","End":"06:27.590","Text":"This is minus 1 because you can take the minus cosine squared plus sine squared."},{"Start":"06:27.590 ","End":"06:31.369","Text":"This is just equal to minus i,"},{"Start":"06:31.369 ","End":"06:32.900","Text":"and I\u0027ll write again,"},{"Start":"06:32.900 ","End":"06:38.510","Text":"b of t. I\u0027ll highlight it,"},{"Start":"06:38.510 ","End":"06:41.850","Text":"and that\u0027s the answer for b and we\u0027re done."}],"ID":9708},{"Watched":false,"Name":"Exercise 11","Duration":"10m 49s","ChapterTopicVideoID":9837,"CourseChapterTopicPlaylistID":57481,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9837.jpeg","UploadDate":"2017-08-06T04:42:45.3000000","DurationForVideoObject":"PT10M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.500","Text":"In this exercise, we have a 3D parameterized curve, r of t,"},{"Start":"00:05.500 ","End":"00:07.585","Text":"and here are the three components,"},{"Start":"00:07.585 ","End":"00:13.000","Text":"and we have to compute the orthonormal frame."},{"Start":"00:13.000 ","End":"00:15.840","Text":"Orthonormal, it\u0027s sometimes called,"},{"Start":"00:15.840 ","End":"00:21.340","Text":"because each of these three is perpendicular to the other and they\u0027re all unit vectors."},{"Start":"00:21.340 ","End":"00:24.785","Text":"We\u0027re to compute basically t the unit tangent,"},{"Start":"00:24.785 ","End":"00:28.480","Text":"the unit normal, and the unit binormal for the curve."},{"Start":"00:28.480 ","End":"00:34.000","Text":"Then in Part b, we have to compute the curvature of this curve at any given point"},{"Start":"00:34.000 ","End":"00:40.524","Text":"t. We\u0027ll start with a and we\u0027ll start with the unit tangent."},{"Start":"00:40.524 ","End":"00:44.300","Text":"First of all, we need any tangent which is"},{"Start":"00:44.300 ","End":"00:49.400","Text":"a prime of t and afterwards we\u0027ll make it into a unit vector."},{"Start":"00:49.400 ","End":"00:53.405","Text":"This is equal to just differentiating each component:"},{"Start":"00:53.405 ","End":"00:59.595","Text":"1 and then 2 cosine t, and then from here,"},{"Start":"00:59.595 ","End":"01:06.075","Text":"minus 2 sine t. We need the magnitude,"},{"Start":"01:06.075 ","End":"01:11.735","Text":"because we\u0027re going to divide by the magnitude to get the unit vector as usual."},{"Start":"01:11.735 ","End":"01:13.310","Text":"This is equal too."},{"Start":"01:13.310 ","End":"01:16.460","Text":"We need the square root and each 1 squared."},{"Start":"01:16.460 ","End":"01:19.585","Text":"1 squared is 1,"},{"Start":"01:19.585 ","End":"01:25.270","Text":"2 cosine of t squared is 4 cosine squared t,"},{"Start":"01:25.270 ","End":"01:35.785","Text":"and here we have a plus 4 sine squared t. Because cosine squared plus sine squared is 1,"},{"Start":"01:35.785 ","End":"01:38.200","Text":"here we have square root of 1 plus 4,"},{"Start":"01:38.200 ","End":"01:42.100","Text":"this boils down to square root of 5."},{"Start":"01:42.100 ","End":"01:51.150","Text":"We can now say that the unit tangent which is this divided by its magnitude,"},{"Start":"01:51.150 ","End":"01:55.749","Text":"well I could write the 1 over square root of 5 in front,"},{"Start":"01:55.749 ","End":"01:58.240","Text":"and then have 1,"},{"Start":"01:58.240 ","End":"02:04.580","Text":"2 cosine t minus 2 sine t."},{"Start":"02:04.580 ","End":"02:11.265","Text":"That\u0027s one part out of three done and now let\u0027s go to the next one, the unit normal."},{"Start":"02:11.265 ","End":"02:13.215","Text":"We basically do the same thing."},{"Start":"02:13.215 ","End":"02:15.240","Text":"Only we start off not with r,"},{"Start":"02:15.240 ","End":"02:20.475","Text":"but with big T. We have"},{"Start":"02:20.475 ","End":"02:27.240","Text":"that big T prime of little t will be the derivative of this,"},{"Start":"02:27.240 ","End":"02:29.790","Text":"1 over square root of 5."},{"Start":"02:29.790 ","End":"02:32.310","Text":"Now 0 from the 1,"},{"Start":"02:32.310 ","End":"02:38.870","Text":"from here minus 2 sine t and from here"},{"Start":"02:38.870 ","End":"02:42.815","Text":"minus 2 cosine t."},{"Start":"02:42.815 ","End":"02:50.090","Text":"The magnitude of t is equal to 1 over the square root of 5."},{"Start":"02:50.090 ","End":"02:52.040","Text":"Then we need to do the square root."},{"Start":"02:52.040 ","End":"02:57.785","Text":"Here we have 4 sine squared t. The 0 squared, that isn\u0027t even write,"},{"Start":"02:57.785 ","End":"03:03.710","Text":"plus 4 cosine squared t. Once again,"},{"Start":"03:03.710 ","End":"03:06.830","Text":"sine squared plus cosine squared is 1."},{"Start":"03:06.830 ","End":"03:09.985","Text":"This is the square root of 4 which is 2,"},{"Start":"03:09.985 ","End":"03:15.495","Text":"and so it just boils down to 2 over the square root of 5."},{"Start":"03:15.495 ","End":"03:20.890","Text":"Now that means that I can get the unit"},{"Start":"03:20.890 ","End":"03:27.310","Text":"normal by taking this and dividing by its magnitude."},{"Start":"03:27.310 ","End":"03:30.175","Text":"If I divide by root 2 over 5,"},{"Start":"03:30.175 ","End":"03:33.870","Text":"it\u0027s like multiplying by root 5 over 2,"},{"Start":"03:33.870 ","End":"03:39.495","Text":"so we just get 1.5 of this."},{"Start":"03:39.495 ","End":"03:47.535","Text":"Yeah, 0 minus 2 sine t minus 2 cosine t,"},{"Start":"03:47.535 ","End":"03:49.755","Text":"and then we write that value again."},{"Start":"03:49.755 ","End":"03:56.940","Text":"The 1 over root 5 divided by the 2 over root"},{"Start":"03:56.940 ","End":"04:06.650","Text":"5 is 1 over root 5 times root 5 over 2 and this cancels and this is 1/2."},{"Start":"04:06.650 ","End":"04:09.515","Text":"Because there\u0027s a two here and here now I see,"},{"Start":"04:09.515 ","End":"04:12.330","Text":"then this is equal to"},{"Start":"04:13.820 ","End":"04:24.630","Text":"0 minus sine t and minus cosine t,"},{"Start":"04:24.630 ","End":"04:28.005","Text":"and that\u0027s 2 out of 3."},{"Start":"04:28.005 ","End":"04:30.985","Text":"Now we need the binormal."},{"Start":"04:30.985 ","End":"04:38.825","Text":"The binormal vector is gotten from the other 2 by using the cross-product."},{"Start":"04:38.825 ","End":"04:41.555","Text":"It\u0027s the unit tangent,"},{"Start":"04:41.555 ","End":"04:46.010","Text":"cross product with the unit normal."},{"Start":"04:46.010 ","End":"04:52.199","Text":"I\u0027m going to use the determinant method."},{"Start":"04:52.310 ","End":"04:55.050","Text":"This of course gives it in the i, j,"},{"Start":"04:55.050 ","End":"04:59.080","Text":"k form rather than component form."},{"Start":"04:59.180 ","End":"05:02.955","Text":"It\u0027s off the screen, let me just scroll back."},{"Start":"05:02.955 ","End":"05:05.040","Text":"I just copy pasted it."},{"Start":"05:05.040 ","End":"05:06.825","Text":"That\u0027s the easiest."},{"Start":"05:06.825 ","End":"05:15.255","Text":"I, j, k and then 1 over root 5,"},{"Start":"05:15.255 ","End":"05:22.365","Text":"2 over root 5, cosine t."},{"Start":"05:22.365 ","End":"05:26.055","Text":"I should have made this a bit bigger and there we are."},{"Start":"05:26.055 ","End":"05:31.410","Text":"Then minus 2 over root 5"},{"Start":"05:31.410 ","End":"05:38.340","Text":"sine t and then in the next row we need the normal."},{"Start":"05:38.340 ","End":"05:39.990","Text":"Here we have a 0."},{"Start":"05:39.990 ","End":"05:41.175","Text":"That\u0027s good."},{"Start":"05:41.175 ","End":"05:45.560","Text":"Then minus sine t and then"},{"Start":"05:45.560 ","End":"05:52.325","Text":"minus cosine t. We do it component by component."},{"Start":"05:52.325 ","End":"05:56.330","Text":"The first component is wherever times i."},{"Start":"05:56.330 ","End":"06:01.190","Text":"For i, we need to cross out the row and column,"},{"Start":"06:01.190 ","End":"06:04.235","Text":"so we need the determinant of this."},{"Start":"06:04.235 ","End":"06:06.320","Text":"Those were checkerboard thing plus,"},{"Start":"06:06.320 ","End":"06:08.920","Text":"minus, plus, so this one is plus."},{"Start":"06:08.920 ","End":"06:11.765","Text":"This times this minus this times this."},{"Start":"06:11.765 ","End":"06:13.220","Text":"Now if you ignore a moment,"},{"Start":"06:13.220 ","End":"06:15.380","Text":"the 2 over root 5,"},{"Start":"06:15.380 ","End":"06:21.540","Text":"we get minus cosine squared, minus sine squared."},{"Start":"06:21.540 ","End":"06:22.690","Text":"It\u0027s minus cosine squared,"},{"Start":"06:22.690 ","End":"06:25.050","Text":"minus sine squared which is minus 1,"},{"Start":"06:25.050 ","End":"06:27.785","Text":"but because of the 2 over root 5,"},{"Start":"06:27.785 ","End":"06:32.015","Text":"it just is 2 over root 5 times 1."},{"Start":"06:32.015 ","End":"06:34.825","Text":"That\u0027s the first component."},{"Start":"06:34.825 ","End":"06:37.935","Text":"Next for the j component,"},{"Start":"06:37.935 ","End":"06:41.240","Text":"the j component is a minus from the checkerboard plus,"},{"Start":"06:41.240 ","End":"06:43.775","Text":"minus, plus, so it\u0027s a minus."},{"Start":"06:43.775 ","End":"06:52.320","Text":"Then I\u0027ll need this and this together determinant."},{"Start":"06:52.320 ","End":"06:54.330","Text":"Now this diagonal is 0,"},{"Start":"06:54.330 ","End":"06:56.370","Text":"so I\u0027ll just need this diagonal\u0027s product,"},{"Start":"06:56.370 ","End":"06:57.630","Text":"but with a minus,"},{"Start":"06:57.630 ","End":"07:04.245","Text":"so it\u0027s just 1 over root 5 cosine t without the minus."},{"Start":"07:04.245 ","End":"07:05.820","Text":"Now the last one,"},{"Start":"07:05.820 ","End":"07:12.705","Text":"the k is a plus and we need the determinant of this."},{"Start":"07:12.705 ","End":"07:15.240","Text":"Once again, one of the diagonals is 0,"},{"Start":"07:15.240 ","End":"07:17.185","Text":"so it\u0027s just this diagonal,"},{"Start":"07:17.185 ","End":"07:22.160","Text":"and it\u0027s going to be minus 1 over root 5"},{"Start":"07:22.160 ","End":"07:31.635","Text":"sine t. That\u0027s the binormal and I\u0027ll highlight it,"},{"Start":"07:31.635 ","End":"07:34.040","Text":"and we\u0027ve actually answered all of Part a."},{"Start":"07:34.040 ","End":"07:35.615","Text":"We have the tangent first,"},{"Start":"07:35.615 ","End":"07:38.750","Text":"then the normal, and then the binormal."},{"Start":"07:38.750 ","End":"07:44.729","Text":"For Part b, we need the formula for the curvature Kappa,"},{"Start":"07:44.729 ","End":"07:45.980","Text":"and here it is."},{"Start":"07:45.980 ","End":"07:50.420","Text":"It\u0027s the magnitude of the derivative of the unit tangent over"},{"Start":"07:50.420 ","End":"07:55.295","Text":"the magnitude of the derivative of the function itself."},{"Start":"07:55.295 ","End":"07:58.380","Text":"Let\u0027s do this computations."},{"Start":"07:59.090 ","End":"08:03.005","Text":"We had some of this computed in Part a."},{"Start":"08:03.005 ","End":"08:06.665","Text":"For example we had a prime of t,"},{"Start":"08:06.665 ","End":"08:08.990","Text":"but it doesn\u0027t hurt to do it again,"},{"Start":"08:08.990 ","End":"08:11.350","Text":"is equal to 1,"},{"Start":"08:11.350 ","End":"08:18.150","Text":"2 cosine t minus 2 sine t,"},{"Start":"08:18.150 ","End":"08:23.884","Text":"and we also computed the unit tangent of t,"},{"Start":"08:23.884 ","End":"08:30.180","Text":"and that came out to be 1 over root 5."},{"Start":"08:30.180 ","End":"08:33.870","Text":"1, the same thing as this,"},{"Start":"08:33.870 ","End":"08:41.010","Text":"2 cosine t minus 2 sine t. That\u0027s right."},{"Start":"08:41.010 ","End":"08:43.385","Text":"We got this from dividing this by its magnitude,"},{"Start":"08:43.385 ","End":"08:46.045","Text":"but we don\u0027t want t,"},{"Start":"08:46.045 ","End":"08:48.020","Text":"we want T prime,"},{"Start":"08:48.020 ","End":"08:51.770","Text":"so we just have to differentiate that,"},{"Start":"08:51.770 ","End":"08:54.140","Text":"and its constant stays."},{"Start":"08:54.140 ","End":"08:56.615","Text":"This comes out to be 0."},{"Start":"08:56.615 ","End":"09:01.205","Text":"This will just be minus 2 sine t,"},{"Start":"09:01.205 ","End":"09:09.040","Text":"and this will be minus 2 cosine t,"},{"Start":"09:09.040 ","End":"09:16.700","Text":"and we need the magnitude of T prime of t. Of course"},{"Start":"09:16.700 ","End":"09:24.315","Text":"we also had the magnitude of r prime of t. We computed that earlier."},{"Start":"09:24.315 ","End":"09:27.230","Text":"Remember we used sine squared plus cosine squared is 1"},{"Start":"09:27.230 ","End":"09:30.640","Text":"and we got that this was the square root of 5,"},{"Start":"09:30.640 ","End":"09:32.090","Text":"so that\u0027s the denominator."},{"Start":"09:32.090 ","End":"09:38.360","Text":"Now we just need the numerator and this is going to equal again using the square root."},{"Start":"09:38.360 ","End":"09:42.840","Text":"Well the 1 over square root of 5 can stay"},{"Start":"09:42.840 ","End":"09:47.990","Text":"outside and now we just have to do 0 squared plus 4,"},{"Start":"09:47.990 ","End":"09:51.805","Text":"sine squared t plus 4,"},{"Start":"09:51.805 ","End":"09:58.860","Text":"cosine squared t. Again sine over cosine squared is 1,"},{"Start":"09:58.860 ","End":"10:00.540","Text":"square root of 4 is 2,"},{"Start":"10:00.540 ","End":"10:05.775","Text":"so this is just 2 over the square root of 5."},{"Start":"10:05.775 ","End":"10:10.780","Text":"It\u0027s a constant, it doesn\u0027t depend on t it seems."},{"Start":"10:11.840 ","End":"10:15.800","Text":"Looks like this also doesn\u0027t depend on t,"},{"Start":"10:15.800 ","End":"10:19.610","Text":"so what we get is that Kappa although"},{"Start":"10:19.610 ","End":"10:23.630","Text":"theoretically a function of t is actually a constant,"},{"Start":"10:23.630 ","End":"10:35.135","Text":"it\u0027s equal to 2 over root 5 divided by root 5,"},{"Start":"10:35.135 ","End":"10:40.610","Text":"and this just comes out to be 2/5."},{"Start":"10:40.610 ","End":"10:50.320","Text":"That\u0027s the answer to Part b. Kappa is 2/5 constant and that completes the exercise."}],"ID":9709}],"Thumbnail":null,"ID":57481},{"Name":"Arc Length with Vector Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"3D Space - Arc Length with Vector Function","Duration":"15m 8s","ChapterTopicVideoID":9868,"CourseChapterTopicPlaylistID":57482,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9868.jpeg","UploadDate":"2020-02-22T16:12:27.0570000","DurationForVideoObject":"PT15M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.595","Text":"We\u0027re continuing with 3 dimensional coordinates and space."},{"Start":"00:05.595 ","End":"00:11.880","Text":"This topic will be arc length with vector functions."},{"Start":"00:11.880 ","End":"00:19.410","Text":"Arc length is also known as curved length and the 2 are practically synonymous."},{"Start":"00:19.410 ","End":"00:22.230","Text":"But mostly here in this chapter I\u0027ll call it"},{"Start":"00:22.230 ","End":"00:25.155","Text":"arc length if only because it\u0027s a shorter word."},{"Start":"00:25.155 ","End":"00:29.490","Text":"We\u0027ve pretty much covered this in 2D space,"},{"Start":"00:29.490 ","End":"00:32.130","Text":"perhaps not with vector functions."},{"Start":"00:32.130 ","End":"00:42.000","Text":"In fact, let me jump to the lecture on the curve length in 2D."},{"Start":"00:42.000 ","End":"00:44.355","Text":"This is a flashback."},{"Start":"00:44.355 ","End":"00:51.080","Text":"We actually talked about 4 separate cases of length of curve or arc length."},{"Start":"00:51.080 ","End":"00:54.115","Text":"One was when y was given as a function of x,"},{"Start":"00:54.115 ","End":"00:57.095","Text":"the other was when x is a function of y."},{"Start":"00:57.095 ","End":"01:01.430","Text":"Then we had the parametric where x and y are both functions of t,"},{"Start":"01:01.430 ","End":"01:03.920","Text":"and then there was the implicit form."},{"Start":"01:03.920 ","End":"01:09.375","Text":"The thing that is most close to the vectors is the parametric form."},{"Start":"01:09.375 ","End":"01:16.929","Text":"Then we gave the following equation for the length of curve from one point to another."},{"Start":"01:16.929 ","End":"01:20.510","Text":"I\u0027m going to use basically this formula only."},{"Start":"01:20.510 ","End":"01:23.090","Text":"I\u0027m going to modify it and give these functions"},{"Start":"01:23.090 ","End":"01:27.510","Text":"names f and g. Let\u0027s get back to the present."},{"Start":"01:28.280 ","End":"01:31.520","Text":"I\u0027m going to talk about 2D,"},{"Start":"01:31.520 ","End":"01:34.280","Text":"which is very similar to 3D,"},{"Start":"01:34.280 ","End":"01:36.575","Text":"and I\u0027ll just generalize."},{"Start":"01:36.575 ","End":"01:41.840","Text":"In 2D, I\u0027ll remind you of the parametric form."},{"Start":"01:41.840 ","End":"01:48.575","Text":"We have x equals f of t,"},{"Start":"01:48.575 ","End":"01:56.090","Text":"y equals g of t. The parametric form is just"},{"Start":"01:56.090 ","End":"02:05.004","Text":"like the vector form where we take r of t is equal to f of t,"},{"Start":"02:05.004 ","End":"02:12.540","Text":"g of t. It\u0027s essentially the same thing just in a different format."},{"Start":"02:14.920 ","End":"02:18.350","Text":"If we take t between a and b,"},{"Start":"02:18.350 ","End":"02:23.250","Text":"let\u0027s say that we want to take t in this range,"},{"Start":"02:23.250 ","End":"02:26.180","Text":"then the arc length is given by"},{"Start":"02:26.180 ","End":"02:30.305","Text":"the formula which I showed you from earlier, just slightly revised,"},{"Start":"02:30.305 ","End":"02:38.015","Text":"is the integral from a to b of the square root of"},{"Start":"02:38.015 ","End":"02:48.625","Text":"f prime of t squared plus g prime of t squared dt."},{"Start":"02:48.625 ","End":"02:51.404","Text":"I want just to be safe,"},{"Start":"02:51.404 ","End":"02:53.030","Text":"so there\u0027s no ambiguity."},{"Start":"02:53.030 ","End":"02:58.890","Text":"I\u0027ll put some brackets around here that this whole thing is squared."},{"Start":"02:59.110 ","End":"03:04.355","Text":"This can now be generalized to 3D."},{"Start":"03:04.355 ","End":"03:07.835","Text":"In 3D, in parametric form,"},{"Start":"03:07.835 ","End":"03:12.335","Text":"we might have x equals same thing."},{"Start":"03:12.335 ","End":"03:17.870","Text":"Let\u0027s say that we have x equals f of t,"},{"Start":"03:17.870 ","End":"03:21.319","Text":"y equals g of t,"},{"Start":"03:21.319 ","End":"03:26.285","Text":"z equals h of t. Next letter in the alphabet,"},{"Start":"03:26.285 ","End":"03:27.490","Text":"natural to use that."},{"Start":"03:27.490 ","End":"03:33.530","Text":"Again we\u0027ll restrict our parameter to be between a and b."},{"Start":"03:33.530 ","End":"03:37.355","Text":"We now get a 3D vector function of t,"},{"Start":"03:37.355 ","End":"03:41.225","Text":"which is going to be f of t,"},{"Start":"03:41.225 ","End":"03:47.060","Text":"g of t, h of t. We\u0027re just concerned about this."},{"Start":"03:47.060 ","End":"03:50.225","Text":"I mentioned in the parametric form because it\u0027s so analogous."},{"Start":"03:50.225 ","End":"03:52.400","Text":"But in any event in 3D,"},{"Start":"03:52.400 ","End":"03:56.525","Text":"the length function is just the generalization that you would expect."},{"Start":"03:56.525 ","End":"04:01.250","Text":"It\u0027s the integral from a to b of the square root sign,"},{"Start":"04:01.250 ","End":"04:05.780","Text":"I make that extra long of f prime,"},{"Start":"04:05.780 ","End":"04:11.870","Text":"of t squared plus g prime of t"},{"Start":"04:11.870 ","End":"04:20.700","Text":"squared plus h prime of t squared dt."},{"Start":"04:21.190 ","End":"04:29.540","Text":"Now note that this expression from here up to"},{"Start":"04:29.540 ","End":"04:37.415","Text":"here is precisely r prime"},{"Start":"04:37.415 ","End":"04:42.225","Text":"of t, the magnitude."},{"Start":"04:42.225 ","End":"04:44.000","Text":"When we take a magnitude,"},{"Start":"04:44.000 ","End":"04:47.010","Text":"we take each of the components squared,"},{"Start":"04:47.010 ","End":"04:48.875","Text":"add and take the square root."},{"Start":"04:48.875 ","End":"04:54.275","Text":"Of course, r prime of t would just be with a prime here and a prime here."},{"Start":"04:54.275 ","End":"04:56.990","Text":"The same thing here."},{"Start":"04:56.990 ","End":"05:04.365","Text":"This is also r prime of t magnitude."},{"Start":"05:04.365 ","End":"05:09.379","Text":"In both cases, in the 2D and the 3D case,"},{"Start":"05:09.379 ","End":"05:14.860","Text":"and actually this generalizes to higher dimensions also in any dimension,"},{"Start":"05:14.860 ","End":"05:21.430","Text":"we have the equation that L is equal to the integral from a to"},{"Start":"05:21.430 ","End":"05:31.515","Text":"b of the magnitude of r prime of t. I keep forgetting to put the arrows."},{"Start":"05:31.515 ","End":"05:35.320","Text":"There\u0027s an arrow here and an arrow here."},{"Start":"05:35.320 ","End":"05:38.410","Text":"In some books they use boldface type instead of"},{"Start":"05:38.410 ","End":"05:45.085","Text":"an arrow but here we use arrows above the letter to show that it\u0027s a vector."},{"Start":"05:45.085 ","End":"05:51.665","Text":"Dt, this is important and I\u0027ll highlight it."},{"Start":"05:51.665 ","End":"05:54.965","Text":"What we need now is an example."},{"Start":"05:54.965 ","End":"05:59.570","Text":"What I\u0027ll do is I\u0027ll just erase stuff that I don\u0027t need first."},{"Start":"05:59.570 ","End":"06:02.930","Text":"As an example I\u0027m going to take the helix we used earlier."},{"Start":"06:02.930 ","End":"06:08.645","Text":"I just did a copy paste of what it was and we\u0027ll limit"},{"Start":"06:08.645 ","End":"06:15.960","Text":"t to between 0 and 2Pi."},{"Start":"06:16.150 ","End":"06:26.400","Text":"What we need is r prime of t. Derivative of cosine is minus sine,"},{"Start":"06:26.400 ","End":"06:29.094","Text":"so minus 4 sinet,"},{"Start":"06:29.094 ","End":"06:34.660","Text":"4cosinet, derivative of t is 1."},{"Start":"06:39.740 ","End":"06:47.725","Text":"I remember now we said sine squared plus cosine squared is 1 and 4 squared is 16."},{"Start":"06:47.725 ","End":"06:51.860","Text":"I have 16 times 1 plus 1 is 17."},{"Start":"06:51.860 ","End":"06:54.405","Text":"It came out square root of 17."},{"Start":"06:54.405 ","End":"06:55.680","Text":"We\u0027ve done it before,"},{"Start":"06:55.680 ","End":"06:57.105","Text":"so I did it quickly."},{"Start":"06:57.105 ","End":"07:02.380","Text":"Actually it comes at a constant which makes it less interesting but easier to do."},{"Start":"07:02.380 ","End":"07:05.060","Text":"What we get is that the length of curve,"},{"Start":"07:05.060 ","End":"07:14.540","Text":"is the integral from 0-2Pi of a constant square root of 17 dt."},{"Start":"07:14.540 ","End":"07:16.985","Text":"Because it\u0027s a constant,"},{"Start":"07:16.985 ","End":"07:24.245","Text":"I can take it in front and the integral of 1 is t. Basically,"},{"Start":"07:24.245 ","End":"07:31.475","Text":"it\u0027s easy to see that what we get is 2Pi square root of 17."},{"Start":"07:31.475 ","End":"07:35.330","Text":"I\u0027ve settled for this simple example because there are more examples in"},{"Start":"07:35.330 ","End":"07:40.155","Text":"the solved exercises after the tutorial."},{"Start":"07:40.155 ","End":"07:43.135","Text":"If instead of 2 Pi here,"},{"Start":"07:43.135 ","End":"07:44.860","Text":"I put a variable,"},{"Start":"07:44.860 ","End":"07:47.515","Text":"I could even use t,"},{"Start":"07:47.515 ","End":"07:51.115","Text":"then we would get that the length would be a function of"},{"Start":"07:51.115 ","End":"07:55.330","Text":"t. This is the next generalization that we\u0027re going to do."},{"Start":"07:55.330 ","End":"08:00.325","Text":"We\u0027re going to take arc length with a fixed lower limit here,"},{"Start":"08:00.325 ","End":"08:04.870","Text":"and a variable upper limit and it\u0027s going to go from 0 to"},{"Start":"08:04.870 ","End":"08:10.150","Text":"t. But then we can\u0027t use the letter t for the variable so we\u0027ll use another one called u."},{"Start":"08:10.150 ","End":"08:17.770","Text":"If I take the integral not from a to b,"},{"Start":"08:17.770 ","End":"08:21.085","Text":"but from 0 to a variable,"},{"Start":"08:21.085 ","End":"08:26.680","Text":"and we\u0027ll use the variable t of the same thing,"},{"Start":"08:26.680 ","End":"08:30.144","Text":"so we have vector r,"},{"Start":"08:30.144 ","End":"08:32.890","Text":"but I can\u0027t use t anymore,"},{"Start":"08:32.890 ","End":"08:36.790","Text":"that\u0027s already taken, so what\u0027s after t?"},{"Start":"08:36.790 ","End":"08:42.355","Text":"U. I\u0027ll use the variable u, duo."},{"Start":"08:42.355 ","End":"08:47.770","Text":"Now, this expression is going to be a function of t. I could have called it l of t,"},{"Start":"08:47.770 ","End":"08:55.705","Text":"but it\u0027s customary to use lowercase s. I\u0027m going to define s of t to equal that."},{"Start":"08:55.705 ","End":"09:01.345","Text":"Basically, s is the curve length or arc length function,"},{"Start":"09:01.345 ","End":"09:06.085","Text":"where the parameter goes from 0 to t. The parameter now is u,"},{"Start":"09:06.085 ","End":"09:08.920","Text":"and that gives us a function of t,"},{"Start":"09:08.920 ","End":"09:19.880","Text":"so it\u0027s a curve length as a function of t. I\u0027ll highlight this one"},{"Start":"09:20.160 ","End":"09:24.700","Text":"but in a different color because this one is"},{"Start":"09:24.700 ","End":"09:34.270","Text":"a definition of the function S. We\u0027ll call it the arc length or curve length,"},{"Start":"09:34.270 ","End":"09:37.400","Text":"the arc length function."},{"Start":"09:37.860 ","End":"09:40.255","Text":"It varies."},{"Start":"09:40.255 ","End":"09:42.265","Text":"As I go along the curve,"},{"Start":"09:42.265 ","End":"09:45.700","Text":"I\u0027m getting a different length."},{"Start":"09:45.700 ","End":"09:47.470","Text":"The more I travel,"},{"Start":"09:47.470 ","End":"09:52.150","Text":"the more length function."},{"Start":"09:52.150 ","End":"09:56.829","Text":"In our case, if we keep to this simple function,"},{"Start":"09:56.829 ","End":"09:59.845","Text":"then what we get is the s of t,"},{"Start":"09:59.845 ","End":"10:03.520","Text":"is the integral from 0,"},{"Start":"10:03.520 ","End":"10:07.030","Text":"to 2 Pi, of a constant."},{"Start":"10:07.030 ","End":"10:10.450","Text":"Although this normally would be a function of u,"},{"Start":"10:10.450 ","End":"10:15.940","Text":"du, sorry, the whole point is that this is the variable."},{"Start":"10:15.940 ","End":"10:18.310","Text":"This is equal to,"},{"Start":"10:18.310 ","End":"10:21.730","Text":"just like above, basically it\u0027s 2 Pi minus 0."},{"Start":"10:21.730 ","End":"10:23.425","Text":"Here\u0027s t minus 0."},{"Start":"10:23.425 ","End":"10:26.900","Text":"We get root 17t."},{"Start":"10:27.450 ","End":"10:34.360","Text":"Very trivial examples but I think it\u0027s better than when it\u0027s too complicated."},{"Start":"10:34.360 ","End":"10:39.055","Text":"Actually it\u0027s good that it came out this simple, because,"},{"Start":"10:39.055 ","End":"10:44.660","Text":"if I say s equals square root of 17t,"},{"Start":"10:45.120 ","End":"10:52.820","Text":"I could reverse this and say that t equals 1 over root 17s."},{"Start":"10:52.980 ","End":"10:58.480","Text":"Now, I can do something called reparameterization."},{"Start":"10:58.480 ","End":"11:02.155","Text":"Up till now we\u0027ve been using t as the parameter,"},{"Start":"11:02.155 ","End":"11:06.880","Text":"but it\u0027s somehow nicer to use s as a parameter."},{"Start":"11:06.880 ","End":"11:10.420","Text":"T is some arbitrary parameter, well,"},{"Start":"11:10.420 ","End":"11:12.430","Text":"in this case it happens to be the angle,"},{"Start":"11:12.430 ","End":"11:15.385","Text":"but in general it doesn\u0027t have much of a meaning."},{"Start":"11:15.385 ","End":"11:19.310","Text":"S is really length of curve."},{"Start":"11:19.650 ","End":"11:25.675","Text":"It\u0027s more natural and it also has uses later on."},{"Start":"11:25.675 ","End":"11:29.545","Text":"What I do is if I switch from t to s,"},{"Start":"11:29.545 ","End":"11:35.560","Text":"the original r is now a function of s and is equal to,"},{"Start":"11:35.560 ","End":"11:37.480","Text":"so it\u0027s going to be, let\u0027s see,"},{"Start":"11:37.480 ","End":"11:47.720","Text":"4 cosine of t is s over root 17,"},{"Start":"11:49.140 ","End":"11:59.500","Text":"4 sine t is 4 sine of s over root 17,"},{"Start":"11:59.500 ","End":"12:09.010","Text":"and t is just s over root 17."},{"Start":"12:10.250 ","End":"12:15.700","Text":"This gives us the position vector of where we are"},{"Start":"12:15.700 ","End":"12:21.100","Text":"after we\u0027ve traveled a distance of s along the curve."},{"Start":"12:21.100 ","End":"12:23.740","Text":"The starting place for measuring distance,"},{"Start":"12:23.740 ","End":"12:27.335","Text":"of course, is where t equals 0."},{"Start":"12:27.335 ","End":"12:30.630","Text":"In this case, where t equals 0,"},{"Start":"12:30.630 ","End":"12:32.280","Text":"then s is also equal 0,"},{"Start":"12:32.280 ","End":"12:33.525","Text":"but that\u0027s not always."},{"Start":"12:33.525 ","End":"12:36.975","Text":"Anyway, we start measuring at the point where t is 0,"},{"Start":"12:36.975 ","End":"12:40.734","Text":"whatever that happens to be in the original,"},{"Start":"12:40.734 ","End":"12:43.030","Text":"and then we travel a distance of s,"},{"Start":"12:43.030 ","End":"12:45.595","Text":"and now we know where we are after we travel."},{"Start":"12:45.595 ","End":"12:47.110","Text":"S could be 2 units,"},{"Start":"12:47.110 ","End":"12:51.000","Text":"4, 7, whatever number of units."},{"Start":"12:51.000 ","End":"12:59.260","Text":"This is the basic example and this is what we call the the reparameterization."},{"Start":"12:59.260 ","End":"13:01.570","Text":"I just had to write that word."},{"Start":"13:01.570 ","End":"13:04.735","Text":"In the UK, it\u0027s spelled with an S,"},{"Start":"13:04.735 ","End":"13:11.920","Text":"US with a Z. I can now ask a question as follows."},{"Start":"13:11.920 ","End":"13:18.265","Text":"Where are we on this curve after we travel a distance of,"},{"Start":"13:18.265 ","End":"13:25.010","Text":"say, 5 Pi root 17?"},{"Start":"13:25.260 ","End":"13:28.900","Text":"What does that mean after we travel a distance of it?"},{"Start":"13:28.900 ","End":"13:31.585","Text":"It just means that the arc length,"},{"Start":"13:31.585 ","End":"13:35.725","Text":"s, is 5 Pi root 17."},{"Start":"13:35.725 ","End":"13:40.690","Text":"Then all we have to do is substitute that value in here,"},{"Start":"13:40.690 ","End":"13:44.690","Text":"and then we\u0027ll get the position vector of where we are."},{"Start":"13:47.520 ","End":"13:56.980","Text":"We get that r will be equal to, I mean,"},{"Start":"13:56.980 ","End":"13:59.305","Text":"when s is this,"},{"Start":"13:59.305 ","End":"14:04.105","Text":"we\u0027ll get, let\u0027s see,"},{"Start":"14:04.105 ","End":"14:09.625","Text":"s over square root of 17 would be 5 Pi,"},{"Start":"14:09.625 ","End":"14:16.075","Text":"so we\u0027d get 4 cosine of 5 Pi,"},{"Start":"14:16.075 ","End":"14:21.805","Text":"4 sine of 5 Pi,"},{"Start":"14:21.805 ","End":"14:29.140","Text":"and just 5 Pi."},{"Start":"14:29.140 ","End":"14:32.695","Text":"This comes out, let\u0027s see,"},{"Start":"14:32.695 ","End":"14:35.410","Text":"multiples of 2 Pi don\u0027t matter,"},{"Start":"14:35.410 ","End":"14:38.500","Text":"so it\u0027s 4 cosine Pi,"},{"Start":"14:38.500 ","End":"14:42.890","Text":"and that is negative 4,"},{"Start":"14:43.500 ","End":"14:47.260","Text":"so we\u0027ve got minus 4,"},{"Start":"14:47.260 ","End":"14:50.740","Text":"sine of 5 Pi is sine of Pi,"},{"Start":"14:50.740 ","End":"14:54.835","Text":"sine of Pi is 0,"},{"Start":"14:54.835 ","End":"14:59.420","Text":"and then 5 Pi is just 5 Pi."},{"Start":"15:01.470 ","End":"15:08.570","Text":"Next clip, we\u0027ll continue 3D space, a different topic."}],"ID":9710},{"Watched":false,"Name":"Exercise 2","Duration":"2m 18s","ChapterTopicVideoID":9843,"CourseChapterTopicPlaylistID":57482,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9843.jpeg","UploadDate":"2017-08-06T04:54:25.5930000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, we have to find the length of curve,"},{"Start":"00:02.970 ","End":"00:05.760","Text":"also known as arc length for"},{"Start":"00:05.760 ","End":"00:12.075","Text":"this parameterized vector function for t between these values."},{"Start":"00:12.075 ","End":"00:19.290","Text":"I wrote down the formula for the length and just have to do the computation."},{"Start":"00:19.290 ","End":"00:22.980","Text":"The first thing we need is r prime of t."},{"Start":"00:22.980 ","End":"00:26.925","Text":"That\u0027s just differentiate each component separately."},{"Start":"00:26.925 ","End":"00:29.490","Text":"This 1 gives me the constant 2."},{"Start":"00:29.490 ","End":"00:31.785","Text":"Here I get the constant minus 5,"},{"Start":"00:31.785 ","End":"00:34.389","Text":"and here I get the constant 4,"},{"Start":"00:34.389 ","End":"00:42.600","Text":"and now the magnitude of r prime of t is just the Pythagoras law."},{"Start":"00:42.600 ","End":"00:49.925","Text":"Square root of 2 squared minus 5 squared is plus 5 squared and then plus 4 squared."},{"Start":"00:49.925 ","End":"00:51.995","Text":"If we do the computation,"},{"Start":"00:51.995 ","End":"00:56.970","Text":"it\u0027s 4 plus 25 plus 16 so that gives"},{"Start":"00:56.970 ","End":"01:04.650","Text":"us 34 and 16 is 20 and 25, 45, root 45."},{"Start":"01:04.650 ","End":"01:06.990","Text":"But, I can take the 9 out,"},{"Start":"01:06.990 ","End":"01:09.165","Text":"but 9 comes out as 3,"},{"Start":"01:09.165 ","End":"01:12.960","Text":"so it\u0027s 3 root 5."},{"Start":"01:12.960 ","End":"01:16.775","Text":"Now what I need is the integral of this."},{"Start":"01:16.775 ","End":"01:20.120","Text":"Our a and b are minus 2 and 5,"},{"Start":"01:20.120 ","End":"01:28.030","Text":"so I need the integral from minus 2-5 of 3 root 5 dt."},{"Start":"01:28.030 ","End":"01:30.740","Text":"Now this is a constant function,"},{"Start":"01:30.740 ","End":"01:38.355","Text":"so it\u0027s just 3 root 5t evaluated between minus 2 and 5."},{"Start":"01:38.355 ","End":"01:40.275","Text":"Maybe I\u0027ll put a brackets here."},{"Start":"01:40.275 ","End":"01:42.420","Text":"What is this equal to?"},{"Start":"01:42.420 ","End":"01:44.625","Text":"If I plug in, well,"},{"Start":"01:44.625 ","End":"01:46.520","Text":"the constants you can take out,"},{"Start":"01:46.520 ","End":"01:48.830","Text":"you don\u0027t have to do the constant."},{"Start":"01:48.830 ","End":"01:51.450","Text":"Then we can just plug in for the t part,"},{"Start":"01:51.450 ","End":"01:53.145","Text":"5 and minus 2,"},{"Start":"01:53.145 ","End":"01:55.320","Text":"so when t is 5, t is 5,"},{"Start":"01:55.320 ","End":"01:57.420","Text":"and when t is minus 2, t is minus 2,"},{"Start":"01:57.420 ","End":"02:02.660","Text":"so we have minus minus 2 so the final answer will be,"},{"Start":"02:02.660 ","End":"02:05.599","Text":"this will be 5 plus 2 is 7,"},{"Start":"02:05.599 ","End":"02:07.625","Text":"7 times 3 is 21."},{"Start":"02:07.625 ","End":"02:11.150","Text":"I make it 21 root 5."},{"Start":"02:11.150 ","End":"02:17.750","Text":"Once again, this is 7, 7 times 3 is 21. That\u0027s it."}],"ID":9711},{"Watched":false,"Name":"Exercise 3","Duration":"3m 12s","ChapterTopicVideoID":9840,"CourseChapterTopicPlaylistID":57482,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9840.jpeg","UploadDate":"2017-08-06T04:53:37.1930000","DurationForVideoObject":"PT3M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to find the length of the curve, given as follows."},{"Start":"00:04.590 ","End":"00:11.385","Text":"It\u0027s parametric, and it\u0027s in 3 dimensions with the i, j, k notation."},{"Start":"00:11.385 ","End":"00:13.650","Text":"I wrote the formula already,"},{"Start":"00:13.650 ","End":"00:16.470","Text":"I copied it from the tutorial."},{"Start":"00:16.470 ","End":"00:23.085","Text":"Here, a and b are 0 and 3, and r is the function here."},{"Start":"00:23.085 ","End":"00:25.530","Text":"We first need r prime,"},{"Start":"00:25.530 ","End":"00:27.840","Text":"so r prime of t,"},{"Start":"00:27.840 ","End":"00:30.705","Text":"we just differentiate component wise."},{"Start":"00:30.705 ","End":"00:32.910","Text":"We get from the first one,"},{"Start":"00:32.910 ","End":"00:34.455","Text":"t squared gives 2t,"},{"Start":"00:34.455 ","End":"00:36.195","Text":"but there\u0027s also a root 2,"},{"Start":"00:36.195 ","End":"00:39.100","Text":"so it\u0027s 2 root 2ti."},{"Start":"00:39.100 ","End":"00:45.525","Text":"The second one, 1/3t cubed gives us t squared, exactly."},{"Start":"00:45.525 ","End":"00:50.710","Text":"J and 4t gives us just 4k."},{"Start":"00:51.010 ","End":"00:54.750","Text":"Now, we need the magnitude of this,"},{"Start":"00:54.750 ","End":"00:58.250","Text":"so the magnitude of r prime of t,"},{"Start":"00:58.250 ","End":"01:02.390","Text":"we just take the sum of the squares of the components, and the square root of that."},{"Start":"01:02.390 ","End":"01:09.620","Text":"I need 2 root 2 squared is 2 squared times 2 is 8."},{"Start":"01:09.620 ","End":"01:16.460","Text":"T-squared squared is t^4"},{"Start":"01:16.460 ","End":"01:25.860","Text":"and 4 squared is 16."},{"Start":"01:25.860 ","End":"01:28.320","Text":"Oh sorry, this is t squared."},{"Start":"01:28.320 ","End":"01:31.660","Text":"This is 8t squared, sorry."},{"Start":"01:32.270 ","End":"01:34.930","Text":"Now, as it happens,"},{"Start":"01:34.930 ","End":"01:36.880","Text":"this has a square root."},{"Start":"01:36.880 ","End":"01:46.060","Text":"Remember the formula, a plus b squared equals a squared plus 2ab plus b squared."},{"Start":"01:46.060 ","End":"01:48.910","Text":"Well, in our case we have this,"},{"Start":"01:48.910 ","End":"01:57.280","Text":"but if I rewrite it as t^4 plus 8t squared plus 16,"},{"Start":"01:57.280 ","End":"02:03.055","Text":"then what we get is this thing is t-squared squared."},{"Start":"02:03.055 ","End":"02:08.410","Text":"This thing is 4 squared and if you check this,"},{"Start":"02:08.410 ","End":"02:10.990","Text":"the first term works out, the last term works out,"},{"Start":"02:10.990 ","End":"02:15.050","Text":"and the middle term is twice this times this works out fine."},{"Start":"02:15.050 ","End":"02:23.120","Text":"This thing is equal to t squared plus 4, but you would have had to know that,"},{"Start":"02:23.120 ","End":"02:27.350","Text":"to notice this, otherwise it would be very messy to integrate."},{"Start":"02:27.350 ","End":"02:30.185","Text":"Now, we just have to take the integral of that,"},{"Start":"02:30.185 ","End":"02:35.990","Text":"which is this, of t squared plus 4dt from a to b,"},{"Start":"02:35.990 ","End":"02:38.910","Text":"which is from 0-3."},{"Start":"02:40.420 ","End":"02:44.180","Text":"This integral is 1/3t cubed,"},{"Start":"02:44.180 ","End":"02:46.355","Text":"this integral is 4t,"},{"Start":"02:46.355 ","End":"02:49.280","Text":"and altogether from 0-3."},{"Start":"02:49.280 ","End":"02:50.780","Text":"Now, when we plug in 0,"},{"Start":"02:50.780 ","End":"02:51.890","Text":"we get 0 everywhere."},{"Start":"02:51.890 ","End":"03:02.430","Text":"So we just have to take the 3, and so it\u0027s equal to 1/3 times 3 cubed is 1/3 of 27 is 9,"},{"Start":"03:02.430 ","End":"03:07.095","Text":"4 times 3 is 12,"},{"Start":"03:07.095 ","End":"03:12.250","Text":"so the answer is 21, and we\u0027re done."}],"ID":9712},{"Watched":false,"Name":"Exercise 4","Duration":"6m 3s","ChapterTopicVideoID":9841,"CourseChapterTopicPlaylistID":57482,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9841.jpeg","UploadDate":"2017-08-06T04:53:53.4570000","DurationForVideoObject":"PT6M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to find the arc length function s"},{"Start":"00:04.530 ","End":"00:08.535","Text":"of t for this parameterized vector function."},{"Start":"00:08.535 ","End":"00:10.680","Text":"Notice that it\u0027s in 2D,"},{"Start":"00:10.680 ","End":"00:15.030","Text":"but the formula works in 2D and in 3D."},{"Start":"00:15.030 ","End":"00:17.760","Text":"I think I\u0027m missing a prime here."},{"Start":"00:17.760 ","End":"00:20.205","Text":"Yeah, this is the formula."},{"Start":"00:20.205 ","End":"00:24.090","Text":"Let\u0027s build up to this step-by-step."},{"Start":"00:24.090 ","End":"00:28.590","Text":"First of all, we\u0027ll need r prime of t in general,"},{"Start":"00:28.590 ","End":"00:31.230","Text":"we\u0027ll need the derivative of the vector function,"},{"Start":"00:31.230 ","End":"00:33.915","Text":"so we just differentiate each component."},{"Start":"00:33.915 ","End":"00:37.435","Text":"1 plus 3t squared gives us 6t,"},{"Start":"00:37.435 ","End":"00:42.910","Text":"and from here we get 6t squared."},{"Start":"00:43.560 ","End":"00:46.330","Text":"I\u0027m going to compute the magnitude of this."},{"Start":"00:46.330 ","End":"00:50.515","Text":"We can do a bit of a shortcut because 6 appears here and here,"},{"Start":"00:50.515 ","End":"00:54.080","Text":"we can actually take 6 outside the vector,"},{"Start":"00:54.080 ","End":"00:56.010","Text":"the scalar times the vector,"},{"Start":"00:56.010 ","End":"00:58.380","Text":"and this will be simpler to compute."},{"Start":"00:58.380 ","End":"01:05.865","Text":"Now, the magnitude of r prime of t,"},{"Start":"01:05.865 ","End":"01:08.280","Text":"I\u0027ll replace t by u afterwards,"},{"Start":"01:08.280 ","End":"01:15.190","Text":"but let\u0027s just say r prime of t would equal the magnitude of this."},{"Start":"01:15.190 ","End":"01:18.340","Text":"Now a positive number goes in front of the magnitude,"},{"Start":"01:18.340 ","End":"01:21.125","Text":"so it\u0027s the magnitude of t,"},{"Start":"01:21.125 ","End":"01:25.030","Text":"t squared, normally would be the absolute value if you take a number out,"},{"Start":"01:25.030 ","End":"01:26.725","Text":"but 6 is positive."},{"Start":"01:26.725 ","End":"01:31.765","Text":"This is equal to 6 times the square root,"},{"Start":"01:31.765 ","End":"01:39.980","Text":"and here we have t squared plus t^4th."},{"Start":"01:40.630 ","End":"01:43.790","Text":"We can actually simplify this further,"},{"Start":"01:43.790 ","End":"01:46.775","Text":"because if you look what\u0027s under the square root sign,"},{"Start":"01:46.775 ","End":"01:52.590","Text":"what we have there is t squared times 1 plus t squared."},{"Start":"01:53.150 ","End":"01:55.500","Text":"Well, t is positive,"},{"Start":"01:55.500 ","End":"02:00.485","Text":"so when I take the square root of t squared, it\u0027s just t,"},{"Start":"02:00.485 ","End":"02:09.060","Text":"not minus t. We can get this as 6t times the square root of 1 plus t squared."},{"Start":"02:09.060 ","End":"02:14.105","Text":"Now, we\u0027re ready to substitute in this expression,"},{"Start":"02:14.105 ","End":"02:16.760","Text":"but because we use the letter t here,"},{"Start":"02:16.760 ","End":"02:18.530","Text":"we have to use a different letter here,"},{"Start":"02:18.530 ","End":"02:26.585","Text":"so what we get is that s of t is equal to the integral from 0 to t,"},{"Start":"02:26.585 ","End":"02:29.120","Text":"this function, but in terms of u,"},{"Start":"02:29.120 ","End":"02:36.125","Text":"so I\u0027ll write 6u square root of 1 plus u squared du."},{"Start":"02:36.125 ","End":"02:39.190","Text":"Now we just have a definite integral problem,"},{"Start":"02:39.190 ","End":"02:41.890","Text":"so I\u0027ll make some space."},{"Start":"02:42.160 ","End":"02:45.650","Text":"I think we can do this by substitution."},{"Start":"02:45.650 ","End":"02:48.004","Text":"You could probably substitute either"},{"Start":"02:48.004 ","End":"02:53.330","Text":"the square root of 1 plus u squared or just 1 plus u squared."},{"Start":"02:53.330 ","End":"02:55.520","Text":"I\u0027m going to go with just 1 plus u squared."},{"Start":"02:55.520 ","End":"02:57.920","Text":"Let\u0027s see, we\u0027ve used t and s, okay,"},{"Start":"02:57.920 ","End":"03:04.375","Text":"I\u0027ll make it v. Let\u0027s say that v is 1 plus u squared,"},{"Start":"03:04.375 ","End":"03:09.390","Text":"and then dv is equal to 2u,"},{"Start":"03:09.390 ","End":"03:12.160","Text":"the derivative of this, du."},{"Start":"03:12.160 ","End":"03:15.860","Text":"But when we substitute,"},{"Start":"03:15.860 ","End":"03:18.860","Text":"we can also substitute the limits of integration so we don\u0027t have"},{"Start":"03:18.860 ","End":"03:21.210","Text":"to come back to t."},{"Start":"03:28.100 ","End":"03:30.735","Text":"We\u0027re substituting u."},{"Start":"03:30.735 ","End":"03:35.050","Text":"When u is 0 you have to figure out what v is,"},{"Start":"03:35.050 ","End":"03:38.140","Text":"and when u is t, you have to figure out what v is."},{"Start":"03:38.140 ","End":"03:43.365","Text":"When u is 0, then v is 1 plus 0 squared,"},{"Start":"03:43.365 ","End":"03:47.985","Text":"so v equals, so I\u0027ll write that 1 plus 0 squared,"},{"Start":"03:47.985 ","End":"03:50.410","Text":"which is just 1."},{"Start":"03:50.900 ","End":"03:53.805","Text":"I wrote it in the wrong place, sorry."},{"Start":"03:53.805 ","End":"04:01.080","Text":"When u is t, then v is just equal to 1 plus t squared."},{"Start":"04:01.080 ","End":"04:04.480","Text":"If we plug everything in here,"},{"Start":"04:04.550 ","End":"04:06.640","Text":"let\u0027s see what we get."},{"Start":"04:06.640 ","End":"04:08.080","Text":"We get the integral."},{"Start":"04:08.080 ","End":"04:13.670","Text":"Now, we know it\u0027s from 1 to 1 plus t squared."},{"Start":"04:13.690 ","End":"04:18.260","Text":"Now, notice that I have here 6u."},{"Start":"04:18.260 ","End":"04:22.055","Text":"6u I can write as 3 times 2u,"},{"Start":"04:22.055 ","End":"04:26.285","Text":"and the 2udu is dv."},{"Start":"04:26.285 ","End":"04:36.570","Text":"I have 3 times the square root of v, and then dv."},{"Start":"04:36.570 ","End":"04:41.895","Text":"I can pull the 3 out in front of the integral,"},{"Start":"04:41.895 ","End":"04:49.145","Text":"and I will get 3 times the integral from 1 to 1 plus t squared,"},{"Start":"04:49.145 ","End":"04:51.335","Text":"and instead of root v,"},{"Start":"04:51.335 ","End":"04:58.025","Text":"I\u0027ll write it as v to the power of a 1/2 dv because then I can use the exponents rule."},{"Start":"04:58.025 ","End":"05:00.125","Text":"This is equal to,"},{"Start":"05:00.125 ","End":"05:02.180","Text":"I raise the power by 1,"},{"Start":"05:02.180 ","End":"05:04.630","Text":"so that\u0027s 3 over 2,"},{"Start":"05:04.630 ","End":"05:06.380","Text":"but I divide by it,"},{"Start":"05:06.380 ","End":"05:12.050","Text":"so it\u0027s times 2/3 V to the power of 3 over 2,"},{"Start":"05:12.050 ","End":"05:18.300","Text":"and all this from 1 to 1 plus t squared."},{"Start":"05:18.300 ","End":"05:22.540","Text":"I can just do it for this because the constants don\u0027t matter,"},{"Start":"05:22.540 ","End":"05:26.560","Text":"and also, I noticed that this 3 cancels with this 3,"},{"Start":"05:26.560 ","End":"05:28.580","Text":"so it\u0027s just 2."},{"Start":"05:28.580 ","End":"05:31.770","Text":"I\u0027m going to need more space."},{"Start":"05:31.770 ","End":"05:40.095","Text":"We get twice, and then when v is 1 plus t squared,"},{"Start":"05:40.095 ","End":"05:48.540","Text":"I get 1 plus t squared to the power of 3 over 2,"},{"Start":"05:48.540 ","End":"05:54.240","Text":"and when v is 1,"},{"Start":"05:54.240 ","End":"05:58.710","Text":"I just get 1 to the 3 over 2,"},{"Start":"05:58.710 ","End":"06:04.630","Text":"which is 1. We\u0027re done."}],"ID":9713},{"Watched":false,"Name":"Exercise 5","Duration":"9m 16s","ChapterTopicVideoID":9842,"CourseChapterTopicPlaylistID":57482,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9842.jpeg","UploadDate":"2017-08-06T04:54:19.2430000","DurationForVideoObject":"PT9M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.205","Text":"In this exercise, we have to find the arc length function s,"},{"Start":"00:05.205 ","End":"00:12.375","Text":"for this vector function parametrized as follows in the ijk notation."},{"Start":"00:12.375 ","End":"00:16.560","Text":"Then there\u0027s a part b where we have to see where we are on"},{"Start":"00:16.560 ","End":"00:21.615","Text":"the curve if we travel a distance of 10 units along it."},{"Start":"00:21.615 ","End":"00:27.720","Text":"I put the formula here so we have something to start with."},{"Start":"00:27.720 ","End":"00:35.470","Text":"We first of all need to find r prime and then take its magnitude, so let\u0027s see."},{"Start":"00:35.750 ","End":"00:41.535","Text":"R prime of t is just"},{"Start":"00:41.535 ","End":"00:47.155","Text":"component-wise we differentiate and we\u0027ll need to use the product rule."},{"Start":"00:47.155 ","End":"00:50.160","Text":"We\u0027re going to get something for the i."},{"Start":"00:50.160 ","End":"00:52.215","Text":"Now, we have this times this,"},{"Start":"00:52.215 ","End":"00:59.640","Text":"the derivative of the first is 2e^2_t, cosine 2_t."},{"Start":"00:59.640 ","End":"01:04.750","Text":"Then we take this as is and differentiate the second."},{"Start":"01:04.750 ","End":"01:07.080","Text":"We\u0027ll get a 2 coming out front,"},{"Start":"01:07.080 ","End":"01:12.985","Text":"see the e^2_t and the derivative of this is minus 2 sine."},{"Start":"01:12.985 ","End":"01:17.490","Text":"I\u0027m going to change this plus to a minus and put a 2 here,"},{"Start":"01:17.490 ","End":"01:21.250","Text":"and then here sine 2_t,"},{"Start":"01:22.130 ","End":"01:25.800","Text":"all this is i."},{"Start":"01:25.800 ","End":"01:29.685","Text":"Now the j part is,"},{"Start":"01:29.685 ","End":"01:35.880","Text":"well, it\u0027s just 0 because 2 is a constant."},{"Start":"01:35.880 ","End":"01:39.045","Text":"Here I have a product rule again."},{"Start":"01:39.045 ","End":"01:47.620","Text":"The derivative of the first 2e^2_t sine of 2_t."},{"Start":"01:48.380 ","End":"01:52.650","Text":"This is in the way, that\u0027s better."},{"Start":"01:52.650 ","End":"01:56.045","Text":"As before, derivative of sine is cosine,"},{"Start":"01:56.045 ","End":"01:58.535","Text":"this time we\u0027ll get plus 2e^2_t."},{"Start":"01:58.535 ","End":"02:01.985","Text":"I\u0027m just differentiating the second part of the product,"},{"Start":"02:01.985 ","End":"02:09.590","Text":"cosine 2t and all this is K. We don\u0027t have a j."},{"Start":"02:09.590 ","End":"02:12.230","Text":"Now we need to figure out the magnitude,"},{"Start":"02:12.230 ","End":"02:14.330","Text":"there\u0027s going to be some computation because we need"},{"Start":"02:14.330 ","End":"02:16.670","Text":"the square root of this squared plus this squared,"},{"Start":"02:16.670 ","End":"02:18.795","Text":"but let\u0027s get to it."},{"Start":"02:18.795 ","End":"02:23.335","Text":"Magnitude of r prime of t is equal to."},{"Start":"02:23.335 ","End":"02:27.315","Text":"Of course, I can take 2 out of everything,"},{"Start":"02:27.315 ","End":"02:30.050","Text":"and if I take 2 out of everything,"},{"Start":"02:30.050 ","End":"02:35.450","Text":"I can also take 2 outside the magnitude because 2 is a positive number, it comes out."},{"Start":"02:35.450 ","End":"02:39.590","Text":"So I get twice and then I\u0027m going to get the square root."},{"Start":"02:39.590 ","End":"02:42.175","Text":"Now I\u0027ve taken the 2 out."},{"Start":"02:42.175 ","End":"02:45.645","Text":"See this 2 gets rid of this 2,"},{"Start":"02:45.645 ","End":"02:47.340","Text":"this 2, this 2,"},{"Start":"02:47.340 ","End":"02:50.325","Text":"and this 2, just once here."},{"Start":"02:50.325 ","End":"02:54.475","Text":"Now I\u0027m going to do a binomial product."},{"Start":"02:54.475 ","End":"02:57.155","Text":"I need this minus this squared,"},{"Start":"02:57.155 ","End":"03:02.435","Text":"but also I can see that e^2_t appears twice,"},{"Start":"03:02.435 ","End":"03:12.350","Text":"so I have e^2_t squared and then cosine t minus sine t squared"},{"Start":"03:12.350 ","End":"03:22.475","Text":"is cosine squared t minus 2 cosine t sine t"},{"Start":"03:22.475 ","End":"03:31.105","Text":"plus sine squared t. The formula I\u0027m using is the 1 from"},{"Start":"03:31.105 ","End":"03:35.980","Text":"basic algebra that a plus or minus b squared is"},{"Start":"03:35.980 ","End":"03:41.075","Text":"a squared plus or minus 2_ab plus b squared,"},{"Start":"03:41.075 ","End":"03:43.000","Text":"I\u0027m using it with the minus."},{"Start":"03:43.000 ","End":"03:46.285","Text":"In the second 1, we\u0027ll be using it with the plus."},{"Start":"03:46.285 ","End":"03:52.705","Text":"Again I get e^2_t comes out the brackets so it\u0027s squared."},{"Start":"03:52.705 ","End":"03:57.240","Text":"This time I have at c sine,"},{"Start":"03:57.240 ","End":"03:59.260","Text":"the first term squared,"},{"Start":"03:59.260 ","End":"04:05.905","Text":"the a squared sine squared 2_t plus 2 sine 2_t,"},{"Start":"04:05.905 ","End":"04:14.185","Text":"cosine 2_t plus cosine squared 2_t."},{"Start":"04:14.185 ","End":"04:19.875","Text":"Oh boy. Now I can\u0027t simplify this,"},{"Start":"04:19.875 ","End":"04:21.800","Text":"when I have something squared,"},{"Start":"04:21.800 ","End":"04:26.280","Text":"something positive squared under the square root sign,"},{"Start":"04:26.280 ","End":"04:29.974","Text":"have it here and here so It\u0027s a common factor."},{"Start":"04:29.974 ","End":"04:32.495","Text":"I can pull it out of the square root,"},{"Start":"04:32.495 ","End":"04:34.975","Text":"but I have to drop the squared."},{"Start":"04:34.975 ","End":"04:41.760","Text":"I\u0027ve got 2_e^2_t square root of."},{"Start":"04:41.760 ","End":"04:46.325","Text":"Now look, I have cosine squared and cosine squared,"},{"Start":"04:46.325 ","End":"04:50.160","Text":"that\u0027s 2 cosine squared."},{"Start":"04:50.300 ","End":"04:55.740","Text":"This is all 2_t, I\u0027m so sorry. Okay, that\u0027s better."},{"Start":"04:55.740 ","End":"05:02.460","Text":"I have 2 cosine squared 2_t and then this term with"},{"Start":"05:02.460 ","End":"05:09.510","Text":"this term cancels minus 2 sine t. This is cosine sine,"},{"Start":"05:09.510 ","End":"05:12.180","Text":"this is sine cosine, same thing."},{"Start":"05:12.180 ","End":"05:16.160","Text":"Plus, and then I have sine squared and sine squared,"},{"Start":"05:16.160 ","End":"05:19.860","Text":"so that\u0027s 2 sine squared 2_t."},{"Start":"05:20.180 ","End":"05:24.230","Text":"Now I want to remind you of the trig formula that sine"},{"Start":"05:24.230 ","End":"05:29.260","Text":"squared alpha plus cosine squared alpha equals 1."},{"Start":"05:29.260 ","End":"05:31.420","Text":"What\u0027s under the square root sign?"},{"Start":"05:31.420 ","End":"05:32.965","Text":"I have 2 of these."},{"Start":"05:32.965 ","End":"05:35.380","Text":"This is just going to equal root 2,"},{"Start":"05:35.380 ","End":"05:39.350","Text":"so I\u0027ve got 2 root 2_e^2_t."},{"Start":"05:42.090 ","End":"05:48.715","Text":"That\u0027s just the magnitude of the derivative."},{"Start":"05:48.715 ","End":"05:51.770","Text":"Now I still have to do an integral."},{"Start":"05:52.140 ","End":"06:01.750","Text":"Lets see, s of t is equal to the integral from 0 to t of this magnitude."},{"Start":"06:01.750 ","End":"06:03.730","Text":"But I have to use it in the latter u,"},{"Start":"06:03.730 ","End":"06:04.990","Text":"because I\u0027ve used the letter t,"},{"Start":"06:04.990 ","End":"06:13.880","Text":"so we replace t with u and 2 root 2_e^2_u du."},{"Start":"06:13.880 ","End":"06:15.110","Text":"Now we\u0027ve got the expression,"},{"Start":"06:15.110 ","End":"06:18.000","Text":"we just have to do the integration."},{"Start":"06:18.890 ","End":"06:22.650","Text":"Root 2 comes outside,"},{"Start":"06:22.650 ","End":"06:30.420","Text":"and now the integral of 2_e^2_u is e^2_u."},{"Start":"06:30.420 ","End":"06:31.850","Text":"If you differentiate this,"},{"Start":"06:31.850 ","End":"06:35.045","Text":"you get the 2 and if you integrate it the 2 get swallowed up,"},{"Start":"06:35.045 ","End":"06:41.640","Text":"and we need this from 0 to t. Let\u0027s see,"},{"Start":"06:41.640 ","End":"06:43.350","Text":"when u is t,"},{"Start":"06:43.350 ","End":"06:46.330","Text":"we\u0027ve got root 2_e^2_t."},{"Start":"06:47.360 ","End":"06:53.655","Text":"When u is 0, we have e^0 is 1,"},{"Start":"06:53.655 ","End":"06:56.930","Text":"so it\u0027s minus root 2 and if you want,"},{"Start":"06:56.930 ","End":"07:02.980","Text":"you can write it as root 2_e^2_t minus 1."},{"Start":"07:02.980 ","End":"07:04.920","Text":"Now that was just part a,"},{"Start":"07:04.920 ","End":"07:08.360","Text":"there was a part b of the exercise which asked,"},{"Start":"07:08.360 ","End":"07:12.680","Text":"where are we on the curve if we travel a distance of 10?"},{"Start":"07:12.680 ","End":"07:16.340","Text":"What this is saying in part b is that we\u0027re"},{"Start":"07:16.340 ","End":"07:23.060","Text":"given S^t is equal to 10 and we have to find where we are otherwise,"},{"Start":"07:23.060 ","End":"07:26.680","Text":"we need rt equals what?"},{"Start":"07:26.680 ","End":"07:28.920","Text":"What we\u0027ll do is first of all,"},{"Start":"07:28.920 ","End":"07:30.830","Text":"find t from this formula,"},{"Start":"07:30.830 ","End":"07:33.380","Text":"I\u0027ll write S^t Again here."},{"Start":"07:33.380 ","End":"07:37.055","Text":"What we can say is that this is equal to 10."},{"Start":"07:37.055 ","End":"07:43.505","Text":"So root 2_e^2_t minus 1 equals 10,"},{"Start":"07:43.505 ","End":"07:49.020","Text":"and try and solve that for t. Let\u0027s see,"},{"Start":"07:49.720 ","End":"07:53.550","Text":"e^2_t is going to equal."},{"Start":"07:53.550 ","End":"08:02.140","Text":"I can divide by root 2 and I can also add 1."},{"Start":"08:02.480 ","End":"08:06.270","Text":"If I take the natural log,"},{"Start":"08:06.270 ","End":"08:09.720","Text":"2_t will be the natural log of this,"},{"Start":"08:09.720 ","End":"08:11.480","Text":"so t will be 1/2,"},{"Start":"08:11.480 ","End":"08:18.650","Text":"the natural log of 10 over root 2 plus 1."},{"Start":"08:18.650 ","End":"08:21.515","Text":"Not an easy expression to do."},{"Start":"08:21.515 ","End":"08:24.085","Text":"I mean, you need a calculator."},{"Start":"08:24.085 ","End":"08:26.780","Text":"Hadn\u0027t planned on it being this messy,"},{"Start":"08:26.780 ","End":"08:29.165","Text":"I\u0027ll just tell you what you would do,"},{"Start":"08:29.165 ","End":"08:30.245","Text":"just the idea of it."},{"Start":"08:30.245 ","End":"08:33.005","Text":"You do find what T equals,"},{"Start":"08:33.005 ","End":"08:37.640","Text":"do it by calculator and after we\u0027ve got the value of T,"},{"Start":"08:37.640 ","End":"08:43.190","Text":"then we plug it in because we know that r of t is"},{"Start":"08:43.190 ","End":"08:52.090","Text":"equal to e^2_t sine 2_t."},{"Start":"08:52.090 ","End":"08:54.360","Text":"I meant cosine 2_t,"},{"Start":"08:54.360 ","End":"09:01.605","Text":"I plus 2_j plus e^2_t sine"},{"Start":"09:01.605 ","End":"09:08.720","Text":"2_t K. Then you would plug this value of t into here,"},{"Start":"09:08.720 ","End":"09:11.810","Text":"again using the calculator."},{"Start":"09:11.810 ","End":"09:16.980","Text":"I\u0027m not going to do it, we got the idea. We\u0027re done."}],"ID":9714}],"Thumbnail":null,"ID":57482},{"Name":"Curvatures in 2D and 3D Space","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"(3D Space - Curvature (of a curve","Duration":"6m 25s","ChapterTopicVideoID":9869,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9869.jpeg","UploadDate":"2020-03-26T10:09:03.0970000","DurationForVideoObject":"PT6M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:07.305","Text":"Here we are again in the 3D coordinate system or 3D space."},{"Start":"00:07.305 ","End":"00:12.225","Text":"In this clip I\u0027m going to be talking about a concept called curvature."},{"Start":"00:12.225 ","End":"00:15.085","Text":"Curvature applies to a curve."},{"Start":"00:15.085 ","End":"00:17.240","Text":"It could be also in 2D,"},{"Start":"00:17.240 ","End":"00:23.105","Text":"so it might actually even be easier sometimes to illustrate in 2D."},{"Start":"00:23.105 ","End":"00:27.430","Text":"We\u0027re talking 2D and 3D."},{"Start":"00:27.950 ","End":"00:33.155","Text":"The curve will usually be a vector curve."},{"Start":"00:33.155 ","End":"00:37.655","Text":"I would like to start by just intuitively"},{"Start":"00:37.655 ","End":"00:44.740","Text":"explaining what it means and later give a more formal definition."},{"Start":"00:44.740 ","End":"00:52.350","Text":"In general, if I have a curve in 2D or 3D,"},{"Start":"00:52.350 ","End":"00:59.990","Text":"just anything, and I can see that at some points it\u0027s more bending."},{"Start":"00:59.990 ","End":"01:04.310","Text":"I would say that here it\u0027s not very curvy for example,"},{"Start":"01:04.310 ","End":"01:07.940","Text":"but here it looks like it\u0027s making a very sharp turn."},{"Start":"01:07.940 ","End":"01:11.750","Text":"You might think of it as a road and you\u0027re traveling along it with the car"},{"Start":"01:11.750 ","End":"01:15.890","Text":"and place where it bends mostly."},{"Start":"01:15.890 ","End":"01:19.130","Text":"This would have a higher curvature."},{"Start":"01:19.130 ","End":"01:21.950","Text":"We\u0027re going to give it a numerical measure to just how"},{"Start":"01:21.950 ","End":"01:26.699","Text":"curvy the curve is at a given point."},{"Start":"01:26.780 ","End":"01:32.790","Text":"I\u0027d like to discuss 2 simple cases: a circle and a line."},{"Start":"01:32.790 ","End":"01:35.565","Text":"Well let me say this."},{"Start":"01:35.565 ","End":"01:40.150","Text":"As for straight lines,"},{"Start":"01:40.150 ","End":"01:49.260","Text":"they will have curvature equals 0 and for circles,"},{"Start":"01:50.660 ","End":"01:54.255","Text":"let me show you a diagram first."},{"Start":"01:54.255 ","End":"01:57.525","Text":"I have here 2 circles."},{"Start":"01:57.525 ","End":"02:00.540","Text":"Let\u0027s call them the large 1 and the small 1."},{"Start":"02:00.540 ","End":"02:04.710","Text":"One has a larger radius and the other has a smaller radius."},{"Start":"02:04.710 ","End":"02:09.835","Text":"Now I\u0027m concerned about what happens at this point of contact."},{"Start":"02:09.835 ","End":"02:11.960","Text":"Well at this point here,"},{"Start":"02:11.960 ","End":"02:13.985","Text":"a circle goes through this point."},{"Start":"02:13.985 ","End":"02:17.825","Text":"I want to know its curvature at this point."},{"Start":"02:17.825 ","End":"02:21.665","Text":"I want it so that the smaller the circle,"},{"Start":"02:21.665 ","End":"02:23.450","Text":"the higher the curvature."},{"Start":"02:23.450 ","End":"02:26.240","Text":"As you can see when the circle is small,"},{"Start":"02:26.240 ","End":"02:29.330","Text":"if you use the car model and you\u0027re driving around,"},{"Start":"02:29.330 ","End":"02:31.220","Text":"it bends more sharply."},{"Start":"02:31.220 ","End":"02:32.420","Text":"The larger the circle,"},{"Start":"02:32.420 ","End":"02:34.850","Text":"the easier the bend or the curving."},{"Start":"02:34.850 ","End":"02:38.460","Text":"So 1 thing that changes large to small is the"},{"Start":"02:38.460 ","End":"02:42.840","Text":"reciprocal and this is what we actually do."},{"Start":"02:43.060 ","End":"02:45.710","Text":"If I call the radius,"},{"Start":"02:45.710 ","End":"02:48.259","Text":"it doesn\u0027t matter of which circle."},{"Start":"02:48.259 ","End":"02:51.995","Text":"It\u0027s like a circle and its radius is capital R,"},{"Start":"02:51.995 ","End":"02:58.190","Text":"then we define the curvature to be"},{"Start":"02:58.190 ","End":"03:04.640","Text":"1 over R. Notice that when R goes to infinity,"},{"Start":"03:04.640 ","End":"03:06.365","Text":"this goes to 0."},{"Start":"03:06.365 ","End":"03:10.640","Text":"In some ways, a line is like a circle that goes to"},{"Start":"03:10.640 ","End":"03:15.410","Text":"infinity so that the curvature is 1 over infinity which is 0."},{"Start":"03:15.410 ","End":"03:19.490","Text":"Now most things like this are not lines or circles,"},{"Start":"03:19.490 ","End":"03:21.350","Text":"so what do we do."},{"Start":"03:21.350 ","End":"03:26.240","Text":"How do we define this in such a way that for lines,"},{"Start":"03:26.240 ","End":"03:28.985","Text":"it will come out 0 and for circles, it will come out this."},{"Start":"03:28.985 ","End":"03:37.895","Text":"Well, there\u0027s a concept of something called an oscillatory circle or a kissing circle."},{"Start":"03:37.895 ","End":"03:42.905","Text":"Here\u0027s our curve and here\u0027s the point we\u0027re interested in,"},{"Start":"03:42.905 ","End":"03:53.270","Text":"and there is a way of finding a best fit circle to the curve at this point."},{"Start":"03:53.270 ","End":"03:57.065","Text":"I won\u0027t say exactly what it means to be a best fit,"},{"Start":"03:57.065 ","End":"04:01.250","Text":"but if this is like a graph in 2-dimensions,"},{"Start":"04:01.250 ","End":"04:02.780","Text":"y as a function of x,"},{"Start":"04:02.780 ","End":"04:09.395","Text":"you would say that this curve and this circle,"},{"Start":"04:09.395 ","End":"04:11.930","Text":"they both go through this point,"},{"Start":"04:11.930 ","End":"04:16.205","Text":"and they have the same derivative or tangent at this point,"},{"Start":"04:16.205 ","End":"04:18.710","Text":"and the second derivative is equal too,"},{"Start":"04:18.710 ","End":"04:21.530","Text":"but that\u0027s already more technical than I intended to."},{"Start":"04:21.530 ","End":"04:25.560","Text":"The idea is just to find a circle that fits best."},{"Start":"04:25.560 ","End":"04:27.490","Text":"When we found this circle,"},{"Start":"04:27.490 ","End":"04:37.210","Text":"we take its radius R. Then we say the curvature is 1 over R for this curve at this point."},{"Start":"04:37.210 ","End":"04:43.615","Text":"If we\u0027re not in the plane and we\u0027re in a 3D space,"},{"Start":"04:43.615 ","End":"04:52.320","Text":"then we can make a plane by taking this to be let\u0027s say a tangent vector,"},{"Start":"04:52.320 ","End":"05:00.700","Text":"it could be the unit tangent vector say T and we would also have a normal vector."},{"Start":"05:00.700 ","End":"05:02.150","Text":"Say it\u0027s the red 1."},{"Start":"05:02.150 ","End":"05:04.725","Text":"Maybe the unit normal or maybe not."},{"Start":"05:04.725 ","End":"05:13.235","Text":"Once we have 2 perpendicular vectors or any 2 vectors that are not parallel,"},{"Start":"05:13.235 ","End":"05:16.810","Text":"then they form a plane and this is the plane that we\u0027re talking about,"},{"Start":"05:16.810 ","End":"05:18.740","Text":"and we get the best fit circle,"},{"Start":"05:18.740 ","End":"05:21.170","Text":"and it\u0027s 1 over the radius."},{"Start":"05:21.170 ","End":"05:24.245","Text":"That\u0027s enough for the intuitive."},{"Start":"05:24.245 ","End":"05:27.565","Text":"Now let\u0027s get more formal."},{"Start":"05:27.565 ","End":"05:32.805","Text":"I\u0027ll just mention still in the intuition phase I\u0027ll introduce a definition,"},{"Start":"05:32.805 ","End":"05:36.095","Text":"there is a letter used for curvature."},{"Start":"05:36.095 ","End":"05:42.560","Text":"Curvature is denoted by; well let me just say that it\u0027s"},{"Start":"05:42.560 ","End":"05:50.550","Text":"the Greek letter Kappa and it has different fonts."},{"Start":"05:50.550 ","End":"05:51.720","Text":"Ways of drawing it,"},{"Start":"05:51.720 ","End":"05:57.950","Text":"it\u0027s pretty much like the letter K. Pick any 1 of these you like,"},{"Start":"05:57.950 ","End":"05:59.450","Text":"I\u0027ll just make an attempt."},{"Start":"05:59.450 ","End":"06:02.780","Text":"Curvature, a little bit of a curved line,"},{"Start":"06:02.780 ","End":"06:04.460","Text":"and then something like this,"},{"Start":"06:04.460 ","End":"06:06.485","Text":"and we\u0027ll know it\u0027s a Greek letter Kappa,"},{"Start":"06:06.485 ","End":"06:08.990","Text":"and that\u0027s what we mean by curvature."},{"Start":"06:08.990 ","End":"06:17.445","Text":"We say something like Kappa equals and then some formula and it means curvature."},{"Start":"06:17.445 ","End":"06:20.080","Text":"Safest writing that."}],"ID":9715},{"Watched":false,"Name":"3D-2D Space - Curvature (of a curve) - continued","Duration":"24m 7s","ChapterTopicVideoID":9870,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9870.jpeg","UploadDate":"2017-08-06T05:40:50.1400000","DurationForVideoObject":"PT24M7S","Description":null,"VideoComments":[],"Subtitles":[],"ID":9716},{"Watched":false,"Name":"(2D Space - Curvature (of a curve","Duration":"51s","ChapterTopicVideoID":9871,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9871.jpeg","UploadDate":"2020-03-26T10:08:14.8200000","DurationForVideoObject":"PT51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.695","Text":"I\u0027m going to clear the board,"},{"Start":"00:01.695 ","End":"00:04.590","Text":"but I think I\u0027ll keep this little picture."},{"Start":"00:04.590 ","End":"00:10.180","Text":"Just 1 more philosophical, not quite."},{"Start":"00:10.430 ","End":"00:13.590","Text":"There are 2 ways to look at curvature,"},{"Start":"00:13.590 ","End":"00:19.350","Text":"1 is just as a magnitude or a number that\u0027s non-negative."},{"Start":"00:19.350 ","End":"00:22.230","Text":"Sometimes there is a signed curvature,"},{"Start":"00:22.230 ","End":"00:24.525","Text":"meaning plus or minus."},{"Start":"00:24.525 ","End":"00:26.430","Text":"It goes like this,"},{"Start":"00:26.430 ","End":"00:29.520","Text":"that if you are going along the curve"},{"Start":"00:29.520 ","End":"00:33.795","Text":"with the direction of the increasing parameter,"},{"Start":"00:33.795 ","End":"00:36.870","Text":"the parameter could be just any T or it could be arc"},{"Start":"00:36.870 ","End":"00:40.635","Text":"length S. If at this point you\u0027re going,"},{"Start":"00:40.635 ","End":"00:45.540","Text":"like this here would be counterclockwise,"},{"Start":"00:45.540 ","End":"00:52.620","Text":"then you would take the curvature to be bigger than 0."}],"ID":9717},{"Watched":false,"Name":"2D Space - Curvature (of a curve) - continued 2","Duration":"10m 46s","ChapterTopicVideoID":9872,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9872.jpeg","UploadDate":"2020-03-26T10:10:51.6030000","DurationForVideoObject":"PT10M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"Next topic, I\u0027m going to concentrate on"},{"Start":"00:04.905 ","End":"00:11.520","Text":"the 2D under a certain special cases here and some extra formulas."},{"Start":"00:11.520 ","End":"00:12.930","Text":"Let me, first of all,"},{"Start":"00:12.930 ","End":"00:15.450","Text":"get rid of what I don\u0027t need."},{"Start":"00:15.450 ","End":"00:18.525","Text":"Then here I\u0027ll write 3D,"},{"Start":"00:18.525 ","End":"00:24.420","Text":"and here I\u0027ll write 2D, underline that."},{"Start":"00:24.420 ","End":"00:30.225","Text":"In 2D, let\u0027s just take a general case where we have"},{"Start":"00:30.225 ","End":"00:39.075","Text":"r as a function of t. R of t,"},{"Start":"00:39.075 ","End":"00:41.850","Text":"and this will be 2 functions;"},{"Start":"00:41.850 ","End":"00:48.225","Text":"let\u0027s just call them x of t and y of t,"},{"Start":"00:48.225 ","End":"00:53.230","Text":"because we\u0027re in 2 dimensions and this is still a vector."},{"Start":"00:53.510 ","End":"00:56.895","Text":"If this is the case,"},{"Start":"00:56.895 ","End":"01:02.885","Text":"and in this case, I just copied the formula from somewhere."},{"Start":"01:02.885 ","End":"01:05.075","Text":"Not even going to go into it,"},{"Start":"01:05.075 ","End":"01:06.755","Text":"I just like you to have it."},{"Start":"01:06.755 ","End":"01:16.020","Text":"It involves the first and second derivatives of x and y and we just substituted in."},{"Start":"01:16.020 ","End":"01:22.830","Text":"This is the absolute value and there\u0027s the curvature."},{"Start":"01:25.310 ","End":"01:34.165","Text":"Another typical case, very common in 2D is when we just have y equals a function of x."},{"Start":"01:34.165 ","End":"01:37.655","Text":"Certainly, we can always write it in vector form."},{"Start":"01:37.655 ","End":"01:40.235","Text":"Because if we take x as a parameter,"},{"Start":"01:40.235 ","End":"01:43.400","Text":"we can write it as r of,"},{"Start":"01:43.400 ","End":"01:45.590","Text":"say a variable x,"},{"Start":"01:45.590 ","End":"01:50.729","Text":"is equal to x, f of x."},{"Start":"01:50.729 ","End":"01:53.975","Text":"Then if I use this formula,"},{"Start":"01:53.975 ","End":"02:03.370","Text":"what happens is that x prime is 1 and x double prime is 0."},{"Start":"02:03.730 ","End":"02:09.340","Text":"What we end up getting is that kappa equals,"},{"Start":"02:09.340 ","End":"02:14.490","Text":"the only thing that\u0027s left is the y double prime because I said x prime is 1,"},{"Start":"02:14.490 ","End":"02:16.215","Text":"x double prime is 0,"},{"Start":"02:16.215 ","End":"02:23.875","Text":"so we get the absolute value of y double prime here over, x prime is 1,"},{"Start":"02:23.875 ","End":"02:26.870","Text":"that\u0027s just 1 plus"},{"Start":"02:26.870 ","End":"02:34.460","Text":"y prime squared to the power of 3 over 2."},{"Start":"02:34.460 ","End":"02:39.510","Text":"But it\u0027s best to write it using the letter f,"},{"Start":"02:39.510 ","End":"02:42.375","Text":"so let\u0027s just replace y by f of x."},{"Start":"02:42.375 ","End":"02:52.625","Text":"The nicer way of writing it is absolute value of f double prime of x,"},{"Start":"02:52.625 ","End":"03:00.240","Text":"over 1 plus f prime of x squared,"},{"Start":"03:00.240 ","End":"03:02.640","Text":"put a brackets here,"},{"Start":"03:02.640 ","End":"03:10.360","Text":"and all of this to the power of 3 over 2."},{"Start":"03:10.790 ","End":"03:17.950","Text":"That\u0027s what we do when we have y as a function of x in the plane."},{"Start":"03:19.550 ","End":"03:24.130","Text":"Those are 2 other formulas for you."},{"Start":"03:24.590 ","End":"03:27.195","Text":"Perhaps let\u0027s just highlight them."},{"Start":"03:27.195 ","End":"03:29.215","Text":"We have this formula,"},{"Start":"03:29.215 ","End":"03:35.300","Text":"and we have this formula depending on the setup."},{"Start":"03:37.580 ","End":"03:44.900","Text":"I also want to give you another intuition which works in the plane;"},{"Start":"03:45.390 ","End":"03:51.595","Text":"it\u0027s not just an intuition it\u0027s actually an alternative definition in some books."},{"Start":"03:51.595 ","End":"03:55.665","Text":"I\u0027m going to bring in another picture,"},{"Start":"03:55.665 ","End":"03:59.570","Text":"and here\u0027s the picture I wanted."},{"Start":"03:59.570 ","End":"04:05.210","Text":"What we can do is we have a curve here and let\u0027s say"},{"Start":"04:05.210 ","End":"04:12.800","Text":"it\u0027s parameterized by s. As we go along the curve,"},{"Start":"04:12.800 ","End":"04:14.315","Text":"we get a curve length,"},{"Start":"04:14.315 ","End":"04:19.130","Text":"and this point that\u0027s moving along the curve"},{"Start":"04:19.130 ","End":"04:25.365","Text":"will also make an angle at each given point of,"},{"Start":"04:25.365 ","End":"04:29.250","Text":"let\u0027s call it the Greek letter, phi."},{"Start":"04:29.680 ","End":"04:34.280","Text":"The sign is like that and it\u0027s Greek letter,"},{"Start":"04:34.280 ","End":"04:37.820","Text":"phi often used for angles."},{"Start":"04:37.820 ","End":"04:43.295","Text":"It turns out that the curvature can be defined"},{"Start":"04:43.295 ","End":"04:48.860","Text":"as the rate of change of this angle relative to curve length."},{"Start":"04:48.860 ","End":"04:57.270","Text":"In other words, that kappa is actually d phi over ds."},{"Start":"04:57.270 ","End":"05:02.500","Text":"You write phi as a function of s and you differentiate it."},{"Start":"05:03.830 ","End":"05:12.650","Text":"Well, I guess we better take the absolute value because we\u0027re using the unsigned version."},{"Start":"05:12.650 ","End":"05:16.655","Text":"That would give another formula and also an intuition,"},{"Start":"05:16.655 ","End":"05:19.550","Text":"because the bigger the curvature,"},{"Start":"05:19.550 ","End":"05:22.810","Text":"the more the angle\u0027s going to be changing as we\u0027re moving along,"},{"Start":"05:22.810 ","End":"05:27.050","Text":"but as long as we\u0027re doing in terms of arc length."},{"Start":"05:27.050 ","End":"05:31.085","Text":"It\u0027s actually not that hard to prove,"},{"Start":"05:31.085 ","End":"05:36.350","Text":"but still I\u0027m just going to leave it like"},{"Start":"05:36.350 ","End":"05:42.840","Text":"that as an extra formula or an intuition of what is curvature."},{"Start":"05:45.230 ","End":"05:56.015","Text":"Now, the final optional topic is getting back to this difference between K and kappa."},{"Start":"05:56.015 ","End":"06:00.300","Text":"I\u0027d like to discuss the signed or oriented curvature."},{"Start":"06:00.300 ","End":"06:02.265","Text":"This is optional."},{"Start":"06:02.265 ","End":"06:05.435","Text":"I haven\u0027t seen it that much."},{"Start":"06:05.435 ","End":"06:09.385","Text":"Anyway, you can decide whether you need this or not."},{"Start":"06:09.385 ","End":"06:13.330","Text":"I\u0027m going to erase most everything."},{"Start":"06:13.370 ","End":"06:16.020","Text":"I got rid of most of the stuff,"},{"Start":"06:16.020 ","End":"06:18.075","Text":"I left the formula for kappa."},{"Start":"06:18.075 ","End":"06:24.380","Text":"Now, we\u0027re going to focus on K. Now in an earlier example,"},{"Start":"06:24.380 ","End":"06:26.060","Text":"in a numerical example,"},{"Start":"06:26.060 ","End":"06:30.915","Text":"we had to compute r prime of s,"},{"Start":"06:30.915 ","End":"06:33.880","Text":"and when we did that,"},{"Start":"06:33.880 ","End":"06:36.600","Text":"we also computed the magnitude."},{"Start":"06:36.600 ","End":"06:37.930","Text":"I remember the example,"},{"Start":"06:37.930 ","End":"06:40.450","Text":"you can just go back and look."},{"Start":"06:40.450 ","End":"06:45.535","Text":"We got that this was equal to 1 in our example."},{"Start":"06:45.535 ","End":"06:48.550","Text":"Well, it turns out that this is always true that when you"},{"Start":"06:48.550 ","End":"06:53.275","Text":"parameterize by arc length or curved length,"},{"Start":"06:53.275 ","End":"07:01.840","Text":"then the derivative has magnitude 1 always."},{"Start":"07:02.120 ","End":"07:04.800","Text":"I could have mentioned that then,"},{"Start":"07:04.800 ","End":"07:08.560","Text":"but we need to practice with the computation."},{"Start":"07:08.560 ","End":"07:10.900","Text":"Now if you remember,"},{"Start":"07:10.900 ","End":"07:16.390","Text":"the unit tangent vector of"},{"Start":"07:16.390 ","End":"07:22.170","Text":"s is equal to r prime of s,"},{"Start":"07:22.170 ","End":"07:25.335","Text":"it\u0027s whatever parameter, it could be t, could be s,"},{"Start":"07:25.335 ","End":"07:35.544","Text":"divided by the magnitude in order to make it a unit vector."},{"Start":"07:35.544 ","End":"07:38.380","Text":"But as I said, this thing is equal to 1,"},{"Start":"07:38.380 ","End":"07:43.870","Text":"so it turns out that the unit tangent is the same"},{"Start":"07:43.870 ","End":"07:50.170","Text":"as the derivative of r. I\u0027m just going to erase the denominator."},{"Start":"07:50.170 ","End":"07:54.670","Text":"Now, t is a unit tangent vector,"},{"Start":"07:54.670 ","End":"08:00.380","Text":"and whenever we have a unit vector and we differentiate it,"},{"Start":"08:00.380 ","End":"08:09.520","Text":"the derivative is always perpendicular to the original."},{"Start":"08:10.460 ","End":"08:14.960","Text":"By the way, we defined the unit normal if you look back,"},{"Start":"08:14.960 ","End":"08:19.310","Text":"the derivative of the tangent is going to be"},{"Start":"08:19.310 ","End":"08:25.995","Text":"some multiple of the unit normal vector."},{"Start":"08:25.995 ","End":"08:30.410","Text":"It\u0027s going to be something here, some scalar."},{"Start":"08:30.410 ","End":"08:36.105","Text":"This is what we call K, not kappa,"},{"Start":"08:36.105 ","End":"08:40.370","Text":"K of s. That\u0027s the number,"},{"Start":"08:40.370 ","End":"08:45.665","Text":"the scalar that you have to multiply this unit vector by to get this derivative."},{"Start":"08:45.665 ","End":"08:48.900","Text":"We know that this is parallel to this,"},{"Start":"08:48.900 ","End":"08:51.165","Text":"it just means that we need a constant."},{"Start":"08:51.165 ","End":"08:57.410","Text":"This actually defines K of s. Once again,"},{"Start":"08:57.410 ","End":"09:01.640","Text":"the derivative of the unit tangent vector is going to be,"},{"Start":"09:01.640 ","End":"09:06.005","Text":"we know, in the direction of the normal vector,"},{"Start":"09:06.005 ","End":"09:09.720","Text":"the unit normal, and it\u0027s a multiple of it,"},{"Start":"09:09.720 ","End":"09:17.795","Text":"and that multiple is what we call K for each s. That\u0027s the definition."},{"Start":"09:17.795 ","End":"09:21.110","Text":"This could come out plus or minus."},{"Start":"09:21.110 ","End":"09:26.100","Text":"Let\u0027s just tie some loose ends together."},{"Start":"09:26.710 ","End":"09:31.650","Text":"Notice that kappa was defined this way."},{"Start":"09:32.950 ","End":"09:37.670","Text":"We\u0027re mixing up the Leibniz\u0027s and the Newton notation,"},{"Start":"09:37.670 ","End":"09:42.155","Text":"but dT over ds is the same as T prime of s,"},{"Start":"09:42.155 ","End":"09:46.715","Text":"so kappa is the magnitude of T prime of"},{"Start":"09:46.715 ","End":"09:52.745","Text":"s. This is equal to the magnitude of this."},{"Start":"09:52.745 ","End":"09:54.620","Text":"But this is a unit vector,"},{"Start":"09:54.620 ","End":"09:57.560","Text":"unit vector has magnitude of 1."},{"Start":"09:57.560 ","End":"10:03.525","Text":"This is just K of s,"},{"Start":"10:03.525 ","End":"10:09.280","Text":"but when we take a magnitude of a scalar times a vector,"},{"Start":"10:09.280 ","End":"10:16.080","Text":"we need to put the absolute value because magnitude is always positive."},{"Start":"10:16.080 ","End":"10:26.500","Text":"This shows really that kappa is actually the absolute value of K. Anyway,"},{"Start":"10:26.540 ","End":"10:36.490","Text":"this line here defines the K and this could have a sign or an orientation."},{"Start":"10:38.030 ","End":"10:42.940","Text":"This is the last topic I wanted to mention on curvature."},{"Start":"10:42.940 ","End":"10:46.370","Text":"We\u0027re finally done with this."}],"ID":9718},{"Watched":false,"Name":"Exercise 6","Duration":"5m 8s","ChapterTopicVideoID":9846,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9846.jpeg","UploadDate":"2017-08-06T05:01:11.1530000","DurationForVideoObject":"PT5M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this exercise, we have to find the curvature of the following curve,"},{"Start":"00:04.380 ","End":"00:08.505","Text":"which is a 3D parametric vector function."},{"Start":"00:08.505 ","End":"00:11.565","Text":"There are 2 formulas for curvature."},{"Start":"00:11.565 ","End":"00:15.749","Text":"Sometimes 1 is easier and sometimes the other is easier."},{"Start":"00:15.749 ","End":"00:18.870","Text":"In both cases, we\u0027re going to have to compute r prime."},{"Start":"00:18.870 ","End":"00:22.680","Text":"Let\u0027s take at that first and then decide which way we want to go."},{"Start":"00:22.680 ","End":"00:28.545","Text":"So r prime of t is equal to 1,"},{"Start":"00:28.545 ","End":"00:32.715","Text":"this one gives me t and this one gives me 2t."},{"Start":"00:32.715 ","End":"00:39.020","Text":"Now, if I take the magnitude of this,"},{"Start":"00:39.020 ","End":"00:42.300","Text":"I\u0027m going to end up with a square root."},{"Start":"00:42.670 ","End":"00:45.140","Text":"To compute the tangent vector,"},{"Start":"00:45.140 ","End":"00:50.045","Text":"I\u0027m going to have to divide these by the square root of 10 plus 1 plus 5t squared,"},{"Start":"00:50.045 ","End":"00:52.460","Text":"then the derivative of that will get messy."},{"Start":"00:52.460 ","End":"00:56.630","Text":"So I think we\u0027ll go with the cross product which won\u0027t be too bad,"},{"Start":"00:56.630 ","End":"01:00.500","Text":"especially since r double prime will have a 0 in it."},{"Start":"01:00.500 ","End":"01:04.695","Text":"I\u0027m going to go with this one and I erase the other."},{"Start":"01:04.695 ","End":"01:12.290","Text":"We\u0027ll need our double-prime and that is equal to 0, 1, 2."},{"Start":"01:12.290 ","End":"01:12.860","Text":"That\u0027s good."},{"Start":"01:12.860 ","End":"01:16.385","Text":"They\u0027re all constants and there\u0027s even a 0."},{"Start":"01:16.385 ","End":"01:21.050","Text":"Now we have to figure out the magnitude of 2 things."},{"Start":"01:21.050 ","End":"01:23.929","Text":"Well, let\u0027s first of all do the cross product,"},{"Start":"01:23.929 ","End":"01:26.960","Text":"and then we\u0027ll compute the magnitudes."},{"Start":"01:26.960 ","End":"01:38.180","Text":"The cross product of r prime,"},{"Start":"01:38.180 ","End":"01:42.610","Text":"I\u0027ll just write it, r prime.r double prime."},{"Start":"01:42.610 ","End":"01:45.225","Text":"This is equal to,"},{"Start":"01:45.225 ","End":"01:50.085","Text":"let\u0027s see, 3-by-3 determinant where we have i, j,"},{"Start":"01:50.085 ","End":"02:03.100","Text":"and k vectors, and then 1, t, 2t and then 0, 1, 2."},{"Start":"02:03.100 ","End":"02:04.820","Text":"Because of the 0 here,"},{"Start":"02:04.820 ","End":"02:07.940","Text":"I\u0027m going to expand either by the last row or the first column."},{"Start":"02:07.940 ","End":"02:10.580","Text":"You know what? I\u0027ll go by this column."},{"Start":"02:10.580 ","End":"02:21.320","Text":"So we get i times the determinant that\u0027s left if I cross out this and this,"},{"Start":"02:21.320 ","End":"02:23.240","Text":"so it\u0027s this here."},{"Start":"02:23.240 ","End":"02:27.945","Text":"So it\u0027s t, 2t, 1, 2."},{"Start":"02:27.945 ","End":"02:30.875","Text":"Then the 1 here,"},{"Start":"02:30.875 ","End":"02:33.825","Text":"it gets a minus sign,"},{"Start":"02:33.825 ","End":"02:35.765","Text":"so it\u0027s minus 1."},{"Start":"02:35.765 ","End":"02:37.865","Text":"Then if I cross out the row and column,"},{"Start":"02:37.865 ","End":"02:41.040","Text":"I\u0027m left with j, k, 1, 2."},{"Start":"02:46.250 ","End":"02:50.370","Text":"Let\u0027s see. This is equal,"},{"Start":"02:50.370 ","End":"02:56.130","Text":"this is 2t minus 2t. That\u0027s lucky."},{"Start":"02:56.130 ","End":"03:04.440","Text":"That comes out to be 0 and this comes out to be 2j minus k,"},{"Start":"03:04.440 ","End":"03:08.670","Text":"so it\u0027s minus 2j plus k."},{"Start":"03:08.670 ","End":"03:17.220","Text":"That\u0027s this."},{"Start":"03:17.220 ","End":"03:22.430","Text":"Now I need to compute its magnitude."},{"Start":"03:23.840 ","End":"03:30.805","Text":"The magnitude of r prime cross r"},{"Start":"03:30.805 ","End":"03:38.680","Text":"double prime is equal to the magnitude of this,"},{"Start":"03:38.680 ","End":"03:46.719","Text":"which will be the square root of 0 squared plus 2 squared plus 1 squared,"},{"Start":"03:46.719 ","End":"03:49.925","Text":"which is the square root of 5."},{"Start":"03:49.925 ","End":"03:52.785","Text":"So that\u0027s the numerator."},{"Start":"03:52.785 ","End":"03:54.230","Text":"As for the denominator,"},{"Start":"03:54.230 ","End":"03:57.515","Text":"I need the magnitude of r prime."},{"Start":"03:57.515 ","End":"04:00.310","Text":"I can get it from here."},{"Start":"04:00.970 ","End":"04:03.890","Text":"I don\u0027t always write the brackets t."},{"Start":"04:03.890 ","End":"04:10.680","Text":"Magnitude of r prime is going to be the square root of,"},{"Start":"04:11.480 ","End":"04:17.385","Text":"it\u0027s going to be 1 plus t squared plus 4t squared,"},{"Start":"04:17.385 ","End":"04:20.845","Text":"I will write straight away, 5t squared."},{"Start":"04:20.845 ","End":"04:25.840","Text":"So we\u0027re ready to substitute in the formula."},{"Start":"04:25.840 ","End":"04:29.135","Text":"What we can get is that"},{"Start":"04:29.135 ","End":"04:37.160","Text":"the curvature kappa is equal to the numerator."},{"Start":"04:37.160 ","End":"04:39.005","Text":"Where is it now?"},{"Start":"04:39.005 ","End":"04:42.300","Text":"Square root of 5."},{"Start":"04:42.370 ","End":"04:49.610","Text":"The denominator is this thing cubed,"},{"Start":"04:49.610 ","End":"04:56.075","Text":"so it\u0027s 1 plus 5t squared."},{"Start":"04:56.075 ","End":"04:59.060","Text":"Now this is to the power of a half and I cube it,"},{"Start":"04:59.060 ","End":"05:04.325","Text":"so I\u0027ll just write it to the power of 3 over 2,"},{"Start":"05:04.325 ","End":"05:06.420","Text":"and that\u0027s the answer."},{"Start":"05:06.420 ","End":"05:08.440","Text":"So we\u0027re done."}],"ID":9719},{"Watched":false,"Name":"Exercise 7","Duration":"4m 29s","ChapterTopicVideoID":9844,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9844.jpeg","UploadDate":"2017-08-06T04:59:41.7870000","DurationForVideoObject":"PT4M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we have to find the curvature of this curve,"},{"Start":"00:04.890 ","End":"00:09.210","Text":"which is a 3D vector function."},{"Start":"00:09.210 ","End":"00:12.240","Text":"It\u0027s an i, j, k notation."},{"Start":"00:12.240 ","End":"00:14.985","Text":"We have 2 formulas."},{"Start":"00:14.985 ","End":"00:20.480","Text":"I\u0027m going to do r prime first because it appears in both,"},{"Start":"00:20.480 ","End":"00:22.925","Text":"and then I\u0027ll see which is easier."},{"Start":"00:22.925 ","End":"00:28.515","Text":"R prime, I don\u0027t always write the parentheses t,"},{"Start":"00:28.515 ","End":"00:38.985","Text":"is equal to 3i plus 4 cosine of t j,"},{"Start":"00:38.985 ","End":"00:46.185","Text":"minus 4 sine t times k. Now,"},{"Start":"00:46.185 ","End":"00:53.075","Text":"what I often do is just think of how complicated the magnitude of r prime is."},{"Start":"00:53.075 ","End":"00:55.835","Text":"It appears both here and here."},{"Start":"00:55.835 ","End":"00:58.070","Text":"I\u0027m looking at it and I see that if I take"},{"Start":"00:58.070 ","End":"01:00.200","Text":"this squared plus this squared plus this squared,"},{"Start":"01:00.200 ","End":"01:04.220","Text":"I\u0027m going to get cosine squared plus sine squared, which is 1."},{"Start":"01:04.220 ","End":"01:08.405","Text":"I think I\u0027m going to go with the second formula this time,"},{"Start":"01:08.405 ","End":"01:10.550","Text":"because as I said,"},{"Start":"01:10.550 ","End":"01:14.015","Text":"the magnitude of r prime is very simple."},{"Start":"01:14.015 ","End":"01:22.940","Text":"Indeed, the magnitude of r prime is the square root of 3 squared,"},{"Start":"01:22.940 ","End":"01:32.700","Text":"plus 4 squared cosine squared t,"},{"Start":"01:32.700 ","End":"01:42.740","Text":"and this is crooked, then plus 4 squared sine squared t. Now look,"},{"Start":"01:42.740 ","End":"01:45.470","Text":"cosine squared plus sine squared is 1."},{"Start":"01:45.470 ","End":"01:50.280","Text":"What I get is the square root of 3 squared plus 4 squared,"},{"Start":"01:50.300 ","End":"01:55.185","Text":"and 3 squared plus 4 squared is 9 plus 16 is 25."},{"Start":"01:55.185 ","End":"01:58.620","Text":"The square root is just 5. How nice."},{"Start":"01:58.620 ","End":"02:02.495","Text":"Now I can compute the unit tangent t,"},{"Start":"02:02.495 ","End":"02:06.140","Text":"which is just r prime divided by its magnitude."},{"Start":"02:06.140 ","End":"02:08.990","Text":"It\u0027s this divided by this."},{"Start":"02:08.990 ","End":"02:10.610","Text":"Well, maybe I\u0027ll just remind you,"},{"Start":"02:10.610 ","End":"02:16.054","Text":"it\u0027s r prime divided by the magnitude of r prime,"},{"Start":"02:16.054 ","End":"02:21.200","Text":"and this is equal to this divided by 5."},{"Start":"02:21.200 ","End":"02:30.875","Text":"It\u0027s 3/5i, plus 4/5 cosine t j,"},{"Start":"02:30.875 ","End":"02:39.735","Text":"minus 4/5 sine t k. That\u0027s t prime."},{"Start":"02:39.735 ","End":"02:43.690","Text":"Now I need the magnitude of t prime."},{"Start":"02:46.220 ","End":"02:50.070","Text":"Sorry, I need to do t prime first."},{"Start":"02:50.070 ","End":"02:52.940","Text":"T prime is equal to,"},{"Start":"02:52.940 ","End":"02:55.220","Text":"well, this is a constant, so that\u0027s nothing."},{"Start":"02:55.220 ","End":"02:58.355","Text":"Here I get minus"},{"Start":"02:58.355 ","End":"03:05.630","Text":"4/5 sine t j and here,"},{"Start":"03:05.630 ","End":"03:15.575","Text":"minus 4/5 cosine of t k. The magnitude of t prime"},{"Start":"03:15.575 ","End":"03:20.030","Text":"is just the square root"},{"Start":"03:20.030 ","End":"03:28.670","Text":"of 4/5 squared, that\u0027s it."},{"Start":"03:28.670 ","End":"03:31.085","Text":"Perhaps I\u0027ll just write it anyway."},{"Start":"03:31.085 ","End":"03:36.380","Text":"I was going to say that sine squared plus cosine squared is 1,"},{"Start":"03:36.380 ","End":"03:38.530","Text":"but let\u0027s just write it in,"},{"Start":"03:38.530 ","End":"03:43.920","Text":"cosine squared t. This thing is 1."},{"Start":"03:43.920 ","End":"03:48.850","Text":"Square root of 4/5 squared is just 4/5."},{"Start":"03:49.670 ","End":"03:52.800","Text":"Look, I have the numerator,"},{"Start":"03:52.800 ","End":"03:55.470","Text":"and that is this."},{"Start":"03:55.470 ","End":"04:01.390","Text":"I have this denominator, which is this."},{"Start":"04:01.390 ","End":"04:04.130","Text":"All I have to do is divide them."},{"Start":"04:04.130 ","End":"04:09.620","Text":"Kappa, the curvature is equal to 5."},{"Start":"04:09.620 ","End":"04:12.830","Text":"No, this over this, sorry,"},{"Start":"04:12.830 ","End":"04:18.935","Text":"4/5 divided by 5,"},{"Start":"04:18.935 ","End":"04:22.830","Text":"and that is equal to 4/25."},{"Start":"04:24.130 ","End":"04:29.130","Text":"That\u0027s the answer. We are done."}],"ID":9720},{"Watched":false,"Name":"Exercise 8","Duration":"11m 11s","ChapterTopicVideoID":9845,"CourseChapterTopicPlaylistID":57483,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9845.jpeg","UploadDate":"2017-08-06T05:00:57.1470000","DurationForVideoObject":"PT11M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we\u0027re given a curve in parametric form,"},{"Start":"00:04.740 ","End":"00:09.525","Text":"r vector as a function of t in 3 dimensions,"},{"Start":"00:09.525 ","End":"00:11.580","Text":"there are sometimes we use angular brackets,"},{"Start":"00:11.580 ","End":"00:14.250","Text":"sometimes round brackets, don\u0027t make a difference."},{"Start":"00:14.250 ","End":"00:17.970","Text":"Part a, we want to know the length of the curve where"},{"Start":"00:17.970 ","End":"00:22.065","Text":"the parameter goes from 1 to e. In the second part,"},{"Start":"00:22.065 ","End":"00:27.240","Text":"we want to compute the curvature of this curve at any given point"},{"Start":"00:27.240 ","End":"00:33.535","Text":"t. Here\u0027s the formula that we\u0027re going to need for part a,"},{"Start":"00:33.535 ","End":"00:38.915","Text":"where of course a and b are 1 and e and r is given here."},{"Start":"00:38.915 ","End":"00:44.240","Text":"First thing we need is our prime and then we need to compute its magnitude,"},{"Start":"00:44.240 ","End":"00:49.805","Text":"but let\u0027s start off our prime of t. We just differentiate component wise."},{"Start":"00:49.805 ","End":"00:52.940","Text":"Here we get 2, here we get t,"},{"Start":"00:52.940 ","End":"00:56.090","Text":"and here we get 2 over t,"},{"Start":"00:56.090 ","End":"01:03.725","Text":"derivative of natural log 1 over t. Now we need the magnitude."},{"Start":"01:03.725 ","End":"01:10.745","Text":"The magnitude of the same thing above is equal to."},{"Start":"01:10.745 ","End":"01:13.580","Text":"We take the square root and the sum of the squares."},{"Start":"01:13.580 ","End":"01:23.520","Text":"We need 2 squared plus t squared plus 2 over t squared."},{"Start":"01:23.520 ","End":"01:26.470","Text":"Let\u0027s see, we can compute it and simplify it."},{"Start":"01:26.470 ","End":"01:29.510","Text":"We have the square root."},{"Start":"01:29.510 ","End":"01:33.340","Text":"Well 2 squared is 4."},{"Start":"01:34.010 ","End":"01:37.500","Text":"Do it bit at a time."},{"Start":"01:37.500 ","End":"01:40.810","Text":"Plus 4 over t squared."},{"Start":"01:40.810 ","End":"01:44.860","Text":"Now let\u0027s put everything over a common denominator, t squared."},{"Start":"01:44.860 ","End":"01:47.005","Text":"We get the square root."},{"Start":"01:47.005 ","End":"01:50.605","Text":"Now here I\u0027ll put a dividing line with t squared,"},{"Start":"01:50.605 ","End":"01:53.455","Text":"and then I need 4t squared."},{"Start":"01:53.455 ","End":"01:55.720","Text":"But I\u0027ll change the order a bit."},{"Start":"01:55.720 ","End":"01:58.180","Text":"Here I\u0027ll need t to the 4,"},{"Start":"01:58.180 ","End":"02:01.490","Text":"and here just 4."},{"Start":"02:01.490 ","End":"02:05.900","Text":"Now, if you remember your algebra as a formula,"},{"Start":"02:05.900 ","End":"02:13.715","Text":"a plus b squared is a squared plus 2ab plus b squared, the binomial expansion."},{"Start":"02:13.715 ","End":"02:18.520","Text":"If we use it here with a is t squared and b is 2,"},{"Start":"02:18.520 ","End":"02:28.280","Text":"then we can rewrite the numerator as t squared plus 2 all squared."},{"Start":"02:28.280 ","End":"02:32.250","Text":"We still have over t squared."},{"Start":"02:32.690 ","End":"02:36.245","Text":"Now we can take the square root."},{"Start":"02:36.245 ","End":"02:42.345","Text":"Notice that t is positive."},{"Start":"02:42.345 ","End":"02:44.570","Text":"When I take the square root,"},{"Start":"02:44.570 ","End":"02:46.535","Text":"I don\u0027t need absolute value,"},{"Start":"02:46.535 ","End":"02:50.600","Text":"I get just t squared plus 2."},{"Start":"02:50.600 ","End":"02:52.925","Text":"Well, this is always positive,"},{"Start":"02:52.925 ","End":"02:54.690","Text":"but t is positive."},{"Start":"02:54.690 ","End":"02:57.410","Text":"The square root of t squared is just t,"},{"Start":"02:57.410 ","End":"03:03.215","Text":"is not the absolute value of t. That\u0027s this bit."},{"Start":"03:03.215 ","End":"03:12.510","Text":"Now we need the integral from 1 to e. I\u0027ll continue over"},{"Start":"03:12.510 ","End":"03:22.664","Text":"here and say that L is equal to the integral from 1 to e of this."},{"Start":"03:22.664 ","End":"03:29.574","Text":"But this I could also write t squared over t is just t,"},{"Start":"03:29.574 ","End":"03:32.695","Text":"and 2 over t. I\u0027ll write it like this,"},{"Start":"03:32.695 ","End":"03:39.440","Text":"divided out, dt, which is equal 2."},{"Start":"03:39.440 ","End":"03:44.490","Text":"Now the integral of t is 1/2t squared."},{"Start":"03:44.490 ","End":"03:53.725","Text":"The integral of 2 over t is twice natural log of t. All this,"},{"Start":"03:53.725 ","End":"03:59.740","Text":"we have to take from 1 to e. If we plug in e,"},{"Start":"03:59.740 ","End":"04:05.475","Text":"we get a 1/2e squared plus 2."},{"Start":"04:05.475 ","End":"04:11.025","Text":"Natural log of e is just 1, subtract."},{"Start":"04:11.025 ","End":"04:15.360","Text":"Now we plug in 1,1/2 times 1 squared is just a 1/2."},{"Start":"04:15.360 ","End":"04:18.105","Text":"I\u0027ll write the 1 squared."},{"Start":"04:18.105 ","End":"04:24.465","Text":"Plus twice natural log of 1 is 0."},{"Start":"04:24.465 ","End":"04:31.950","Text":"What we get is a 1/2e squared and then plus 2,"},{"Start":"04:31.950 ","End":"04:35.070","Text":"and then minus a 1/2,"},{"Start":"04:35.070 ","End":"04:37.260","Text":"That\u0027s plus 1 and 1/2."},{"Start":"04:37.260 ","End":"04:40.810","Text":"We\u0027ll write it as 3 over 2."},{"Start":"04:41.420 ","End":"04:45.330","Text":"That\u0027s the curve length."},{"Start":"04:45.330 ","End":"04:49.380","Text":"That\u0027s the answer to part a."},{"Start":"04:49.380 ","End":"04:52.665","Text":"Now on to part b to find the curvature."},{"Start":"04:52.665 ","End":"04:54.180","Text":"This is the formula,"},{"Start":"04:54.180 ","End":"04:58.550","Text":"actually it should be k of t. Now we can reuse some of"},{"Start":"04:58.550 ","End":"05:03.995","Text":"the work we did in part a. I copy pasted here."},{"Start":"05:03.995 ","End":"05:07.415","Text":"That means that we\u0027re really have r prime."},{"Start":"05:07.415 ","End":"05:10.655","Text":"We also have the magnitude of r prime."},{"Start":"05:10.655 ","End":"05:14.975","Text":"What we need is r double-prime."},{"Start":"05:14.975 ","End":"05:19.265","Text":"R double-prime is the next thing to compute."},{"Start":"05:19.265 ","End":"05:22.550","Text":"Just differentiate this, 2 gives us 0,"},{"Start":"05:22.550 ","End":"05:29.180","Text":"t gives us 1 and 2 over t gives minus 2 over t squared."},{"Start":"05:29.180 ","End":"05:35.630","Text":"Next, we need to compute the cross-product so that r prime of"},{"Start":"05:35.630 ","End":"05:42.350","Text":"t crossed with r double prime of t. There several ways to do this,"},{"Start":"05:42.350 ","End":"05:46.310","Text":"1 of them is using the determinant."},{"Start":"05:46.310 ","End":"05:48.740","Text":"We put here the i,"},{"Start":"05:48.740 ","End":"05:51.680","Text":"here vector j, here vector k,"},{"Start":"05:51.680 ","End":"05:57.590","Text":"then the coordinates of one of them 2t and 2 over t,"},{"Start":"05:57.590 ","End":"06:01.070","Text":"and then the coordinates of the second 1, 0,"},{"Start":"06:01.070 ","End":"06:05.495","Text":"1, and minus 2 over t squared."},{"Start":"06:05.495 ","End":"06:07.415","Text":"This will continue over here."},{"Start":"06:07.415 ","End":"06:09.260","Text":"This is equal 2."},{"Start":"06:09.260 ","End":"06:14.000","Text":"We\u0027ll put it back into the components with brackets."},{"Start":"06:14.000 ","End":"06:16.970","Text":"Let\u0027s see for I, which is the first component,"},{"Start":"06:16.970 ","End":"06:23.735","Text":"we get the determinant of this times this minus this times this."},{"Start":"06:23.735 ","End":"06:26.915","Text":"This times this is minus 2 over t,"},{"Start":"06:26.915 ","End":"06:29.600","Text":"minus 2 over t,"},{"Start":"06:29.600 ","End":"06:34.830","Text":"so minus 4 over t. For j,"},{"Start":"06:34.830 ","End":"06:36.230","Text":"because of the checkerboard,"},{"Start":"06:36.230 ","End":"06:41.855","Text":"we need a minus and then we need the determinant of these."},{"Start":"06:41.855 ","End":"06:45.860","Text":"Highlight them, see them better. This and this."},{"Start":"06:45.860 ","End":"06:49.175","Text":"2 times this minus 0 times this,"},{"Start":"06:49.175 ","End":"06:50.645","Text":"but with a minus."},{"Start":"06:50.645 ","End":"06:51.860","Text":"It\u0027s this times this,"},{"Start":"06:51.860 ","End":"06:53.680","Text":"but forget the minus."},{"Start":"06:53.680 ","End":"06:58.760","Text":"We get 4 over t squared in the j components is like the I,"},{"Start":"06:58.760 ","End":"07:01.615","Text":"the J the K. Now for the k,"},{"Start":"07:01.615 ","End":"07:05.315","Text":"we\u0027re going to need the determinant of this bit,"},{"Start":"07:05.315 ","End":"07:09.260","Text":"but it\u0027s going to be with a plus 2 times 1 minus 0 times"},{"Start":"07:09.260 ","End":"07:14.460","Text":"t. That makes it just 2 in the k place."},{"Start":"07:14.460 ","End":"07:20.175","Text":"Now we still need the magnitude of this."},{"Start":"07:20.175 ","End":"07:24.190","Text":"What we get is the following."},{"Start":"07:24.190 ","End":"07:28.870","Text":"We get the Kappa that\u0027s like the Greek K. Kappa of t is"},{"Start":"07:28.870 ","End":"07:35.020","Text":"equal to t squared is 16 over t squared,"},{"Start":"07:35.020 ","End":"07:42.230","Text":"and then 16 over t to the 4."},{"Start":"07:42.860 ","End":"07:48.480","Text":"Here we have 4 under the square root."},{"Start":"07:48.480 ","End":"07:55.895","Text":"All this is over this thing which is this cubed."},{"Start":"07:55.895 ","End":"08:01.530","Text":"It\u0027s t plus 2 over t cubed."},{"Start":"08:01.530 ","End":"08:05.090","Text":"That\u0027s an answer, but we can simplify it."},{"Start":"08:05.090 ","End":"08:11.060","Text":"Tell you what, let\u0027s just do this bit under the square root of the side exercise."},{"Start":"08:11.060 ","End":"08:15.585","Text":"Now what I\u0027m going to do is take 4 over t to the 4,"},{"Start":"08:15.585 ","End":"08:20.130","Text":"outside brackets, 4 over t to the 4."},{"Start":"08:20.130 ","End":"08:24.160","Text":"What I\u0027m left with is,"},{"Start":"08:24.380 ","End":"08:28.050","Text":"let\u0027s see 4t squared."},{"Start":"08:28.050 ","End":"08:30.930","Text":"Because if I take out t to the 4,"},{"Start":"08:30.930 ","End":"08:33.115","Text":"put it over t to the 4 basically,"},{"Start":"08:33.115 ","End":"08:37.025","Text":"multiply top and bottom by t squared the level has take in the 4 out."},{"Start":"08:37.025 ","End":"08:42.060","Text":"Now here I get just 4."},{"Start":"08:42.410 ","End":"08:49.620","Text":"Here I get taken out the 4,"},{"Start":"08:49.620 ","End":"08:53.170","Text":"but it\u0027s 4t to the 4. This is what we get."},{"Start":"08:53.170 ","End":"08:57.265","Text":"Multiply it out and see that you get this."},{"Start":"08:57.265 ","End":"09:03.775","Text":"Now, if I rearrange the order a bit, this first,"},{"Start":"09:03.775 ","End":"09:06.360","Text":"this second, and this third,"},{"Start":"09:06.360 ","End":"09:10.585","Text":"then it\u0027s one of those binomial expansions."},{"Start":"09:10.585 ","End":"09:12.910","Text":"It just, I\u0027ll tell you the answer."},{"Start":"09:12.910 ","End":"09:15.270","Text":"T squared plus 2 squared."},{"Start":"09:15.270 ","End":"09:18.670","Text":"Multiplied out t squared squared is t to the 4 plus twice"},{"Start":"09:18.670 ","End":"09:22.975","Text":"this times this is 4t squared plus this squared."},{"Start":"09:22.975 ","End":"09:29.555","Text":"Back here, I get that Kappa is equal to the square root of this."},{"Start":"09:29.555 ","End":"09:31.910","Text":"Take the square root of each bit separately."},{"Start":"09:31.910 ","End":"09:36.260","Text":"This would be 2 over t squared."},{"Start":"09:36.260 ","End":"09:38.030","Text":"If I take the square root of this,"},{"Start":"09:38.030 ","End":"09:39.530","Text":"if I take the square root of this,"},{"Start":"09:39.530 ","End":"09:42.695","Text":"I just throw out the 2."},{"Start":"09:42.695 ","End":"09:47.435","Text":"We get t squared plus 2."},{"Start":"09:47.435 ","End":"09:50.255","Text":"We still have the denominator,"},{"Start":"09:50.255 ","End":"09:53.360","Text":"which is t plus 2 over t,"},{"Start":"09:53.360 ","End":"09:58.580","Text":"which I can also write as t squared"},{"Start":"09:58.580 ","End":"10:04.265","Text":"plus 2 over t cubed."},{"Start":"10:04.265 ","End":"10:08.750","Text":"Now, this t squared plus 2 occurs more than 1."},{"Start":"10:08.750 ","End":"10:12.590","Text":"I think we go for simplification. Let\u0027s see."},{"Start":"10:12.590 ","End":"10:16.715","Text":"We can write this 2 over t squared first,"},{"Start":"10:16.715 ","End":"10:19.970","Text":"and then we can put the t cubed from"},{"Start":"10:19.970 ","End":"10:25.495","Text":"the denominator of the denominator into the numerator."},{"Start":"10:25.495 ","End":"10:30.660","Text":"Then we have a t squared plus 2 from here."},{"Start":"10:30.660 ","End":"10:36.795","Text":"We still have 1 over t squared plus 2 cubed."},{"Start":"10:36.795 ","End":"10:41.745","Text":"Start to cancel, t squared goes into t cubed, just t times."},{"Start":"10:41.745 ","End":"10:49.470","Text":"Here, t squared plus 2 will knock this 3 down to a 2, and so."},{"Start":"10:49.470 ","End":"10:51.390","Text":"Let\u0027s see, putting it all together,"},{"Start":"10:51.390 ","End":"10:53.659","Text":"we have a 2 from here,"},{"Start":"10:53.659 ","End":"10:56.425","Text":"we have a t from here,"},{"Start":"10:56.425 ","End":"11:04.080","Text":"and we still have t squared plus 2 squared."},{"Start":"11:04.080 ","End":"11:11.590","Text":"That\u0027s our curvature, and that\u0027s the answer to part b. We\u0027re done."}],"ID":9721}],"Thumbnail":null,"ID":57483},{"Name":"Velocity and Acceleration in Space","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"3D Space - Velocity and Acceleration Physics","Duration":"10m 22s","ChapterTopicVideoID":9874,"CourseChapterTopicPlaylistID":57484,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9874.jpeg","UploadDate":"2020-03-26T10:14:12.3300000","DurationForVideoObject":"PT10M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"We\u0027re still with the 3D coordinate system or 3D space."},{"Start":"00:04.500 ","End":"00:08.340","Text":"This time the topic is velocity and acceleration,"},{"Start":"00:08.340 ","End":"00:16.800","Text":"which really belongs to physics or at least to applied mathematics."},{"Start":"00:16.800 ","End":"00:20.580","Text":"1 of the reasons that it\u0027s physics is that we\u0027re talking"},{"Start":"00:20.580 ","End":"00:25.300","Text":"about time and space, reality, existence."},{"Start":"00:25.430 ","End":"00:32.880","Text":"In this case, the parameter t that we\u0027ll take will be time,"},{"Start":"00:32.880 ","End":"00:36.490","Text":"and we\u0027ll usually be given,"},{"Start":"00:36.500 ","End":"00:42.085","Text":"or we\u0027ll have to find it, a position vector."},{"Start":"00:42.085 ","End":"00:45.365","Text":"This is the position as a function of time."},{"Start":"00:45.365 ","End":"00:51.510","Text":"The other useful thing will be the vector velocity,"},{"Start":"00:51.510 ","End":"00:54.000","Text":"also the function of time."},{"Start":"00:54.000 ","End":"01:00.769","Text":"We\u0027re also going to be talking about acceleration as a function of time."},{"Start":"01:00.769 ","End":"01:07.670","Text":"Now the velocity is just the derivative of the position."},{"Start":"01:07.670 ","End":"01:13.895","Text":"That\u0027s pretty much the same like we did before we had vector calculus."},{"Start":"01:13.895 ","End":"01:19.350","Text":"Velocity is the derivative of the position."},{"Start":"01:19.350 ","End":"01:25.380","Text":"An acceleration is the derivative of the velocity."},{"Start":"01:25.400 ","End":"01:30.170","Text":"If you like, we could also say it\u0027s the second derivative"},{"Start":"01:30.170 ","End":"01:34.610","Text":"of the position because the derivative,"},{"Start":"01:34.610 ","End":"01:37.509","Text":"derivative, double derivative."},{"Start":"01:37.509 ","End":"01:39.840","Text":"Sometimes in problems,"},{"Start":"01:39.840 ","End":"01:41.890","Text":"we\u0027re given the velocity,"},{"Start":"01:41.890 ","End":"01:44.380","Text":"and we have to find the position."},{"Start":"01:44.380 ","End":"01:47.390","Text":"We would use the opposite of differentiation,"},{"Start":"01:47.390 ","End":"01:51.650","Text":"which is integration to get from velocity to position,"},{"Start":"01:51.650 ","End":"01:55.145","Text":"and similarly, if we have acceleration and we want velocity,"},{"Start":"01:55.145 ","End":"01:59.615","Text":"then we\u0027ll do the opposite of differentiation, which is integration."},{"Start":"01:59.615 ","End":"02:03.450","Text":"Let\u0027s start straight away with an example problem."},{"Start":"02:04.210 ","End":"02:09.395","Text":"In our example, we\u0027ll be discussing a spaceship,"},{"Start":"02:09.395 ","End":"02:11.210","Text":"but doesn\u0027t have to be a spaceship."},{"Start":"02:11.210 ","End":"02:15.890","Text":"It could be abstractly an object moving in space."},{"Start":"02:15.890 ","End":"02:26.425","Text":"We are given its acceleration as a function of time to equal 2_t,"},{"Start":"02:26.425 ","End":"02:32.415","Text":"0 minus sine t, with 3-dimensional space."},{"Start":"02:32.415 ","End":"02:35.810","Text":"I want to mention most of the stuff will work for 2D also,"},{"Start":"02:35.810 ","End":"02:37.460","Text":"but unless I say otherwise,"},{"Start":"02:37.460 ","End":"02:41.730","Text":"we\u0027ll assume that we\u0027re working in 3D space."},{"Start":"02:42.170 ","End":"02:46.465","Text":"We\u0027re given another 2 conditions."},{"Start":"02:46.465 ","End":"02:49.940","Text":"I forgot to say we start counting time at 0."},{"Start":"02:49.940 ","End":"02:53.120","Text":"We don\u0027t go back in time indefinitely."},{"Start":"02:53.120 ","End":"02:58.040","Text":"Anyway, we start at time 0. Who knows how long the spaceship will carry on."},{"Start":"02:58.040 ","End":"02:59.360","Text":"Anyway, like I was saying,"},{"Start":"02:59.360 ","End":"03:00.830","Text":"we\u0027re given some extra condition,"},{"Start":"03:00.830 ","End":"03:05.750","Text":"is that the velocity at the very start at time 0"},{"Start":"03:05.750 ","End":"03:11.815","Text":"is equal to 0,0,1."},{"Start":"03:11.815 ","End":"03:18.155","Text":"We\u0027re also given the initial position of the rocket, that at time 0,"},{"Start":"03:18.155 ","End":"03:21.430","Text":"it\u0027s situated at,"},{"Start":"03:21.430 ","End":"03:26.805","Text":"see, 1, 2, 300."},{"Start":"03:26.805 ","End":"03:31.835","Text":"It\u0027s quite common in this physics problem to have what are called initial conditions."},{"Start":"03:31.835 ","End":"03:34.550","Text":"The reason is is that when we integrate"},{"Start":"03:34.550 ","End":"03:38.600","Text":"acceleration to get velocity or when we integrate velocity to get position,"},{"Start":"03:38.600 ","End":"03:40.505","Text":"there\u0027s a constant of integration,"},{"Start":"03:40.505 ","End":"03:44.000","Text":"so we need an extra piece of information to find that constant."},{"Start":"03:44.000 ","End":"03:45.770","Text":"Quite typically, we\u0027re given,"},{"Start":"03:45.770 ","End":"03:48.590","Text":"and it\u0027s quite often at time t equals 0."},{"Start":"03:48.590 ","End":"03:50.300","Text":"So far, it\u0027s just the data."},{"Start":"03:50.300 ","End":"03:51.965","Text":"Now, what are the questions?"},{"Start":"03:51.965 ","End":"03:53.900","Text":"There are 2 questions."},{"Start":"03:53.900 ","End":"03:59.525","Text":"1, to compute what is the velocity in general of time t?"},{"Start":"03:59.525 ","End":"04:01.055","Text":"That\u0027s the first question."},{"Start":"04:01.055 ","End":"04:05.630","Text":"The second question is to find the position of the rocket at"},{"Start":"04:05.630 ","End":"04:10.395","Text":"the time when t is Pi over 2."},{"Start":"04:10.395 ","End":"04:13.300","Text":"That\u0027s the second question."},{"Start":"04:13.700 ","End":"04:17.180","Text":"1 way to solve this is as follows."},{"Start":"04:17.180 ","End":"04:19.100","Text":"To find v of t,"},{"Start":"04:19.100 ","End":"04:24.340","Text":"we remember that the velocity is a derivative of the acceleration,"},{"Start":"04:24.340 ","End":"04:27.335","Text":"and we\u0027re given the acceleration."},{"Start":"04:27.335 ","End":"04:36.510","Text":"The velocity would be the integral of the acceleration,"},{"Start":"04:36.510 ","End":"04:38.610","Text":"which we already have,"},{"Start":"04:38.610 ","End":"04:41.425","Text":"which is 2t,"},{"Start":"04:41.425 ","End":"04:49.800","Text":"0 minus sine t, dt."},{"Start":"04:49.800 ","End":"04:51.690","Text":"This is an easy integration."},{"Start":"04:51.690 ","End":"04:55.040","Text":"When we integrate, we just do it component-wise."},{"Start":"04:55.040 ","End":"04:59.644","Text":"We get the integral of 2t is t squared,"},{"Start":"04:59.644 ","End":"05:04.490","Text":"the integral of 0 is 0,"},{"Start":"05:04.490 ","End":"05:08.870","Text":"and the integral of minus sine t is"},{"Start":"05:08.870 ","End":"05:12.980","Text":"cosine t. All these were up to"},{"Start":"05:12.980 ","End":"05:18.500","Text":"constants but what we do is we just put 1 single vector constant,"},{"Start":"05:18.500 ","End":"05:23.210","Text":"let\u0027s call it C. To find this constant,"},{"Start":"05:23.210 ","End":"05:26.690","Text":"we substitute t equals 0"},{"Start":"05:26.690 ","End":"05:31.270","Text":"because we are already given a condition for the velocity at time 0."},{"Start":"05:31.270 ","End":"05:34.275","Text":"Let me just copy this line here."},{"Start":"05:34.275 ","End":"05:37.080","Text":"If t is 0, here on the left,"},{"Start":"05:37.080 ","End":"05:38.850","Text":"we get v of 0."},{"Start":"05:38.850 ","End":"05:40.710","Text":"I\u0027m I forgetting the arrows?"},{"Start":"05:40.710 ","End":"05:46.740","Text":"Yes. V of 0 is 0,0,1,"},{"Start":"05:46.740 ","End":"05:49.155","Text":"so that\u0027s the left-hand side."},{"Start":"05:49.155 ","End":"05:51.180","Text":"The right-hand side, let\u0027s see."},{"Start":"05:51.180 ","End":"05:56.045","Text":"If t is 0, we get 0 squared is 0,"},{"Start":"05:56.045 ","End":"06:00.640","Text":"0 is 0, and cosine of 0 is 1,"},{"Start":"06:00.640 ","End":"06:05.370","Text":"plus the C. Since this equals this,"},{"Start":"06:05.370 ","End":"06:09.820","Text":"C is the 0 vector, 0,0,0, so C,"},{"Start":"06:09.820 ","End":"06:14.360","Text":"I\u0027ll just call it the 0 vector rather than writing 0,0,0 which"},{"Start":"06:14.360 ","End":"06:21.120","Text":"means that the answer for the velocity if C is 0,"},{"Start":"06:21.120 ","End":"06:24.590","Text":"is v of t,"},{"Start":"06:24.590 ","End":"06:32.220","Text":"is equal to t squared 0 cosine t,"},{"Start":"06:32.220 ","End":"06:36.620","Text":"and this is the answer to this part."},{"Start":"06:36.620 ","End":"06:39.470","Text":"Now let\u0027s look at the second part."},{"Start":"06:39.470 ","End":"06:47.570","Text":"My strategy will be to find r of t in general and then substitute t equals Pi over 2."},{"Start":"06:47.570 ","End":"06:51.690","Text":"It\u0027s very similar to the first part because when we may go from"},{"Start":"06:51.690 ","End":"06:56.060","Text":"v to r since velocity is the derivative of the position,"},{"Start":"06:56.060 ","End":"06:58.460","Text":"then we just do another integration."},{"Start":"06:58.460 ","End":"07:06.545","Text":"What I would say is that r of t is the integral of the velocity."},{"Start":"07:06.545 ","End":"07:08.555","Text":"We already have the velocity,"},{"Start":"07:08.555 ","End":"07:12.650","Text":"so it\u0027s the integral of t squared 0,"},{"Start":"07:12.650 ","End":"07:16.435","Text":"cosine t, dt."},{"Start":"07:16.435 ","End":"07:21.210","Text":"Once again, we do it component-wise,"},{"Start":"07:21.210 ","End":"07:27.010","Text":"so we get t cubed over 3,"},{"Start":"07:27.190 ","End":"07:34.295","Text":"integral of 0,0 integral of cosine is sine up to a constant."},{"Start":"07:34.295 ","End":"07:38.210","Text":"Again, not the same constant here,"},{"Start":"07:38.210 ","End":"07:41.060","Text":"but I\u0027ll use the same letter C again won\u0027t hurt."},{"Start":"07:41.060 ","End":"07:45.995","Text":"Once again, we want to put an initial condition,"},{"Start":"07:45.995 ","End":"07:47.705","Text":"and we can use this 1."},{"Start":"07:47.705 ","End":"07:53.110","Text":"Here I\u0027ll also let t equals 0 and substitute that."},{"Start":"07:53.110 ","End":"07:55.435","Text":"It\u0027s like we substituted t equals 0 here."},{"Start":"07:55.435 ","End":"07:56.965","Text":"We can do it here."},{"Start":"07:56.965 ","End":"08:01.980","Text":"We know that r of 0 is given by this,"},{"Start":"08:01.980 ","End":"08:06.600","Text":"so we have 1,2,300."},{"Start":"08:06.600 ","End":"08:16.135","Text":"The other hand, plugging t equals 0 here gives us 0 cubed over 3 is 0,"},{"Start":"08:16.135 ","End":"08:22.525","Text":"0 is 0, and sine 0 is 0 plus"},{"Start":"08:22.525 ","End":"08:31.325","Text":"C. If I bring this to the other side and subtract C is just equal to this."},{"Start":"08:31.325 ","End":"08:36.170","Text":"If C is equal to this here,"},{"Start":"08:36.170 ","End":"08:38.530","Text":"then I can put it back in here,"},{"Start":"08:38.530 ","End":"08:46.360","Text":"and I can get that the position vector at time t is equal to whatever was here,"},{"Start":"08:46.360 ","End":"08:48.575","Text":"t cubed over 3,"},{"Start":"08:48.575 ","End":"08:53.350","Text":"0 sine t plus the C,"},{"Start":"08:53.350 ","End":"08:57.140","Text":"which we get from here is 1,2,300."},{"Start":"08:57.140 ","End":"09:04.090","Text":"Let me just get some more space here."},{"Start":"09:04.450 ","End":"09:10.830","Text":"I could simplify this by doing the addition component-wise."},{"Start":"09:10.830 ","End":"09:14.520","Text":"Here I have 1/3t cubed,"},{"Start":"09:14.520 ","End":"09:17.055","Text":"that\u0027s this, plus 1,"},{"Start":"09:17.055 ","End":"09:19.680","Text":"0 plus 2 is 2,"},{"Start":"09:19.680 ","End":"09:25.000","Text":"sine t plus 300."},{"Start":"09:25.000 ","End":"09:26.860","Text":"Now, to answer the question,"},{"Start":"09:26.860 ","End":"09:30.730","Text":"all I have to do is substitute instead of t,"},{"Start":"09:30.730 ","End":"09:32.710","Text":"Pi over 2,"},{"Start":"09:32.710 ","End":"09:38.780","Text":"and then I get Pi over 2 that\u0027s a 3."},{"Start":"09:39.290 ","End":"09:45.405","Text":"Pi over 2 cubed is Pi cubed over 8,"},{"Start":"09:45.405 ","End":"09:53.375","Text":"with the 3 makes it over 24 plus 1,"},{"Start":"09:53.375 ","End":"10:00.140","Text":"and then here just 2 sine Pi over 2 is 1,"},{"Start":"10:00.140 ","End":"10:01.770","Text":"sine of 90 degrees."},{"Start":"10:01.770 ","End":"10:05.525","Text":"1 plus 300 is 301."},{"Start":"10:05.525 ","End":"10:10.190","Text":"Now we have the answer to the second question too."},{"Start":"10:10.190 ","End":"10:13.835","Text":"I\u0027m going to return to this example later,"},{"Start":"10:13.835 ","End":"10:16.830","Text":"now back to a bit more theory."},{"Start":"10:17.720 ","End":"10:22.020","Text":"I think I\u0027ll erase what I don\u0027t need."}],"ID":9722},{"Watched":false,"Name":"3D Space - Velocity and Acceleration","Duration":"20m 3s","ChapterTopicVideoID":9873,"CourseChapterTopicPlaylistID":57484,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9873.jpeg","UploadDate":"2020-03-26T10:18:12.4570000","DurationForVideoObject":"PT20M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.479","Text":"I\u0027d like to relate this stuff to what we learned earlier about the unit tangent,"},{"Start":"00:06.479 ","End":"00:09.030","Text":"unit normals, curvature and all that."},{"Start":"00:09.030 ","End":"00:13.470","Text":"Let me insert a diagram here."},{"Start":"00:13.470 ","End":"00:17.580","Text":"Here\u0027s the picture I wanted to show you,"},{"Start":"00:17.580 ","End":"00:21.195","Text":"and now I\u0027ll talk some and we\u0027ll relate to the picture."},{"Start":"00:21.195 ","End":"00:26.445","Text":"There are 3 vectors in question here."},{"Start":"00:26.445 ","End":"00:28.545","Text":"List as the position,"},{"Start":"00:28.545 ","End":"00:31.900","Text":"the velocity, and the acceleration."},{"Start":"00:32.180 ","End":"00:40.880","Text":"Let\u0027s say this gray curve is where the spaceship or particle,"},{"Start":"00:40.880 ","End":"00:42.680","Text":"whatever is moving along,"},{"Start":"00:42.680 ","End":"00:51.030","Text":"and maybe it\u0027s going in this direction where time is increasing this way,"},{"Start":"00:51.030 ","End":"00:55.069","Text":"and at a given point,"},{"Start":"00:55.069 ","End":"01:01.225","Text":"remember that we had 2 special unit vectors."},{"Start":"01:01.225 ","End":"01:03.900","Text":"We had at any given point t,"},{"Start":"01:03.900 ","End":"01:06.315","Text":"we had a unit normal,"},{"Start":"01:06.315 ","End":"01:08.685","Text":"I meant the other way around."},{"Start":"01:08.685 ","End":"01:13.635","Text":"We had a unit tangent vector as a function of t,"},{"Start":"01:13.635 ","End":"01:21.515","Text":"and we also had a unit normal vector also as a function of t. There was also a bi-normal,"},{"Start":"01:21.515 ","End":"01:23.990","Text":"but we\u0027re not going to be using that here."},{"Start":"01:23.990 ","End":"01:25.490","Text":"I\u0027ll write it down,"},{"Start":"01:25.490 ","End":"01:29.510","Text":"but which is perpendicular to them both"},{"Start":"01:29.510 ","End":"01:34.475","Text":"and each of them are perpendicular to the other and are of unit length."},{"Start":"01:34.475 ","End":"01:39.110","Text":"Now, that\u0027s for r of t,"},{"Start":"01:39.110 ","End":"01:41.210","Text":"that\u0027s less interesting to us."},{"Start":"01:41.210 ","End":"01:42.560","Text":"We\u0027re not going to do anything with that."},{"Start":"01:42.560 ","End":"01:45.470","Text":"It\u0027s some vector, maybe there\u0027s an origin somewhere."},{"Start":"01:45.470 ","End":"01:47.420","Text":"Let\u0027s say if this was the origin,"},{"Start":"01:47.420 ","End":"01:51.650","Text":"then the position vector might be something like this."},{"Start":"01:51.650 ","End":"01:52.940","Text":"It\u0027s not very interesting."},{"Start":"01:52.940 ","End":"01:55.400","Text":"Just doing it for completeness that we have r,"},{"Start":"01:55.400 ","End":"01:58.720","Text":"this would be the origin o."},{"Start":"01:58.720 ","End":"02:02.090","Text":"Then there\u0027s the velocity vector V,"},{"Start":"02:02.090 ","End":"02:04.744","Text":"which is also called r prime."},{"Start":"02:04.744 ","End":"02:10.594","Text":"I won\u0027t do a double labeling and that would be this 1 here."},{"Start":"02:10.594 ","End":"02:12.740","Text":"Maybe I will label it."},{"Start":"02:12.740 ","End":"02:18.785","Text":"This is v of t. Then there\u0027s an acceleration vector,"},{"Start":"02:18.785 ","End":"02:22.150","Text":"and that\u0027s this 1 here."},{"Start":"02:22.150 ","End":"02:23.540","Text":"I hope I get it straight."},{"Start":"02:23.540 ","End":"02:25.980","Text":"If not, you\u0027ll forgive me."},{"Start":"02:26.020 ","End":"02:29.855","Text":"Now, as I said, we\u0027re not going to do anything with the position vector."},{"Start":"02:29.855 ","End":"02:33.904","Text":"Not interesting, velocity, less interesting."},{"Start":"02:33.904 ","End":"02:37.520","Text":"I\u0027ll just mention that the velocity vector is"},{"Start":"02:37.520 ","End":"02:42.330","Text":"always in the same direction as the tangent."},{"Start":"02:42.650 ","End":"02:48.440","Text":"I say this, and you\u0027ll see why I say this when I come to acceleration,"},{"Start":"02:48.440 ","End":"02:52.630","Text":"because acceleration is not"},{"Start":"02:52.740 ","End":"02:57.310","Text":"in the direction of the tangent or in the direction of the normal."},{"Start":"02:57.310 ","End":"03:01.000","Text":"One thing we do know about the acceleration vector"},{"Start":"03:01.000 ","End":"03:06.100","Text":"is that it is in the plane defined by the tangent."},{"Start":"03:06.100 ","End":"03:08.755","Text":"This is the tangent, this is the normal."},{"Start":"03:08.755 ","End":"03:11.120","Text":"It is in the plane."},{"Start":"03:11.120 ","End":"03:13.960","Text":"I\u0027m just mentioning this, you don\u0027t have to know this."},{"Start":"03:13.960 ","End":"03:15.880","Text":"It\u0027s called an osculating plane."},{"Start":"03:15.880 ","End":"03:22.765","Text":"Remember we talked about the osculating circle that kisses the curve at that point."},{"Start":"03:22.765 ","End":"03:25.700","Text":"Anyway, I\u0027m just mentioning it,"},{"Start":"03:25.700 ","End":"03:30.345","Text":"but you don\u0027t have to know the term."},{"Start":"03:30.345 ","End":"03:36.260","Text":"Since it\u0027s in the plane defined by the normal and the tangent,"},{"Start":"03:36.260 ","End":"03:38.090","Text":"we can break it up into components,"},{"Start":"03:38.090 ","End":"03:40.420","Text":"and that\u0027s something we do a lot in physics,"},{"Start":"03:40.420 ","End":"03:45.439","Text":"where we have 2 perpendiculars and another diagonal,"},{"Start":"03:45.439 ","End":"03:50.435","Text":"so to speak, we break it up and these dotted lines show how we break it up."},{"Start":"03:50.435 ","End":"03:55.230","Text":"This vector would be this vector,"},{"Start":"03:55.230 ","End":"03:56.750","Text":"don\u0027t have enough colors,"},{"Start":"03:56.750 ","End":"04:03.800","Text":"but it\u0027s just up to where the curly braces are from here to here,"},{"Start":"04:03.800 ","End":"04:07.890","Text":"maybe I\u0027ll just put an arrow there."},{"Start":"04:07.890 ","End":"04:09.815","Text":"From here to here,"},{"Start":"04:09.815 ","End":"04:16.610","Text":"that\u0027s the component of the acceleration in the tangential direction."},{"Start":"04:16.610 ","End":"04:19.940","Text":"Tangential from tangent. In fact,"},{"Start":"04:19.940 ","End":"04:24.210","Text":"I\u0027ll write the word tangential."},{"Start":"04:25.150 ","End":"04:29.860","Text":"This direction is called the normal direction."},{"Start":"04:29.860 ","End":"04:31.635","Text":"We get a component,"},{"Start":"04:31.635 ","End":"04:36.125","Text":"which is we if we project this line onto this line,"},{"Start":"04:36.125 ","End":"04:38.210","Text":"we get this vector here,"},{"Start":"04:38.210 ","End":"04:43.560","Text":"so that this 1 is equal to this plus this."},{"Start":"04:43.940 ","End":"04:51.590","Text":"Here it\u0027s written down that the coefficient of the normal,"},{"Start":"04:51.590 ","End":"04:55.280","Text":"it was how many times the normal goes into this vector."},{"Start":"04:55.280 ","End":"04:57.695","Text":"We\u0027ll call that a_N,"},{"Start":"04:57.695 ","End":"04:59.765","Text":"the normal component of a,"},{"Start":"04:59.765 ","End":"05:05.055","Text":"and a_T will be the tangential component."},{"Start":"05:05.055 ","End":"05:12.310","Text":"This vector here, this 1 is how many times this goes into this?"},{"Start":"05:12.310 ","End":"05:15.630","Text":"In other words, it\u0027s the magnitude, basically."},{"Start":"05:15.680 ","End":"05:22.030","Text":"So a_T is the magnitude of this vector,"},{"Start":"05:22.030 ","End":"05:27.070","Text":"and a_N, is the magnitude of this vector, the 2 components."},{"Start":"05:27.070 ","End":"05:31.615","Text":"What I can do is write the acceleration."},{"Start":"05:31.615 ","End":"05:33.620","Text":"Now, everything is a function of t,"},{"Start":"05:33.620 ","End":"05:37.325","Text":"but just for simplicity I won\u0027t write the brackets t all the time,"},{"Start":"05:37.325 ","End":"05:40.430","Text":"is equal to, I prefer the other order,"},{"Start":"05:40.430 ","End":"05:44.600","Text":"the tangential first, the tangential component,"},{"Start":"05:44.600 ","End":"05:46.085","Text":"which is a scalar,"},{"Start":"05:46.085 ","End":"05:49.850","Text":"times the unit tangent vector plus"},{"Start":"05:49.850 ","End":"05:56.870","Text":"the normal component of the acceleration that\u0027s in the normal direction."},{"Start":"05:56.870 ","End":"06:01.065","Text":"I\u0027m breaking this up into 2 bits, this and this."},{"Start":"06:01.065 ","End":"06:10.430","Text":"Now there are formulas for a_T and a_N which are illustrated with these curly braces,"},{"Start":"06:10.430 ","End":"06:13.625","Text":"these 2 magnitudes, and here\u0027s the formulas."},{"Start":"06:13.625 ","End":"06:18.950","Text":"The formula for the tangential component of"},{"Start":"06:18.950 ","End":"06:26.570","Text":"the acceleration is equal to r double-prime,"},{"Start":"06:26.570 ","End":"06:30.180","Text":"well, let\u0027s put the t in,"},{"Start":"06:33.290 ","End":"06:35.460","Text":"this is a vector,"},{"Start":"06:35.460 ","End":"06:42.020","Text":"dot product with r prime of t over"},{"Start":"06:42.020 ","End":"06:49.980","Text":"the magnitude of the r prime"},{"Start":"06:49.980 ","End":"06:55.695","Text":"of t vector, vector."},{"Start":"06:55.695 ","End":"06:59.905","Text":"Strictly speaking, this should really be a function of t also."},{"Start":"06:59.905 ","End":"07:05.450","Text":"I\u0027ll write that. This is a function of t. There are many books omit that."},{"Start":"07:05.450 ","End":"07:12.860","Text":"The other formula for the normal component of the acceleration."},{"Start":"07:12.860 ","End":"07:17.340","Text":"Yeah, the function of t is equal to,"},{"Start":"07:17.590 ","End":"07:21.229","Text":"it\u0027s very similar to this formula."},{"Start":"07:21.229 ","End":"07:26.165","Text":"We have also r double prime of t,"},{"Start":"07:26.165 ","End":"07:27.710","Text":"instead of a dot product,"},{"Start":"07:27.710 ","End":"07:30.280","Text":"we have a cross product,"},{"Start":"07:30.280 ","End":"07:34.470","Text":"that\u0027s 1 of the basic differences."},{"Start":"07:34.470 ","End":"07:37.399","Text":"Because the cross product is a vector,"},{"Start":"07:37.399 ","End":"07:38.660","Text":"but this is a scalar,"},{"Start":"07:38.660 ","End":"07:44.870","Text":"we\u0027ll take the magnitude of this and we divide it by the same thing as here."},{"Start":"07:44.870 ","End":"07:52.580","Text":"The magnitude of r prime of t. Notice that,"},{"Start":"07:52.580 ","End":"07:54.785","Text":"you could rewrite this."},{"Start":"07:54.785 ","End":"07:56.480","Text":"I don\u0027t know if it\u0027s simpler,"},{"Start":"07:56.480 ","End":"08:00.860","Text":"but because r double-prime is acceleration and r prime and velocity,"},{"Start":"08:00.860 ","End":"08:07.475","Text":"I suppose I could have written this as the acceleration dot"},{"Start":"08:07.475 ","End":"08:14.790","Text":"with the velocity over the magnitude of the velocity."},{"Start":"08:14.790 ","End":"08:18.815","Text":"Similarly here, I could have written It\u0027s the acceleration,"},{"Start":"08:18.815 ","End":"08:26.930","Text":"cross the velocity then take the magnitude and over the same thing,"},{"Start":"08:26.930 ","End":"08:30.860","Text":"magnitude of the velocity."},{"Start":"08:30.860 ","End":"08:33.485","Text":"But for some reason,"},{"Start":"08:33.485 ","End":"08:37.500","Text":"the books presented this way."},{"Start":"08:37.730 ","End":"08:41.050","Text":"This is the way we\u0027ll do it."},{"Start":"08:43.850 ","End":"08:50.655","Text":"There is also another set of formulas. You know what?"},{"Start":"08:50.655 ","End":"08:55.775","Text":"Perhaps I\u0027ll do the example first and then I\u0027ll give you the alternate formulas."},{"Start":"08:55.775 ","End":"08:59.900","Text":"I put these 2 important formulas in a box."},{"Start":"08:59.900 ","End":"09:04.789","Text":"I guess I don\u0027t really need this."},{"Start":"09:04.789 ","End":"09:07.310","Text":"The example I\u0027m going to use is the same as"},{"Start":"09:07.310 ","End":"09:10.565","Text":"the previous example that we had with the rocket."},{"Start":"09:10.565 ","End":"09:20.090","Text":"I just copied the acceleration and velocity from the previous example."},{"Start":"09:20.090 ","End":"09:21.380","Text":"You can go back and look."},{"Start":"09:21.380 ","End":"09:24.095","Text":"We don\u0027t actually need r. Notice that"},{"Start":"09:24.095 ","End":"09:28.820","Text":"the position vector itself doesn\u0027t appear only as double-prime or prime,"},{"Start":"09:28.820 ","End":"09:31.860","Text":"meaning velocity and acceleration."},{"Start":"09:32.240 ","End":"09:35.059","Text":"We can conform with this formula."},{"Start":"09:35.059 ","End":"09:36.290","Text":"I\u0027ll write this 1,"},{"Start":"09:36.290 ","End":"09:43.560","Text":"this is r double prime of t and the velocity is r prime of t. Now,"},{"Start":"09:44.260 ","End":"09:52.815","Text":"the question is to find what is a _T and what is a_N."},{"Start":"09:52.815 ","End":"09:59.070","Text":"What are the tangential and normal components of this acceleration?"},{"Start":"09:59.930 ","End":"10:04.575","Text":"Just substitute it into the formula."},{"Start":"10:04.575 ","End":"10:14.310","Text":"We get that a tangential really as a function of t is equal to,"},{"Start":"10:14.350 ","End":"10:17.390","Text":"I have the dot product of these 2,"},{"Start":"10:17.390 ","End":"10:20.375","Text":"which is these 2, and you know what?"},{"Start":"10:20.375 ","End":"10:23.350","Text":"I think I\u0027d like to have the prime before the double-prime,"},{"Start":"10:23.350 ","End":"10:27.885","Text":"it doesn\u0027t make any difference I just like it better that way."},{"Start":"10:27.885 ","End":"10:31.680","Text":"I have this, dot product with this."},{"Start":"10:31.680 ","End":"10:33.330","Text":"Why don\u0027t I just do it right here?"},{"Start":"10:33.330 ","End":"10:37.070","Text":"The dot product means this times this plus this times this plus this times this."},{"Start":"10:37.070 ","End":"10:46.140","Text":"T squared times 2t is 2t cubed,"},{"Start":"10:46.140 ","End":"10:50.745","Text":"and this with this is minus sine t,"},{"Start":"10:50.745 ","End":"10:55.380","Text":"cosine t, then I need a dividing line,"},{"Start":"10:55.380 ","End":"10:57.960","Text":"and it\u0027s over this."},{"Start":"10:57.960 ","End":"11:00.200","Text":"Now, this appears here and here."},{"Start":"11:00.200 ","End":"11:03.210","Text":"Why don\u0027t I just compute it once?"},{"Start":"11:04.300 ","End":"11:11.250","Text":"The magnitude of r prime of t will equal,"},{"Start":"11:11.250 ","End":"11:12.500","Text":"if I do this, it\u0027s going to be"},{"Start":"11:12.500 ","End":"11:17.395","Text":"the square root of this squared plus this squared plus this squared."},{"Start":"11:17.395 ","End":"11:28.460","Text":"It\u0027s t to the 4th plus 0 plus cosine squared t. I can put that in here."},{"Start":"11:31.340 ","End":"11:37.740","Text":"Plus cosine squared t. That\u0027s the first 1."},{"Start":"11:37.740 ","End":"11:40.230","Text":"Now, the next 1,"},{"Start":"11:40.230 ","End":"11:44.340","Text":"the normal component of the acceleration."},{"Start":"11:44.340 ","End":"11:47.850","Text":"This time, I need the cross-product."},{"Start":"11:47.850 ","End":"11:51.120","Text":"I\u0027m not going to waste time with cross-products."},{"Start":"11:51.120 ","End":"11:56.805","Text":"I\u0027ll tell you the answer to the cross-product of this cross with this."},{"Start":"11:56.805 ","End":"12:00.240","Text":"It comes out to be 0,"},{"Start":"12:00.240 ","End":"12:06.315","Text":"t squared sine t"},{"Start":"12:06.315 ","End":"12:15.930","Text":"plus 2t cosine t, 0."},{"Start":"12:15.930 ","End":"12:17.670","Text":"It\u0027s a vector what we get,"},{"Start":"12:17.670 ","End":"12:19.950","Text":"and we have to put that in bars."},{"Start":"12:19.950 ","End":"12:26.139","Text":"I mean, take the magnitude and it\u0027s over the same thing."},{"Start":"12:26.240 ","End":"12:32.595","Text":"So once again, the square root of t to the 4 plus cosine"},{"Start":"12:32.595 ","End":"12:39.180","Text":"squared t. The only thing we have left to do then is to compute this."},{"Start":"12:39.180 ","End":"12:45.765","Text":"But look, if I have the magnitude of 0,"},{"Start":"12:45.765 ","End":"12:50.610","Text":"something 0, what I\u0027ll get is"},{"Start":"12:50.610 ","End":"12:53.460","Text":"the square root of 0 squared plus"},{"Start":"12:53.460 ","End":"12:56.850","Text":"a squared plus 0 squared is the square root of a squared,"},{"Start":"12:56.850 ","End":"13:00.970","Text":"and that is just the absolute value of a."},{"Start":"13:01.280 ","End":"13:05.790","Text":"So what I have here,"},{"Start":"13:05.790 ","End":"13:10.485","Text":"I can just throw everything out."},{"Start":"13:10.485 ","End":"13:13.330","Text":"Maybe I\u0027ll write it again."},{"Start":"13:14.210 ","End":"13:19.230","Text":"So this equals dividing line,"},{"Start":"13:19.230 ","End":"13:28.860","Text":"square root of t to the 4 plus cosine squared t. Here just this expression,"},{"Start":"13:28.860 ","End":"13:40.260","Text":"t squared sine t plus 2t cosine t. Extend this a bit,"},{"Start":"13:40.260 ","End":"13:43.500","Text":"and technically we should put this in bars,"},{"Start":"13:43.500 ","End":"13:46.765","Text":"even though I happen to know this is always positive."},{"Start":"13:46.765 ","End":"13:49.930","Text":"So that answers that."},{"Start":"13:49.930 ","End":"13:53.370","Text":"Now the only thing that I still have left, as I mentioned,"},{"Start":"13:53.370 ","End":"13:58.245","Text":"that there is another alternative form of these,"},{"Start":"13:58.245 ","End":"14:03.120","Text":"and just to get the space,"},{"Start":"14:03.120 ","End":"14:07.125","Text":"I will, again whether this diagram."},{"Start":"14:07.125 ","End":"14:09.420","Text":"Before I give you the alternative formula,"},{"Start":"14:09.420 ","End":"14:11.790","Text":"just want to note something."},{"Start":"14:11.790 ","End":"14:13.860","Text":"When we say velocity,"},{"Start":"14:13.860 ","End":"14:18.480","Text":"velocity is a vector and we write it with an arrow."},{"Start":"14:18.480 ","End":"14:23.325","Text":"In this case, velocity is a function of time."},{"Start":"14:23.325 ","End":"14:26.175","Text":"There was also a concept called speed."},{"Start":"14:26.175 ","End":"14:31.365","Text":"The difference between speed and velocity is that velocity has direction,"},{"Start":"14:31.365 ","End":"14:34.755","Text":"it\u0027s a vector, but speed has just the magnitude."},{"Start":"14:34.755 ","End":"14:40.200","Text":"So in fact, V without the arrow on"},{"Start":"14:40.200 ","End":"14:49.380","Text":"top is just the magnitude of the velocity vector."},{"Start":"14:49.380 ","End":"14:51.075","Text":"So I\u0027ll just write that."},{"Start":"14:51.075 ","End":"14:55.575","Text":"That\u0027s what\u0027s inside the bars is velocity."},{"Start":"14:55.575 ","End":"14:58.050","Text":"But this is speed,"},{"Start":"14:58.050 ","End":"15:02.220","Text":"and here vector here, no vector."},{"Start":"15:02.220 ","End":"15:05.370","Text":"Now I tend to sometimes forget to write the vector sign,"},{"Start":"15:05.370 ","End":"15:07.289","Text":"but here it\u0027s important."},{"Start":"15:07.289 ","End":"15:10.170","Text":"So now that I\u0027ve given you this,"},{"Start":"15:10.170 ","End":"15:14.355","Text":"now I\u0027m going to give you an alternative formula for this,"},{"Start":"15:14.355 ","End":"15:18.540","Text":"and it turns out that they look simpler anyway,"},{"Start":"15:18.540 ","End":"15:20.010","Text":"doesn\u0027t mean that they\u0027re easier."},{"Start":"15:20.010 ","End":"15:25.935","Text":"That this tangential acceleration is just V prime of t,"},{"Start":"15:25.935 ","End":"15:29.820","Text":"V being the speed, no vector here."},{"Start":"15:29.820 ","End":"15:37.470","Text":"The normal component of the acceleration is equal to Kappa,"},{"Start":"15:37.470 ","End":"15:43.065","Text":"the curvature times V squared."},{"Start":"15:43.065 ","End":"15:47.440","Text":"Well, I should really say V of t squared."},{"Start":"15:47.660 ","End":"15:52.020","Text":"Now, notice that in our case we do already have"},{"Start":"15:52.020 ","End":"15:56.130","Text":"the speed because the velocity is r prime."},{"Start":"15:56.130 ","End":"16:02.175","Text":"So this is actually equal to V of t. We already have that."},{"Start":"16:02.175 ","End":"16:10.440","Text":"So what I\u0027d like to do besides just to add 1 more question and to say,"},{"Start":"16:10.440 ","End":"16:13.650","Text":"what is Kappa equal to."},{"Start":"16:13.650 ","End":"16:16.560","Text":"I guess I\u0027d also like to check"},{"Start":"16:16.560 ","End":"16:26.050","Text":"that this formula for the tangential acceleration comes out the same."},{"Start":"16:26.600 ","End":"16:31.350","Text":"Let\u0027s do that, because all I have to do is differentiate."},{"Start":"16:31.350 ","End":"16:33.750","Text":"If I have V of t, is this,"},{"Start":"16:33.750 ","End":"16:43.380","Text":"then V prime of t is equal to the derivative of this."},{"Start":"16:43.380 ","End":"16:45.585","Text":"So because of the square root,"},{"Start":"16:45.585 ","End":"16:52.800","Text":"I have to put 1 over twice the square root of whatever it is t to the"},{"Start":"16:52.800 ","End":"17:00.410","Text":"fourth plus cosine squared t. But because it\u0027s not t,"},{"Start":"17:00.410 ","End":"17:01.430","Text":"that\u0027s the inner function,"},{"Start":"17:01.430 ","End":"17:03.050","Text":"I need the derivative of this."},{"Start":"17:03.050 ","End":"17:07.330","Text":"The derivative of this is 4t cubed,"},{"Start":"17:07.330 ","End":"17:11.790","Text":"and derivative of cosine squared t"},{"Start":"17:11.790 ","End":"17:18.690","Text":"is 2 cosine t times the derivative of cosine t,"},{"Start":"17:18.690 ","End":"17:23.385","Text":"which is minus sine t. Now,"},{"Start":"17:23.385 ","End":"17:28.530","Text":"if you divide top and bottom by 2,"},{"Start":"17:28.530 ","End":"17:30.975","Text":"this goes, this goes."},{"Start":"17:30.975 ","End":"17:33.330","Text":"2 into 4 goes twice,"},{"Start":"17:33.330 ","End":"17:35.505","Text":"and this here becomes a minus,"},{"Start":"17:35.505 ","End":"17:41.490","Text":"and you see that what we have here is exactly what we have here."},{"Start":"17:41.490 ","End":"17:42.825","Text":"So it\u0027s the same."},{"Start":"17:42.825 ","End":"17:45.640","Text":"So that\u0027s check."},{"Start":"17:45.800 ","End":"17:50.850","Text":"Let\u0027s just compute the curvature and then we\u0027ll be done with this."},{"Start":"17:50.850 ","End":"17:54.060","Text":"I\u0027ll delete this arrow just getting in the way."},{"Start":"17:54.060 ","End":"18:00.585","Text":"I just realized to be precise or Kappa is also a function of t. So put that in there."},{"Start":"18:00.585 ","End":"18:04.740","Text":"Now to find Kappa of t,"},{"Start":"18:04.740 ","End":"18:12.960","Text":"I can write it as just bring this over to the other side as"},{"Start":"18:12.960 ","End":"18:23.800","Text":"a_N of t divided by V of t squared."},{"Start":"18:32.240 ","End":"18:36.165","Text":"I\u0027ll tidy this up by getting rid of the middle term."},{"Start":"18:36.165 ","End":"18:39.495","Text":"Now, notice that this denominator,"},{"Start":"18:39.495 ","End":"18:48.750","Text":"just the denominator part is V of t. So if I\u0027m going to take a_N,"},{"Start":"18:48.750 ","End":"18:51.540","Text":"which is this, and divide it by V squared."},{"Start":"18:51.540 ","End":"18:54.645","Text":"I\u0027m going to get it over V cubed."},{"Start":"18:54.645 ","End":"19:00.660","Text":"So Kappa of t is equal to the numerator from here,"},{"Start":"19:00.660 ","End":"19:09.195","Text":"absolute value of t squared sine t plus 2t cosine t, I say,"},{"Start":"19:09.195 ","End":"19:11.040","Text":"may not need the absolute value,"},{"Start":"19:11.040 ","End":"19:16.200","Text":"but this just to be safe over now as we said,"},{"Start":"19:16.200 ","End":"19:19.815","Text":"this is V squared and this is V so we get V cubed."},{"Start":"19:19.815 ","End":"19:22.605","Text":"Now the square root of something cubed,"},{"Start":"19:22.605 ","End":"19:26.730","Text":"say square root of x and I cube it."},{"Start":"19:26.730 ","End":"19:28.110","Text":"This is to the power of half,"},{"Start":"19:28.110 ","End":"19:29.220","Text":"this is the power of 3."},{"Start":"19:29.220 ","End":"19:32.955","Text":"I can write it as x to the power of 3 over 2."},{"Start":"19:32.955 ","End":"19:34.860","Text":"So in our case,"},{"Start":"19:34.860 ","End":"19:45.210","Text":"we just get t to the 4 plus cosine squared t to the power of 3 over 2,"},{"Start":"19:45.210 ","End":"19:48.195","Text":"and then we found Kappa also."},{"Start":"19:48.195 ","End":"19:55.270","Text":"So we\u0027ve really solved just about everything there is to do about in this problem."},{"Start":"19:55.640 ","End":"20:03.280","Text":"That\u0027s it for Velocity and Acceleration in 3D space. We\u0027re done."}],"ID":9723},{"Watched":false,"Name":"Exercise 9","Duration":"10m 16s","ChapterTopicVideoID":9850,"CourseChapterTopicPlaylistID":57484,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9850.jpeg","UploadDate":"2017-08-06T05:11:51.5730000","DurationForVideoObject":"PT10M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"Here we have one of these velocity and acceleration problems."},{"Start":"00:03.450 ","End":"00:11.475","Text":"We\u0027re given the acceleration of a body by this formula in the ijk notation."},{"Start":"00:11.475 ","End":"00:16.425","Text":"T is time in these problems, a for acceleration."},{"Start":"00:16.425 ","End":"00:19.560","Text":"Now we\u0027re given the body\u0027s initial velocity."},{"Start":"00:19.560 ","End":"00:21.510","Text":"Initial means at time 0,"},{"Start":"00:21.510 ","End":"00:23.445","Text":"is given by this,"},{"Start":"00:23.445 ","End":"00:25.350","Text":"and we have its initial position,"},{"Start":"00:25.350 ","End":"00:26.955","Text":"r is the position vector,"},{"Start":"00:26.955 ","End":"00:28.500","Text":"time 0 is this."},{"Start":"00:28.500 ","End":"00:33.075","Text":"We have to find the body\u0027s velocity and position functions."},{"Start":"00:33.075 ","End":"00:35.220","Text":"That is, v of t,"},{"Start":"00:35.220 ","End":"00:42.465","Text":"and r of t. Let\u0027s start off by computing v of t. Now,"},{"Start":"00:42.465 ","End":"00:46.580","Text":"v of t, the velocity is the integral of the acceleration."},{"Start":"00:46.580 ","End":"00:50.779","Text":"So we want the integral of this,"},{"Start":"00:50.779 ","End":"00:57.859","Text":"I\u0027ll write it 3ti minus"},{"Start":"00:57.859 ","End":"01:08.130","Text":"4e^ minus t j plus 12t squared k dt."},{"Start":"01:08.230 ","End":"01:10.490","Text":"What is this equal to?"},{"Start":"01:10.490 ","End":"01:19.755","Text":"Well, this gives me 3/2 t squared i minus,"},{"Start":"01:19.755 ","End":"01:22.995","Text":"now e^ minus t,"},{"Start":"01:22.995 ","End":"01:26.250","Text":"its integral is minus e^ minus t,"},{"Start":"01:26.250 ","End":"01:33.300","Text":"so I\u0027ll make that a plus 4e^ minus t j."},{"Start":"01:33.300 ","End":"01:36.950","Text":"Here, I raised by 1 is 3 divided by 3."},{"Start":"01:36.950 ","End":"01:44.340","Text":"It\u0027s 4t cubed k. These arrows."},{"Start":"01:44.920 ","End":"01:49.010","Text":"That\u0027s not all because we need a constant of integration,"},{"Start":"01:49.010 ","End":"01:52.010","Text":"but the constant here is a vector constant."},{"Start":"01:52.010 ","End":"01:53.810","Text":"What I\u0027m going to do?"},{"Start":"01:53.810 ","End":"02:00.030","Text":"I\u0027m going to replace this by an ijk notation."},{"Start":"02:00.030 ","End":"02:09.750","Text":"Instead of the c, I\u0027m going to put c_1 i plus c_2 j plus c_3 k. Now,"},{"Start":"02:09.750 ","End":"02:11.900","Text":"how am I going to find c_1, c_2, c_3?"},{"Start":"02:11.900 ","End":"02:15.350","Text":"Well, I\u0027m going to use the fact that I have the initial velocity."},{"Start":"02:15.350 ","End":"02:18.290","Text":"If I substitute 0 on both sides,"},{"Start":"02:18.290 ","End":"02:25.040","Text":"v prime of 0 is j minus 3k."},{"Start":"02:25.040 ","End":"02:27.110","Text":"On the right-hand side,"},{"Start":"02:27.110 ","End":"02:33.430","Text":"I just have to substitute 0 for t. This part is 0."},{"Start":"02:33.430 ","End":"02:36.850","Text":"Here I get e^ minus 0 is 1,"},{"Start":"02:36.850 ","End":"02:38.205","Text":"so this is 4j."},{"Start":"02:38.205 ","End":"02:43.200","Text":"Here, well, t is 0,"},{"Start":"02:43.200 ","End":"02:44.910","Text":"so I get again plus 0."},{"Start":"02:44.910 ","End":"02:46.970","Text":"I\u0027m writing it just to show I haven\u0027t forgotten it."},{"Start":"02:46.970 ","End":"02:52.450","Text":"Plus c_1 i plus c_2"},{"Start":"02:52.450 ","End":"02:59.520","Text":"j plus c_3 k. Now what we do here is we compare i,"},{"Start":"02:59.520 ","End":"03:03.105","Text":"j and k. Each component has to be equal."},{"Start":"03:03.105 ","End":"03:05.405","Text":"Let\u0027s go for the i component."},{"Start":"03:05.405 ","End":"03:06.860","Text":"On the left-hand side,"},{"Start":"03:06.860 ","End":"03:13.580","Text":"I have 0, and on the right-hand side,"},{"Start":"03:13.580 ","End":"03:14.855","Text":"how many i do I have?"},{"Start":"03:14.855 ","End":"03:18.260","Text":"Just c_1. If I compare j,"},{"Start":"03:18.260 ","End":"03:20.105","Text":"on the left, I have 1."},{"Start":"03:20.105 ","End":"03:23.815","Text":"On the right, I have 4 from here,"},{"Start":"03:23.815 ","End":"03:26.355","Text":"plus c_2 from here."},{"Start":"03:26.355 ","End":"03:30.095","Text":"As for k, I have here minus 3,"},{"Start":"03:30.095 ","End":"03:34.950","Text":"and here I have c_3."},{"Start":"03:34.970 ","End":"03:40.205","Text":"Well, we already have c_1 and c_3 given to us,"},{"Start":"03:40.205 ","End":"03:43.600","Text":"so we really only need to work on this 1."},{"Start":"03:43.600 ","End":"03:49.829","Text":"I have that c_2 is equal to 1 minus 4,"},{"Start":"03:49.829 ","End":"03:53.100","Text":"which is minus 3."},{"Start":"03:53.100 ","End":"04:04.205","Text":"Now I can fully write v of t. Now that I know the constants is equal to,"},{"Start":"04:04.205 ","End":"04:09.905","Text":"let me just write down here what c_1 was equal to 0,"},{"Start":"04:09.905 ","End":"04:14.235","Text":"c_2 was minus 3,"},{"Start":"04:14.235 ","End":"04:19.215","Text":"and c_3 was also minus 3."},{"Start":"04:19.215 ","End":"04:21.760","Text":"Yeah, just a coincidence."},{"Start":"04:22.100 ","End":"04:25.190","Text":"Now if I do the computation,"},{"Start":"04:25.190 ","End":"04:27.585","Text":"I\u0027ve got to take the i terms,"},{"Start":"04:27.585 ","End":"04:33.970","Text":"3/2 t squared plus 0 is just 3/2 t squared i."},{"Start":"04:33.970 ","End":"04:42.039","Text":"Next the j, 4e^ minus t minus 3,"},{"Start":"04:42.039 ","End":"04:47.340","Text":"4e^ minus t minus 3 for j."},{"Start":"04:47.340 ","End":"04:51.175","Text":"Now, how many k do I get?"},{"Start":"04:51.175 ","End":"04:59.255","Text":"I have 4t cubed minus"},{"Start":"04:59.255 ","End":"05:09.085","Text":"3 k. That\u0027s the velocity function and that\u0027s taken care of this one."},{"Start":"05:09.085 ","End":"05:12.235","Text":"Now we need the position function."},{"Start":"05:12.235 ","End":"05:19.325","Text":"For the position, I need to take the integral of this."},{"Start":"05:19.325 ","End":"05:24.065","Text":"Let me just get some more space;"},{"Start":"05:24.065 ","End":"05:30.730","Text":"r of t will be the integral."},{"Start":"05:33.380 ","End":"05:39.010","Text":"You know what? I\u0027ll just write ditto, dt."},{"Start":"05:39.010 ","End":"05:42.710","Text":"I don\u0027t want to copy it all again, I have it right here."},{"Start":"05:46.370 ","End":"05:54.450","Text":"Let\u0027s see. This, I raised by 1 and divide by it,"},{"Start":"05:54.450 ","End":"05:59.380","Text":"so I get 1/2 t cubed i."},{"Start":"05:59.420 ","End":"06:03.660","Text":"Here I get, let\u0027s see how many j,"},{"Start":"06:03.660 ","End":"06:13.140","Text":"minus 4e^ minus t minus 3t j."},{"Start":"06:13.140 ","End":"06:16.605","Text":"For k, let\u0027s see, t^4 over 4,"},{"Start":"06:16.605 ","End":"06:27.810","Text":"that\u0027s just t^4 minus 3t k,"},{"Start":"06:27.810 ","End":"06:29.435","Text":"plus the constant."},{"Start":"06:29.435 ","End":"06:34.970","Text":"The constant again, I\u0027m going to write a c_1 i plus c_2 j"},{"Start":"06:34.970 ","End":"06:43.079","Text":"plus c_3 k. Now,"},{"Start":"06:43.079 ","End":"06:47.405","Text":"we need to plug in r of 0."},{"Start":"06:47.405 ","End":"06:49.130","Text":"Well, I\u0027ll just plug it in."},{"Start":"06:49.130 ","End":"06:52.655","Text":"Just indicate that I\u0027m letting t equal 0,"},{"Start":"06:52.655 ","End":"07:00.420","Text":"r of 0 is"},{"Start":"07:00.420 ","End":"07:04.690","Text":"minus 5i plus 2j minus 3k."},{"Start":"07:06.620 ","End":"07:16.630","Text":"Minus 5i plus 2k minus 3j."},{"Start":"07:16.820 ","End":"07:20.580","Text":"That\u0027s r of 0 from the left-hand side."},{"Start":"07:20.580 ","End":"07:23.570","Text":"Now, on the right-hand side, t equals 0."},{"Start":"07:23.570 ","End":"07:25.940","Text":"Well, this one is 0 for the"},{"Start":"07:25.940 ","End":"07:34.890","Text":"i. I really should write 0i, vector 0."},{"Start":"07:37.670 ","End":"07:40.170","Text":"Let\u0027s see, t is 0,"},{"Start":"07:40.170 ","End":"07:43.320","Text":"this one is 0, e^ minus 0 is 1."},{"Start":"07:43.320 ","End":"07:48.270","Text":"It\u0027s just minus 4j."},{"Start":"07:48.270 ","End":"07:54.730","Text":"For k, everything is 0, so 0k,"},{"Start":"07:54.920 ","End":"08:04.470","Text":"then plus c_1 i plus c_2 j plus c_3 k. Once again,"},{"Start":"08:04.470 ","End":"08:08.475","Text":"we compare i, j and k. For i,"},{"Start":"08:08.475 ","End":"08:16.540","Text":"we get minus 5 equals 0 plus c_1 is c_1."},{"Start":"08:19.100 ","End":"08:21.885","Text":"I wrote these backwards, sorry."},{"Start":"08:21.885 ","End":"08:26.110","Text":"This is j and this is k. For j,"},{"Start":"08:26.110 ","End":"08:30.080","Text":"we get that 2 is equal"},{"Start":"08:30.080 ","End":"08:37.880","Text":"to minus 4 plus c_2."},{"Start":"08:37.880 ","End":"08:46.740","Text":"For k, we get minus 3 equals 0 plus c_3."},{"Start":"08:46.970 ","End":"08:51.105","Text":"We have c_1, we have c_3,"},{"Start":"08:51.105 ","End":"08:58.800","Text":"and from here we get that c_2 is 2 minus minus 4 is 6."},{"Start":"08:58.800 ","End":"09:01.220","Text":"I\u0027ll just make a note of that,"},{"Start":"09:01.220 ","End":"09:07.300","Text":"that c_1 is minus 5,"},{"Start":"09:07.300 ","End":"09:14.505","Text":"c_2 is 6, and c_3 is minus 3."},{"Start":"09:14.505 ","End":"09:22.059","Text":"Now I can write the position function r of t equals."},{"Start":"09:22.059 ","End":"09:24.610","Text":"Let\u0027s see, how many i do I have?"},{"Start":"09:24.610 ","End":"09:30.770","Text":"I have 1/2 t cubed minus 5i."},{"Start":"09:32.130 ","End":"09:35.440","Text":"Next I need the j."},{"Start":"09:35.440 ","End":"09:40.450","Text":"For j, I have minus 4e^ minus t minus"},{"Start":"09:40.450 ","End":"09:47.065","Text":"3t plus 6 j."},{"Start":"09:47.065 ","End":"09:50.090","Text":"Finally, k;"},{"Start":"09:55.140 ","End":"10:03.810","Text":"t^4 minus 3t minus 3 k,"},{"Start":"10:03.810 ","End":"10:06.620","Text":"and so we have the answers."},{"Start":"10:06.620 ","End":"10:09.679","Text":"This is v, the velocity function,"},{"Start":"10:09.679 ","End":"10:13.010","Text":"and this is the position function;"},{"Start":"10:13.010 ","End":"10:16.770","Text":"both parts answered, and we\u0027re done."}],"ID":9724},{"Watched":false,"Name":"Exercise 10","Duration":"7m 18s","ChapterTopicVideoID":9847,"CourseChapterTopicPlaylistID":57484,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9847.jpeg","UploadDate":"2017-08-06T05:09:53.8770000","DurationForVideoObject":"PT7M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we\u0027re given the position of a body or at least"},{"Start":"00:03.570 ","End":"00:08.760","Text":"its position vector is given by the parametric vector function as follows,"},{"Start":"00:08.760 ","End":"00:11.175","Text":"where t is time."},{"Start":"00:11.175 ","End":"00:13.050","Text":"We have to determine"},{"Start":"00:13.050 ","End":"00:18.870","Text":"the tangential and normal components of its acceleration, the acceleration vector."},{"Start":"00:18.870 ","End":"00:21.670","Text":"It\u0027s really just a matter of using the formulas,"},{"Start":"00:21.670 ","End":"00:24.735","Text":"and I copied this formula from the tutorial."},{"Start":"00:24.735 ","End":"00:26.940","Text":"The T is the tangential,"},{"Start":"00:26.940 ","End":"00:28.695","Text":"N is for normal."},{"Start":"00:28.695 ","End":"00:31.995","Text":"We have to compute both of these quantities."},{"Start":"00:31.995 ","End":"00:35.010","Text":"Let\u0027s see, do it Lego style,"},{"Start":"00:35.010 ","End":"00:36.795","Text":"we\u0027ll get the building blocks."},{"Start":"00:36.795 ","End":"00:40.200","Text":"Obviously we need r prime of t,"},{"Start":"00:40.200 ","End":"00:44.565","Text":"then we\u0027ll need the magnitude of that and we\u0027ll need our double prime. You know what?"},{"Start":"00:44.565 ","End":"00:49.655","Text":"Let\u0027s just first compute the derivatives and then we\u0027ll see about magnitudes."},{"Start":"00:49.655 ","End":"00:56.210","Text":"r prime of t is equal to the derivative of this,"},{"Start":"00:56.210 ","End":"01:00.800","Text":"which is 3 cosine 3t."},{"Start":"01:00.800 ","End":"01:03.905","Text":"The derivative of minus cosine is plus sign,"},{"Start":"01:03.905 ","End":"01:07.595","Text":"so we have 3 sine 3t,"},{"Start":"01:07.595 ","End":"01:11.460","Text":"and we get 0 in the last component."},{"Start":"01:13.120 ","End":"01:16.745","Text":"You know what? Let\u0027s compute the magnitude already."},{"Start":"01:16.745 ","End":"01:24.320","Text":"Magnitude of r prime of t is the square root of this squared plus this squared."},{"Start":"01:24.320 ","End":"01:27.500","Text":"We\u0027ve seen this many times before."},{"Start":"01:27.500 ","End":"01:29.870","Text":"Cosine squared plus sine squared is 1,"},{"Start":"01:29.870 ","End":"01:31.970","Text":"so we end up with just 3."},{"Start":"01:31.970 ","End":"01:33.140","Text":"I\u0027ll leave you to check that."},{"Start":"01:33.140 ","End":"01:35.815","Text":"The square root of this squared plus this squared."},{"Start":"01:35.815 ","End":"01:40.545","Text":"Now we need r double prime of t,"},{"Start":"01:40.545 ","End":"01:43.050","Text":"just differentiate this again."},{"Start":"01:43.050 ","End":"01:52.100","Text":"We get what? Minus 9 sine 3t because we get 3 times 3 and cosine gets minus sine."},{"Start":"01:52.100 ","End":"01:59.490","Text":"Sine gives cosine, but again at the multiply by 3 and it\u0027s still 0 here."},{"Start":"01:59.490 ","End":"02:02.345","Text":"Now, what else do we need?"},{"Start":"02:02.345 ","End":"02:09.550","Text":"We need the dot product and the cross product of this with this."},{"Start":"02:09.550 ","End":"02:13.350","Text":"Let\u0027s go for the dot product, it\u0027s easier."},{"Start":"02:14.750 ","End":"02:18.345","Text":"I should put the arrows on."},{"Start":"02:18.345 ","End":"02:21.030","Text":"Supposed to do that."},{"Start":"02:21.030 ","End":"02:24.205","Text":"This dot this."},{"Start":"02:24.205 ","End":"02:28.760","Text":"I\u0027ll just write r prime cross r double prime."},{"Start":"02:28.760 ","End":"02:32.900","Text":"This is equal to the dot product of"},{"Start":"02:32.900 ","End":"02:38.860","Text":"2 vectors is this times this plus this times this plus this times this."},{"Start":"02:38.860 ","End":"02:44.580","Text":"We get 3 times minus 9 minus"},{"Start":"02:44.580 ","End":"02:50.730","Text":"27 cosine 3t, sine 3t."},{"Start":"02:50.730 ","End":"02:53.130","Text":"Then this times this,"},{"Start":"02:53.130 ","End":"03:01.330","Text":"so plus 27 sine 3t."},{"Start":"03:01.460 ","End":"03:08.530","Text":"Cosine 3t plus 0 times 0 is 0."},{"Start":"03:09.500 ","End":"03:13.340","Text":"Well, this works out nicely because"},{"Start":"03:13.340 ","End":"03:18.740","Text":"cosine 3t sine 3t is the same as sine 3t cosine 3t in as a minus and a plus."},{"Start":"03:18.740 ","End":"03:21.325","Text":"This is just equal to 0."},{"Start":"03:21.325 ","End":"03:24.800","Text":"Good. Next we need"},{"Start":"03:24.800 ","End":"03:32.599","Text":"the cross product of these 2 vectors and a double prime."},{"Start":"03:32.599 ","End":"03:34.850","Text":"A cross product is a vector,"},{"Start":"03:34.850 ","End":"03:37.730","Text":"not a scalar so we need to do the determinant."},{"Start":"03:37.730 ","End":"03:40.430","Text":"We always put i, j,"},{"Start":"03:40.430 ","End":"03:45.360","Text":"k vector, vector, vector."},{"Start":"03:45.360 ","End":"03:47.250","Text":"Then let\u0027s see."},{"Start":"03:47.250 ","End":"03:51.300","Text":"I need make this bigger."},{"Start":"03:51.300 ","End":"03:54.850","Text":"I don\u0027t why don\u0027t I just stretch it a bit."},{"Start":"03:55.640 ","End":"04:01.095","Text":"3 cosine 3t. I\u0027m copying from here,"},{"Start":"04:01.095 ","End":"04:06.675","Text":"3 sine 3t 0."},{"Start":"04:06.675 ","End":"04:13.590","Text":"Then this 1, minus 9 sine 3t."},{"Start":"04:13.590 ","End":"04:17.850","Text":"I forgot 3 here."},{"Start":"04:17.850 ","End":"04:25.860","Text":"9 cosine 3t and 0."},{"Start":"04:25.860 ","End":"04:30.304","Text":"Obviously I\u0027m going to expand this by the 1/3 column,"},{"Start":"04:30.304 ","End":"04:32.090","Text":"because it\u0027s got 2 zeros in it."},{"Start":"04:32.090 ","End":"04:34.895","Text":"I just need k times this."},{"Start":"04:34.895 ","End":"04:44.530","Text":"I get the determinant of 3 cosine 3t,3 sine 3t"},{"Start":"04:47.090 ","End":"04:54.495","Text":"minus 9 sine 3t 9 cosine 3t"},{"Start":"04:54.495 ","End":"05:00.360","Text":"k. This is 3"},{"Start":"05:00.360 ","End":"05:05.445","Text":"times 9 is 27 cosine squared 3t."},{"Start":"05:05.445 ","End":"05:08.220","Text":"This times this is minus 27,"},{"Start":"05:08.220 ","End":"05:11.200","Text":"sine squared 3t, but it\u0027s subtracted,"},{"Start":"05:11.200 ","End":"05:18.290","Text":"so it\u0027s plus 27 sine squared 3t k. Now,"},{"Start":"05:18.290 ","End":"05:20.690","Text":"cosine squared plus sine squared is 1,"},{"Start":"05:20.690 ","End":"05:25.740","Text":"so this is just equal to 27k."},{"Start":"05:26.120 ","End":"05:29.610","Text":"Well, let me highlight some things."},{"Start":"05:29.610 ","End":"05:34.970","Text":"In both formulas, I need this quantity and this quantity,"},{"Start":"05:34.970 ","End":"05:37.140","Text":"I have it here."},{"Start":"05:45.680 ","End":"05:53.610","Text":"This quantity is what we have, where is it?"},{"Start":"05:53.610 ","End":"05:56.890","Text":"Is here, it\u0027s just 0."},{"Start":"06:00.110 ","End":"06:03.320","Text":"Let\u0027s quickly do that. The magnitude of"},{"Start":"06:03.320 ","End":"06:11.130","Text":"r prime cross r double prime is just the magnitude of 27k,"},{"Start":"06:13.010 ","End":"06:17.535","Text":"and obviously this is just equal to 27."},{"Start":"06:17.535 ","End":"06:21.750","Text":"I can highlight this."},{"Start":"06:21.750 ","End":"06:24.450","Text":"Now I need to do the computations."},{"Start":"06:24.450 ","End":"06:28.800","Text":"Let\u0027s see where I have space to do that."},{"Start":"06:28.800 ","End":"06:32.145","Text":"We can just about see."},{"Start":"06:32.145 ","End":"06:39.755","Text":"The acceleration tangential component is equal to green over yellow."},{"Start":"06:39.755 ","End":"06:47.640","Text":"It\u0027s 0 over 3, which equals 0."},{"Start":"06:47.640 ","End":"06:55.580","Text":"The normal component of the acceleration is this light blue over yellow,"},{"Start":"06:55.580 ","End":"07:05.640","Text":"which is 27 over 3, which equals 9."},{"Start":"07:07.040 ","End":"07:12.510","Text":"That\u0027s it. Tangential 0, normal is 9."},{"Start":"07:12.510 ","End":"07:15.740","Text":"I don\u0027t know why I highlighted that."},{"Start":"07:15.740 ","End":"07:18.150","Text":"Anyway, we\u0027re done."}],"ID":9725},{"Watched":false,"Name":"Exercise 11","Duration":"7m 28s","ChapterTopicVideoID":9848,"CourseChapterTopicPlaylistID":57484,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9848.jpeg","UploadDate":"2017-08-06T05:10:15.5430000","DurationForVideoObject":"PT7M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.590 ","End":"00:04.455","Text":"This exercise is very straightforward."},{"Start":"00:04.455 ","End":"00:07.320","Text":"Of course, there\u0027s a story wrapped around it about"},{"Start":"00:07.320 ","End":"00:10.695","Text":"the alien spaceship flying through the galaxy,"},{"Start":"00:10.695 ","End":"00:14.565","Text":"but really, we just want the mathematical part."},{"Start":"00:14.565 ","End":"00:18.210","Text":"We\u0027re given the position vector as follows;"},{"Start":"00:18.210 ","End":"00:21.450","Text":"it\u0027s in 3 dimensions, x, y, and z,"},{"Start":"00:21.450 ","End":"00:23.729","Text":"and we want to compute"},{"Start":"00:23.729 ","End":"00:29.580","Text":"its tangential and normal acceleration at the time when t is equal to Pi."},{"Start":"00:29.580 ","End":"00:32.340","Text":"Of course, I\u0027ll need to remind you of the formulas."},{"Start":"00:32.340 ","End":"00:35.400","Text":"I brought them with me from the tutorial,"},{"Start":"00:35.400 ","End":"00:37.875","Text":"and here are the formulas,"},{"Start":"00:37.875 ","End":"00:41.050","Text":"T for tangential, N for normal."},{"Start":"00:41.050 ","End":"00:46.890","Text":"Other than that, it all depends on this function r,"},{"Start":"00:46.890 ","End":"00:49.785","Text":"the position vector function."},{"Start":"00:49.785 ","End":"00:52.710","Text":"Let\u0027s do all the computations."},{"Start":"00:52.710 ","End":"00:57.255","Text":"I see we need r prime and r double prime,"},{"Start":"00:57.255 ","End":"00:59.340","Text":"and then we\u0027ll have to start doing dot product,"},{"Start":"00:59.340 ","End":"01:02.140","Text":"cross product, and taking magnitude."},{"Start":"01:02.140 ","End":"01:04.550","Text":"Let\u0027s begin. Here we have r,"},{"Start":"01:04.550 ","End":"01:09.515","Text":"so r prime is just differentiating each component."},{"Start":"01:09.515 ","End":"01:12.725","Text":"It\u0027s almost cosine 2t,"},{"Start":"01:12.725 ","End":"01:15.620","Text":"but the inner derivative is 2, so that\u0027s that."},{"Start":"01:15.620 ","End":"01:18.950","Text":"3 gives me 0, and cosine 2t,"},{"Start":"01:18.950 ","End":"01:22.665","Text":"you start off with sine 2t,"},{"Start":"01:22.665 ","End":"01:25.190","Text":"the derivative of cosine is minus sine."},{"Start":"01:25.190 ","End":"01:29.490","Text":"There is also a 2 so it\u0027s minus 2 sine 2t."},{"Start":"01:32.240 ","End":"01:36.890","Text":"I\u0027m leaving space here for the magnitude of this."},{"Start":"01:36.890 ","End":"01:40.700","Text":"Let\u0027s continue to the second derivative."},{"Start":"01:40.700 ","End":"01:42.925","Text":"This is equal to,"},{"Start":"01:42.925 ","End":"01:44.280","Text":"from here is similar."},{"Start":"01:44.280 ","End":"01:47.585","Text":"Cosine is minus sine and we get an extra 2,"},{"Start":"01:47.585 ","End":"01:51.545","Text":"so it\u0027s minus 4 sine 2t."},{"Start":"01:51.545 ","End":"01:54.650","Text":"The 0 stays 0,"},{"Start":"01:54.650 ","End":"02:03.680","Text":"and here we have minus 4 cosine 2t."},{"Start":"02:03.680 ","End":"02:09.630","Text":"Let\u0027s compute the magnitude of this."},{"Start":"02:09.630 ","End":"02:16.910","Text":"The magnitude of r prime of t"},{"Start":"02:16.910 ","End":"02:24.700","Text":"is equal to the square root of this squared plus this squared plus this squared."},{"Start":"02:24.700 ","End":"02:33.345","Text":"4 cosine squared 2t plus 4 sine squared 2t,"},{"Start":"02:33.345 ","End":"02:36.150","Text":"I didn\u0027t write the plus 0."},{"Start":"02:36.150 ","End":"02:39.120","Text":"Cosine squared plus sine squared is 1,"},{"Start":"02:39.120 ","End":"02:42.760","Text":"square root of 4 is 2."},{"Start":"02:43.100 ","End":"02:46.750","Text":"We need the dot product."},{"Start":"02:49.940 ","End":"02:58.020","Text":"R first derivative dot product with r second derivative,"},{"Start":"02:58.020 ","End":"03:03.450","Text":"that would be this with this dot product."},{"Start":"03:03.450 ","End":"03:08.460","Text":"We just multiply component-wise and add. What do we get?"},{"Start":"03:08.460 ","End":"03:11.985","Text":"This with this gives us minus"},{"Start":"03:11.985 ","End":"03:20.925","Text":"8 sine 2t cosine 2t,"},{"Start":"03:20.925 ","End":"03:23.730","Text":"then 0 with 0 is 0."},{"Start":"03:23.730 ","End":"03:29.505","Text":"Then this with this is plus"},{"Start":"03:29.505 ","End":"03:37.420","Text":"8 sine 2t cosine 2t."},{"Start":"03:38.510 ","End":"03:44.040","Text":"It cancels out, this thing is 0."},{"Start":"03:44.040 ","End":"03:50.110","Text":"Already I can say that the tangential component of the acceleration is,"},{"Start":"03:50.110 ","End":"03:52.085","Text":"this with this is 0,"},{"Start":"03:52.085 ","End":"03:55.620","Text":"it doesn\u0027t matter when dividing 0 by anything,"},{"Start":"03:55.620 ","End":"03:59.565","Text":"but it\u0027s 0 over 2 it\u0027s still 0."},{"Start":"03:59.565 ","End":"04:05.220","Text":"Now we need a cross product for the normal."},{"Start":"04:05.220 ","End":"04:10.800","Text":"The cross product r of t,"},{"Start":"04:10.800 ","End":"04:13.385","Text":"with this instead of dot we need a cross,"},{"Start":"04:13.385 ","End":"04:17.660","Text":"that\u0027s going to give us a vector double prime of t,"},{"Start":"04:17.660 ","End":"04:22.850","Text":"and I\u0027ll use the determinant method."},{"Start":"04:22.850 ","End":"04:24.630","Text":"We\u0027re going to I put here i,"},{"Start":"04:24.630 ","End":"04:27.629","Text":"j, k. Remember the ijk notation,"},{"Start":"04:27.629 ","End":"04:30.870","Text":"instead of on each component with i, j,"},{"Start":"04:30.870 ","End":"04:34.865","Text":"k this is the x-component, y-component, z-component."},{"Start":"04:34.865 ","End":"04:37.850","Text":"Now we write each of these. Let\u0027s see."},{"Start":"04:37.850 ","End":"04:40.730","Text":"The first 1 is r prime,"},{"Start":"04:40.730 ","End":"04:45.330","Text":"which is 2 cosine 2t."},{"Start":"04:45.330 ","End":"04:47.520","Text":"Oh, I didn\u0027t plan this right,"},{"Start":"04:47.520 ","End":"04:49.620","Text":"I\u0027ll fix it in a more moment."},{"Start":"04:49.930 ","End":"04:58.110","Text":"Then 0, then minus 2 sine 2t."},{"Start":"04:59.560 ","End":"05:03.060","Text":"I\u0027ll just adjust the size a bit."},{"Start":"05:03.060 ","End":"05:05.070","Text":"Now the r double prime,"},{"Start":"05:05.070 ","End":"05:10.380","Text":"so here minus 4 sine 2t, once again,"},{"Start":"05:10.380 ","End":"05:16.780","Text":"0 and minus 4 cosine 2t."},{"Start":"05:17.120 ","End":"05:20.450","Text":"We\u0027ll pick out the i component separately, the j,"},{"Start":"05:20.450 ","End":"05:23.145","Text":"then the k, remember there\u0027s a checkerboard thing."},{"Start":"05:23.145 ","End":"05:25.990","Text":"That there is a plus, minus, plus,"},{"Start":"05:25.990 ","End":"05:33.135","Text":"and then for each 1 we do the minor by crossing out row and column."},{"Start":"05:33.135 ","End":"05:40.495","Text":"The i component is gotten by the determinant of this 2 by 2,"},{"Start":"05:40.495 ","End":"05:44.980","Text":"so this means that that\u0027s 0 for the i."},{"Start":"05:44.980 ","End":"05:51.380","Text":"Now let\u0027s do the j. This time we cross out the j row and column."},{"Start":"05:51.380 ","End":"05:55.625","Text":"We\u0027ve got the 2 by 2 determinant of this,"},{"Start":"05:55.625 ","End":"05:57.670","Text":"and it\u0027s with a minus."},{"Start":"05:57.670 ","End":"06:00.785","Text":"What we do is this times this minus this times this,"},{"Start":"06:00.785 ","End":"06:07.530","Text":"and we\u0027ve got minus 8 cosine squared 2t,"},{"Start":"06:07.530 ","End":"06:09.630","Text":"and then minus, minus,"},{"Start":"06:09.630 ","End":"06:14.080","Text":"minus 8 sine squared 2t."},{"Start":"06:14.260 ","End":"06:17.440","Text":"Because sine squared plus cosine squared is 1,"},{"Start":"06:17.440 ","End":"06:20.635","Text":"we just get the minus 8, basically."},{"Start":"06:20.635 ","End":"06:24.150","Text":"Just wrote that for reference that this what I used,"},{"Start":"06:24.150 ","End":"06:25.710","Text":"or any angle that works."},{"Start":"06:25.710 ","End":"06:28.365","Text":"Now the k component."},{"Start":"06:28.365 ","End":"06:31.095","Text":"With this and with this."},{"Start":"06:31.095 ","End":"06:33.615","Text":"Again, we\u0027ve got 2 zeros,"},{"Start":"06:33.615 ","End":"06:37.620","Text":"so that\u0027s just a 0 there also,"},{"Start":"06:37.620 ","End":"06:41.540","Text":"but what we need is the magnitude of that."},{"Start":"06:41.540 ","End":"06:47.040","Text":"I copy that, put bars around it."},{"Start":"06:48.230 ","End":"06:51.990","Text":"Obviously, the magnitude of this is just 8."},{"Start":"06:51.990 ","End":"06:55.310","Text":"You could take 0 squared plus this squared plus this squared,"},{"Start":"06:55.310 ","End":"06:58.740","Text":"take the square root, obviously, it\u0027s 8."},{"Start":"06:59.180 ","End":"07:06.440","Text":"We can now say that the normal component of acceleration,"},{"Start":"07:06.440 ","End":"07:09.815","Text":"we have the numerator now which is 8,"},{"Start":"07:09.815 ","End":"07:13.475","Text":"we have the denominator which is 2,"},{"Start":"07:13.475 ","End":"07:16.410","Text":"and this is equal to 4."},{"Start":"07:17.360 ","End":"07:22.819","Text":"I guess that answers the question tangential is the constant 0,"},{"Start":"07:22.819 ","End":"07:28.140","Text":"and the normal acceleration is the constant 4. We\u0027re done."}],"ID":9726},{"Watched":false,"Name":"Exercise 12","Duration":"15m 17s","ChapterTopicVideoID":9849,"CourseChapterTopicPlaylistID":57484,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9849.jpeg","UploadDate":"2017-08-06T05:11:25.4170000","DurationForVideoObject":"PT15M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.914","Text":"This exercise boy, it\u0027s a long word problem."},{"Start":"00:03.914 ","End":"00:07.305","Text":"I won\u0027t read it all and let you pause the clip and read it."},{"Start":"00:07.305 ","End":"00:10.335","Text":"I\u0027ll just go over the main points."},{"Start":"00:10.335 ","End":"00:15.390","Text":"Essentially it involves the flight of a football,"},{"Start":"00:15.390 ","End":"00:21.855","Text":"in this case from Tom Brady to a receiver and a little sketch will help."},{"Start":"00:21.855 ","End":"00:29.740","Text":"Let\u0027s see, let\u0027s do a y-axis and an x-axis,"},{"Start":"00:30.740 ","End":"00:33.750","Text":"x and y, we\u0027ll call them."},{"Start":"00:33.750 ","End":"00:37.995","Text":"Let say this is the position where"},{"Start":"00:37.995 ","End":"00:44.030","Text":"Tom is and somewhere here is where the receiver is."},{"Start":"00:44.030 ","End":"00:45.110","Text":"We said they have the same height,"},{"Start":"00:45.110 ","End":"00:47.090","Text":"let\u0027s call that height 0."},{"Start":"00:47.090 ","End":"00:50.050","Text":"He\u0027s throwing at a certain angle."},{"Start":"00:50.050 ","End":"00:54.290","Text":"It\u0027s going to go up in the air and come down to the receiver."},{"Start":"00:54.290 ","End":"00:59.045","Text":"Just label this as 0 and this is 40. It\u0027s easier."},{"Start":"00:59.045 ","End":"01:01.790","Text":"Now I draw a tangent line here,"},{"Start":"01:01.790 ","End":"01:09.360","Text":"and this also represents the direction of the initial throw."},{"Start":"01:09.360 ","End":"01:11.330","Text":"If there wasn\u0027t any gravity,"},{"Start":"01:11.330 ","End":"01:13.730","Text":"that would be a no air resistance,"},{"Start":"01:13.730 ","End":"01:17.920","Text":"then that would be the path that the ball would take."},{"Start":"01:17.920 ","End":"01:24.095","Text":"We know the initial speed is 30 yards per second."},{"Start":"01:24.095 ","End":"01:27.815","Text":"Yards per second."},{"Start":"01:27.815 ","End":"01:31.050","Text":"Here\u0027s, 40 yards."},{"Start":"01:32.540 ","End":"01:35.015","Text":"Well, we don\u0027t know."},{"Start":"01:35.015 ","End":"01:41.125","Text":"What we want to know is this initial angle, this, this."},{"Start":"01:41.125 ","End":"01:44.610","Text":"Let\u0027s call the angle Theta."},{"Start":"01:44.610 ","End":"01:46.099","Text":"That\u0027s our unknown."},{"Start":"01:46.099 ","End":"01:48.680","Text":"We\u0027ll also given another piece of information,"},{"Start":"01:48.680 ","End":"01:52.040","Text":"g, that\u0027s the acceleration due to gravity."},{"Start":"01:52.040 ","End":"01:53.690","Text":"Gravity is downwards."},{"Start":"01:53.690 ","End":"01:54.919","Text":"When it\u0027s an acceleration,"},{"Start":"01:54.919 ","End":"01:56.030","Text":"we put a double arrow."},{"Start":"01:56.030 ","End":"01:59.170","Text":"When it\u0027s just a velocity, a single arrow."},{"Start":"01:59.170 ","End":"02:02.840","Text":"This gravity acceleration is"},{"Start":"02:02.840 ","End":"02:13.145","Text":"11.25 units or yards per second, per second or per second squared."},{"Start":"02:13.145 ","End":"02:16.405","Text":"That\u0027s basically it."},{"Start":"02:16.405 ","End":"02:24.155","Text":"The basic setup for this kind of question in general with speed and acceleration,"},{"Start":"02:24.155 ","End":"02:26.240","Text":"is that we have 3 vectors."},{"Start":"02:26.240 ","End":"02:30.260","Text":"You have a position vector, r of t,"},{"Start":"02:30.260 ","End":"02:33.050","Text":"which describes the position at"},{"Start":"02:33.050 ","End":"02:37.910","Text":"any given moment t. We\u0027re going to use 2-dimensional vectors,"},{"Start":"02:37.910 ","End":"02:40.070","Text":"of course, they\u0027re just x and y."},{"Start":"02:40.070 ","End":"02:45.575","Text":"We also have the velocity vector,"},{"Start":"02:45.575 ","End":"02:49.250","Text":"which is the derivative of the position vector."},{"Start":"02:49.250 ","End":"02:53.240","Text":"The third 1 is the acceleration vector,"},{"Start":"02:53.240 ","End":"02:59.805","Text":"which is the derivative of the velocity vector."},{"Start":"02:59.805 ","End":"03:04.650","Text":"Since we said we\u0027re working in 2-dimensions, r of t,"},{"Start":"03:04.650 ","End":"03:10.010","Text":"we\u0027ll write it as x of t,"},{"Start":"03:10.010 ","End":"03:16.775","Text":"y of t. I\u0027m going to write all these in more mathematical terms."},{"Start":"03:16.775 ","End":"03:20.090","Text":"Let\u0027s say we\u0027ll start with the gravity thing."},{"Start":"03:20.090 ","End":"03:28.940","Text":"What that says is that the acceleration vector is 11.25 downwards,"},{"Start":"03:28.940 ","End":"03:38.855","Text":"which means that the x component is 0 and the y component is minus 11.25."},{"Start":"03:38.855 ","End":"03:44.270","Text":"That\u0027s a constant, doesn\u0027t depend on t. That\u0027s 1 thing we know."},{"Start":"03:44.270 ","End":"03:48.215","Text":"The other thing we know is the initial velocity."},{"Start":"03:48.215 ","End":"03:52.295","Text":"We know that v at time 0,"},{"Start":"03:52.295 ","End":"03:55.040","Text":"it\u0027s 30 yards per second,"},{"Start":"03:55.040 ","End":"03:59.810","Text":"but in a direction given by the angle Theta."},{"Start":"03:59.810 ","End":"04:02.120","Text":"So if we break it up into components,"},{"Start":"04:02.120 ","End":"04:06.255","Text":"it\u0027s going to be equal using the trigonometry."},{"Start":"04:06.255 ","End":"04:08.275","Text":"Maybe a little sketch."},{"Start":"04:08.275 ","End":"04:13.405","Text":"See we have here is a right angle triangle."},{"Start":"04:13.405 ","End":"04:18.805","Text":"This is 30 and this is 90 degrees,"},{"Start":"04:18.805 ","End":"04:20.340","Text":"and this is Theta."},{"Start":"04:20.340 ","End":"04:23.245","Text":"The question is, what is this and what is this?"},{"Start":"04:23.245 ","End":"04:25.165","Text":"That\u0027s the y and that\u0027s the x."},{"Start":"04:25.165 ","End":"04:28.140","Text":"So the x part is going to be"},{"Start":"04:28.140 ","End":"04:36.275","Text":"30 cosine Theta and the y part is going to be 30 sine Theta."},{"Start":"04:36.275 ","End":"04:41.560","Text":"That\u0027s how we break it up into components horizontal and vertical."},{"Start":"04:42.910 ","End":"04:50.690","Text":"Another initial condition that we have is that the position at time 0 is the origin."},{"Start":"04:50.690 ","End":"04:57.990","Text":"Let\u0027s say we start counting time from the moment he throws the football."},{"Start":"04:58.060 ","End":"05:05.315","Text":"Now let\u0027s see what we can do with the main trick is the integration."},{"Start":"05:05.315 ","End":"05:11.555","Text":"Because if v is a derivative of r and a is a derivative of v,"},{"Start":"05:11.555 ","End":"05:14.855","Text":"we can work our way backwards and say that v is the integral of a and"},{"Start":"05:14.855 ","End":"05:19.385","Text":"r is the integral of v. Let\u0027s start doing that."},{"Start":"05:19.385 ","End":"05:22.850","Text":"Later we\u0027ll get into a condition on Theta"},{"Start":"05:22.850 ","End":"05:25.490","Text":"because there\u0027s 1 piece of information that we haven\u0027t used,"},{"Start":"05:25.490 ","End":"05:29.905","Text":"and that\u0027s the 40 yard piece of information that will come in later."},{"Start":"05:29.905 ","End":"05:32.270","Text":"Let\u0027s start with the acceleration,"},{"Start":"05:32.270 ","End":"05:34.920","Text":"integrate it to get velocity."},{"Start":"05:35.200 ","End":"05:45.125","Text":"Let\u0027s make a note, let\u0027s just say that v is the integral of a, just in shorthand."},{"Start":"05:45.125 ","End":"05:54.340","Text":"Later on we\u0027ll also say that r is the integral of v. It\u0027s just the reverse of these 2."},{"Start":"05:54.340 ","End":"06:02.915","Text":"V of t is"},{"Start":"06:02.915 ","End":"06:09.250","Text":"the integral of the acceleration dt,"},{"Start":"06:09.740 ","End":"06:19.170","Text":"which is the integral of a constant 0 minus 11.25 dt,"},{"Start":"06:19.170 ","End":"06:26.165","Text":"which is equal to the integral of 0 is nothing."},{"Start":"06:26.165 ","End":"06:29.000","Text":"But we have to add the constant."},{"Start":"06:29.000 ","End":"06:30.770","Text":"We actually have to add the vector constant,"},{"Start":"06:30.770 ","End":"06:33.215","Text":"which is C_1, C_2."},{"Start":"06:33.215 ","End":"06:37.215","Text":"We\u0027ll say this is equal to constant 1."},{"Start":"06:37.215 ","End":"06:44.700","Text":"Here the integral of this is minus 11.25t plus another constant."},{"Start":"06:44.840 ","End":"06:47.675","Text":"Now we have a condition."},{"Start":"06:47.675 ","End":"06:52.145","Text":"We\u0027ll apply this, that at time 0,"},{"Start":"06:52.145 ","End":"06:59.360","Text":"v of 0 is this."},{"Start":"06:59.360 ","End":"07:01.010","Text":"On the 1 hand,"},{"Start":"07:01.010 ","End":"07:09.380","Text":"it\u0027s 30 cosine Theta, 30 sine Theta."},{"Start":"07:09.380 ","End":"07:11.075","Text":"But on the other hand,"},{"Start":"07:11.075 ","End":"07:12.830","Text":"it\u0027s also equal to,"},{"Start":"07:12.830 ","End":"07:15.095","Text":"if I plug in 0 here,"},{"Start":"07:15.095 ","End":"07:19.620","Text":"I\u0027ll get C_1, C_2."},{"Start":"07:19.620 ","End":"07:27.070","Text":"That means that C_1 is this and C_2 is this."},{"Start":"07:27.070 ","End":"07:31.450","Text":"Now I can totally get what v of t is,"},{"Start":"07:31.450 ","End":"07:34.280","Text":"because if I plug in to here,"},{"Start":"07:36.150 ","End":"07:39.955","Text":"C_1 actually, then it comes out is just this,"},{"Start":"07:39.955 ","End":"07:42.340","Text":"and C_2 is just this."},{"Start":"07:42.340 ","End":"07:49.610","Text":"We get in general that the velocity function without constants is equal to"},{"Start":"07:50.010 ","End":"08:00.550","Text":"30 cosine Theta and then minus 11.25t plus C_2,"},{"Start":"08:00.550 ","End":"08:05.215","Text":"which is 30 sine Theta."},{"Start":"08:05.215 ","End":"08:06.729","Text":"That\u0027s the velocity."},{"Start":"08:06.729 ","End":"08:11.050","Text":"Now another integration will give us the position with a constant,"},{"Start":"08:11.050 ","End":"08:13.690","Text":"but we have an initial condition and we\u0027ll find the constants"},{"Start":"08:13.690 ","End":"08:16.990","Text":"and that\u0027s how much of calculus works."},{"Start":"08:16.990 ","End":"08:22.645","Text":"r of t, is going to be the integral of"},{"Start":"08:22.645 ","End":"08:28.735","Text":"v of t, dt and this will equal,"},{"Start":"08:28.735 ","End":"08:36.015","Text":"the integral of this will be 30 cosine Theta t,"},{"Start":"08:36.015 ","End":"08:37.960","Text":"maybe put brackets here."},{"Start":"08:37.960 ","End":"08:46.915","Text":"Then here I need to make it t squared and divide by 2. Let\u0027s see."},{"Start":"08:46.915 ","End":"08:54.880","Text":"This divided by 2 is minus 5.625t squared."},{"Start":"08:54.880 ","End":"08:58.435","Text":"Integral of t is t squared over 2 I divided this by 2."},{"Start":"08:58.435 ","End":"09:03.910","Text":"Then here plus this thing times t, I mean that\u0027s a constant,"},{"Start":"09:03.910 ","End":"09:13.885","Text":"Theta\u0027s not the variable sine Theta t. Now we\u0027ll make a condition on r of 0."},{"Start":"09:13.885 ","End":"09:16.074","Text":"On the 1 hand,"},{"Start":"09:16.074 ","End":"09:18.790","Text":"we now at 0, 0."},{"Start":"09:18.790 ","End":"09:20.710","Text":"On the other hand,"},{"Start":"09:20.710 ","End":"09:24.980","Text":"it\u0027s equal to this when I plug in t equals 0,"},{"Start":"09:25.560 ","End":"09:30.050","Text":"I forgot to put the constants in, didn\u0027t I?"},{"Start":"09:30.050 ","End":"09:32.970","Text":"I can use C_1 and C_2 again,"},{"Start":"09:32.970 ","End":"09:37.245","Text":"don\u0027t matter no confusion, C2."},{"Start":"09:37.245 ","End":"09:42.600","Text":"Sorry about that and now if we plug in t equals 0,"},{"Start":"09:42.600 ","End":"09:45.340","Text":"we get that t is 0,"},{"Start":"09:45.340 ","End":"09:48.505","Text":"then this is just equal to C_1,"},{"Start":"09:48.505 ","End":"09:51.460","Text":"t is 0, t is 0 C_2."},{"Start":"09:51.460 ","End":"09:55.780","Text":"Once again, we find C_1 and C_2 this time each of them is 0,"},{"Start":"09:55.780 ","End":"09:58.165","Text":"C_1 is 0, C_2 is 0,"},{"Start":"09:58.165 ","End":"10:00.820","Text":"which means that I can now go back to"},{"Start":"10:00.820 ","End":"10:04.285","Text":"r of t what I actually wrote was correct, the constants were 0,"},{"Start":"10:04.285 ","End":"10:12.370","Text":"that r of t in general is equal to the x-component is"},{"Start":"10:12.370 ","End":"10:20.830","Text":"30 cosine Theta t. The y component is minus 5.625t"},{"Start":"10:20.830 ","End":"10:27.860","Text":"squared plus 30 sine Theta t."},{"Start":"10:28.380 ","End":"10:33.160","Text":"Now we haven\u0027t used the 40 yards."},{"Start":"10:33.160 ","End":"10:36.910","Text":"Let me scroll back up to show you what I mean."},{"Start":"10:36.910 ","End":"10:38.620","Text":"We know that when we throw it,"},{"Start":"10:38.620 ","End":"10:41.035","Text":"it goes 40 yards."},{"Start":"10:41.035 ","End":"10:46.870","Text":"What I\u0027m going to do is see for which value of t I"},{"Start":"10:46.870 ","End":"10:52.900","Text":"get that y equals 0 that will actually be 2 places where y is 0 at the very beginning,"},{"Start":"10:52.900 ","End":"10:58.270","Text":"but we know that and there\u0027ll be another solution of t where y is 0."},{"Start":"10:58.270 ","End":"11:01.390","Text":"I\u0027ll take the other solution of t, not the 0,"},{"Start":"11:01.390 ","End":"11:05.860","Text":"and then I\u0027ll plug it into x and set that equal to 40."},{"Start":"11:05.860 ","End":"11:12.430","Text":"What I\u0027m basically saying is that when y is 0,"},{"Start":"11:12.430 ","End":"11:14.890","Text":"then x is 0 or 40,"},{"Start":"11:14.890 ","End":"11:18.380","Text":"let me just go back down and write that."},{"Start":"11:18.720 ","End":"11:24.805","Text":"I\u0027II use different color. When y of t equals 0,"},{"Start":"11:24.805 ","End":"11:34.750","Text":"then we have that x of t is equal to 0, or 40."},{"Start":"11:34.750 ","End":"11:36.820","Text":"But that\u0027s the interesting 1."},{"Start":"11:36.820 ","End":"11:44.290","Text":"From this, I\u0027ll be able to get the t when x is 40."},{"Start":"11:44.450 ","End":"11:51.640","Text":"No, I didn\u0027t say that right. I get the expression in terms of Theta,"},{"Start":"11:51.640 ","End":"11:54.040","Text":"which would be this expression,"},{"Start":"11:54.040 ","End":"11:56.364","Text":"that\u0027s the x and that\u0027s the y."},{"Start":"11:56.364 ","End":"11:58.585","Text":"If y of t is 0,"},{"Start":"11:58.585 ","End":"12:02.395","Text":"then we get that minus"},{"Start":"12:02.395 ","End":"12:12.565","Text":"5.625 t squared plus 30 sine Theta t,"},{"Start":"12:12.565 ","End":"12:14.560","Text":"brackets are wrong here,"},{"Start":"12:14.560 ","End":"12:16.810","Text":"that this is equal to 0."},{"Start":"12:16.810 ","End":"12:20.200","Text":"Now look, we said we don\u0027t want the t equals 0 solution."},{"Start":"12:20.200 ","End":"12:22.090","Text":"That\u0027s the starting point."},{"Start":"12:22.090 ","End":"12:26.020","Text":"I can divide both sides by t because I don\u0027t want that solution."},{"Start":"12:26.020 ","End":"12:33.055","Text":"I want the non-zero solution that gives me this and now I can solve this for t,"},{"Start":"12:33.055 ","End":"12:35.740","Text":"and t comes out to be,"},{"Start":"12:35.740 ","End":"12:41.290","Text":"let\u0027s see, 30 sine Theta."},{"Start":"12:41.290 ","End":"12:42.935","Text":"Put this on the other side and,"},{"Start":"12:42.935 ","End":"12:46.840","Text":"they\u0027re both minus, divide by 5.625."},{"Start":"12:48.210 ","End":"12:57.175","Text":"Now from this, I can get that x of t is equal to this,"},{"Start":"12:57.175 ","End":"13:03.565","Text":"which is 30 cosine Theta and then"},{"Start":"13:03.565 ","End":"13:13.610","Text":"t is 30 sine Theta over 5.625."},{"Start":"13:13.920 ","End":"13:23.980","Text":"I can get an equation in Theta because this is equal also to 40."},{"Start":"13:23.980 ","End":"13:26.920","Text":"If I take this equal to this,"},{"Start":"13:26.920 ","End":"13:28.540","Text":"I\u0027ll get an equation in Theta,"},{"Start":"13:28.540 ","End":"13:33.250","Text":"which I\u0027ll now solve cosine Theta sine Theta."},{"Start":"13:33.250 ","End":"13:35.290","Text":"I\u0027ll keep that on the 1 side,"},{"Start":"13:35.290 ","End":"13:48.190","Text":"but everything else on the other side is equal to 40 times 5.625."},{"Start":"13:48.190 ","End":"13:53.050","Text":"Over 30 times 30 is 900."},{"Start":"13:53.050 ","End":"13:56.720","Text":"Well, 1 of the 0s you can cancel."},{"Start":"13:58.710 ","End":"14:04.360","Text":"But wait, I just realized we have cosine Theta sine Theta."},{"Start":"14:04.360 ","End":"14:06.820","Text":"How about, I\u0027m going to emphasize,"},{"Start":"14:06.820 ","End":"14:09.595","Text":"I\u0027m going to put a 2 here and a 2 here."},{"Start":"14:09.595 ","End":"14:11.635","Text":"Now, why on earth did I do that?"},{"Start":"14:11.635 ","End":"14:13.960","Text":"That\u0027s because there\u0027s a formula that"},{"Start":"14:13.960 ","End":"14:22.540","Text":"2 cosine Theta sine Theta is equal to sine of 2 Theta."},{"Start":"14:22.540 ","End":"14:27.370","Text":"I\u0027m going to use that and then we can get"},{"Start":"14:27.370 ","End":"14:34.435","Text":"that sine of 2 Theta equals 2 times 4 is 8,"},{"Start":"14:34.435 ","End":"14:40.945","Text":"8 times 5, and 5,8s comes out to be 45,"},{"Start":"14:40.945 ","End":"14:45.640","Text":"45 over 90, which is 1.5."},{"Start":"14:45.640 ","End":"14:50.435","Text":"Now 1.5 is 1 of those angles that we know is sine 30."},{"Start":"14:50.435 ","End":"14:55.220","Text":"We get that 2 Theta equals"},{"Start":"14:55.220 ","End":"15:01.100","Text":"30 degrees on the calculator you would do and the arc sine,"},{"Start":"15:01.100 ","End":"15:04.475","Text":"shift sine or inverse sine of 0.5,"},{"Start":"15:04.475 ","End":"15:06.845","Text":"you would get 30 degrees."},{"Start":"15:06.845 ","End":"15:10.120","Text":"If that\u0027s the case, then Theta is"},{"Start":"15:10.120 ","End":"15:12.965","Text":"15 degrees and that solves"},{"Start":"15:12.965 ","End":"15:17.430","Text":"our problem because we wanted to know the angle, so we\u0027re done."}],"ID":9727}],"Thumbnail":null,"ID":57484},{"Name":"Cylindrical and Spherical Coordinate Systems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Cylindrical - Cartesian , Spherical","Duration":"7m 31s","ChapterTopicVideoID":9875,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9875.jpeg","UploadDate":"2020-01-16T07:12:39.5900000","DurationForVideoObject":"PT7M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"We\u0027re continuing with 3D space and the 3D coordinate system."},{"Start":"00:04.545 ","End":"00:09.825","Text":"The next topic will be cylindrical coordinates."},{"Start":"00:09.825 ","End":"00:14.640","Text":"Whereas previously we had Cartesian,"},{"Start":"00:14.640 ","End":"00:17.415","Text":"that\u0027s the x, y, z."},{"Start":"00:17.415 ","End":"00:21.210","Text":"The next one, not now,"},{"Start":"00:21.210 ","End":"00:23.685","Text":"the next one will be spherical."},{"Start":"00:23.685 ","End":"00:26.805","Text":"Then we\u0027ll be done with this whole topic,"},{"Start":"00:26.805 ","End":"00:28.740","Text":"the Cartesian is our usual x,"},{"Start":"00:28.740 ","End":"00:34.665","Text":"y, z cylindrical is an analog of polar from 2D."},{"Start":"00:34.665 ","End":"00:40.410","Text":"I would strongly recommend reviewing the 2D coordinate system,"},{"Start":"00:40.410 ","End":"00:44.310","Text":"both the Cartesian, which is the x,"},{"Start":"00:44.310 ","End":"00:48.105","Text":"y, and the polar,"},{"Start":"00:48.105 ","End":"00:50.715","Text":"which is r Theta."},{"Start":"00:50.715 ","End":"00:53.700","Text":"You should review those."},{"Start":"00:53.700 ","End":"00:57.220","Text":"Let me introduce a diagram."},{"Start":"00:57.680 ","End":"01:00.599","Text":"If we have a point in space,"},{"Start":"01:00.599 ","End":"01:05.280","Text":"we drop a perpendicular to the xy-plane."},{"Start":"01:05.280 ","End":"01:12.795","Text":"In the xy-plane, we can use polar coordinates and get r and Theta."},{"Start":"01:12.795 ","End":"01:16.230","Text":"Then we add an extra coordinate,"},{"Start":"01:16.230 ","End":"01:18.225","Text":"which is the height,"},{"Start":"01:18.225 ","End":"01:22.800","Text":"which is just the z, the same as in Cartesian coordinates."},{"Start":"01:22.800 ","End":"01:25.785","Text":"In Cartesian, we would have x, y,"},{"Start":"01:25.785 ","End":"01:30.975","Text":"and z because we would just drop perpendiculars."},{"Start":"01:30.975 ","End":"01:34.170","Text":"This is y, this is x,"},{"Start":"01:34.170 ","End":"01:37.815","Text":"and this here would be z."},{"Start":"01:37.815 ","End":"01:40.170","Text":"Then in polar or cylindrical,"},{"Start":"01:40.170 ","End":"01:41.310","Text":"we use r and Theta,"},{"Start":"01:41.310 ","End":"01:43.410","Text":"but z is the same."},{"Start":"01:43.410 ","End":"01:50.850","Text":"Now, just as we had formulas to convert from Cartesian to polar and back,"},{"Start":"01:50.850 ","End":"01:57.015","Text":"we have formulas from Cartesian to cylindrical and back."},{"Start":"01:57.015 ","End":"02:00.840","Text":"The conversion formulas from cylindrical to"},{"Start":"02:00.840 ","End":"02:06.000","Text":"Cartesian are exactly the same as from Cartesian to polar and back,"},{"Start":"02:06.000 ","End":"02:11.790","Text":"if we just ignore the z, the top 2 in each case are the formulas we use then."},{"Start":"02:11.790 ","End":"02:16.845","Text":"But we just add an extra line at z equals z and z is unchanged."},{"Start":"02:16.845 ","End":"02:18.810","Text":"Here we get x, y,"},{"Start":"02:18.810 ","End":"02:23.370","Text":"z from r Theta and z."},{"Start":"02:23.480 ","End":"02:28.005","Text":"Here we get r Theta and z given x, y,"},{"Start":"02:28.005 ","End":"02:31.350","Text":"and z. Oh, by the way,"},{"Start":"02:31.350 ","End":"02:35.490","Text":"the reason it\u0027s called cylindrical coordinates."},{"Start":"02:35.490 ","End":"02:45.060","Text":"This can be explained with another diagram that if we keep one of these constant,"},{"Start":"02:45.060 ","End":"02:48.735","Text":"one of the r, Theta or z,"},{"Start":"02:48.735 ","End":"02:51.855","Text":"then if we keep z constant,"},{"Start":"02:51.855 ","End":"02:56.325","Text":"we get a plane through a given point z."},{"Start":"02:56.325 ","End":"02:59.595","Text":"If we keep r constant,"},{"Start":"02:59.595 ","End":"03:04.020","Text":"then r is just the distance from the point to the z-axis,"},{"Start":"03:04.020 ","End":"03:06.030","Text":"then we get a cylinder."},{"Start":"03:06.030 ","End":"03:09.570","Text":"If Theta\u0027s constant, well here it\u0027s called Phi."},{"Start":"03:09.570 ","End":"03:12.810","Text":"I just want to say that in some places instead of Theta,"},{"Start":"03:12.810 ","End":"03:17.865","Text":"they use Greek letter Phi or even capital."},{"Start":"03:17.865 ","End":"03:19.890","Text":"Here\u0027s a capital phi,"},{"Start":"03:19.890 ","End":"03:21.975","Text":"which is more like this."},{"Start":"03:21.975 ","End":"03:24.420","Text":"Anyway, we\u0027re going to use the Theta,"},{"Start":"03:24.420 ","End":"03:28.245","Text":"but whatever Greek letter works for you."},{"Start":"03:28.245 ","End":"03:32.880","Text":"If r is constantly, we see cylinders and that reminded someone of cylinders and"},{"Start":"03:32.880 ","End":"03:38.400","Text":"so cylindrical coordinates. Now an example."},{"Start":"03:38.400 ","End":"03:44.955","Text":"I\u0027m not going to give examples of how to convert from r Theta z to x, y, z."},{"Start":"03:44.955 ","End":"03:49.350","Text":"Because you have all the examples you want from the polar coordinates."},{"Start":"03:49.350 ","End":"03:52.395","Text":"The z just stays the same when we convert."},{"Start":"03:52.395 ","End":"03:55.890","Text":"We just have to do polar conversion for the first 2 coordinates."},{"Start":"03:55.890 ","End":"03:58.440","Text":"Instead, I\u0027ll give examples of how to identify"},{"Start":"03:58.440 ","End":"04:04.270","Text":"surfaces described in cylindrical coordinates."},{"Start":"04:04.580 ","End":"04:07.860","Text":"Take the following example first."},{"Start":"04:07.860 ","End":"04:12.990","Text":"I\u0027ll take the equation r equals 3."},{"Start":"04:12.990 ","End":"04:16.305","Text":"No Theta and no z here."},{"Start":"04:16.305 ","End":"04:18.945","Text":"Well, this is what I was saying earlier,"},{"Start":"04:18.945 ","End":"04:22.360","Text":"is that when r is constant,"},{"Start":"04:22.430 ","End":"04:24.600","Text":"then it means, well,"},{"Start":"04:24.600 ","End":"04:25.650","Text":"if it was in the plane,"},{"Start":"04:25.650 ","End":"04:27.855","Text":"it would be a circle of radius 3,"},{"Start":"04:27.855 ","End":"04:29.280","Text":"a distance to the center,"},{"Start":"04:29.280 ","End":"04:34.230","Text":"but in 3D r is not the distance to the origin,"},{"Start":"04:34.230 ","End":"04:36.360","Text":"but the distance to the z-axis."},{"Start":"04:36.360 ","End":"04:38.820","Text":"Like I was saying here,"},{"Start":"04:38.820 ","End":"04:41.835","Text":"constant r just gives us a cylinder."},{"Start":"04:41.835 ","End":"04:50.195","Text":"This would be a cylinder of radius 3."},{"Start":"04:50.195 ","End":"04:51.590","Text":"It intersects the x,"},{"Start":"04:51.590 ","End":"04:53.975","Text":"y plane in a circle of radius 3."},{"Start":"04:53.975 ","End":"04:58.555","Text":"Then we just extended in the z direction both ways infinitely."},{"Start":"04:58.555 ","End":"05:01.140","Text":"Let\u0027s take another example."},{"Start":"05:01.140 ","End":"05:05.050","Text":"This one will be example 2."},{"Start":"05:05.780 ","End":"05:15.465","Text":"I\u0027m going to take r squared plus z squared equals 36."},{"Start":"05:15.465 ","End":"05:18.780","Text":"I\u0027m going to write this in Cartesian coordinates."},{"Start":"05:18.780 ","End":"05:23.775","Text":"Here we have r but we also have r squared,"},{"Start":"05:23.775 ","End":"05:27.670","Text":"which is x squared plus y squared."},{"Start":"05:27.920 ","End":"05:33.150","Text":"What we get here is x squared plus y squared,"},{"Start":"05:33.150 ","End":"05:37.905","Text":"that\u0027s instead of the r squared plus z squared as it is equals 36,"},{"Start":"05:37.905 ","End":"05:42.105","Text":"and allow me to write 36 as 6 squared."},{"Start":"05:42.105 ","End":"05:47.910","Text":"This will be the equation in the chapter on quadric surfaces."},{"Start":"05:47.910 ","End":"05:57.270","Text":"This is a sphere and its radius is 6."},{"Start":"05:57.270 ","End":"05:59.940","Text":"Last example."},{"Start":"05:59.940 ","End":"06:05.710","Text":"This one will be number 3,"},{"Start":"06:05.780 ","End":"06:12.765","Text":"and this one will be z equals r. No mention of Theta."},{"Start":"06:12.765 ","End":"06:15.465","Text":"Again, actually, in all of these, there wasn\u0027t."},{"Start":"06:15.465 ","End":"06:21.465","Text":"What we do here is to square both sides."},{"Start":"06:21.465 ","End":"06:28.455","Text":"We get z squared is equal to r-squared."},{"Start":"06:28.455 ","End":"06:29.820","Text":"Now we already had this,"},{"Start":"06:29.820 ","End":"06:35.500","Text":"that r squared is x squared plus y squared."},{"Start":"06:35.630 ","End":"06:39.915","Text":"If you look at the section on quadratic surfaces,"},{"Start":"06:39.915 ","End":"06:43.350","Text":"this is a cone."},{"Start":"06:43.350 ","End":"06:46.770","Text":"In fact, it\u0027s a circular cone."},{"Start":"06:46.770 ","End":"06:48.300","Text":"If you look it up."},{"Start":"06:48.300 ","End":"06:50.820","Text":"Now normally, a cone has 2 parts."},{"Start":"06:50.820 ","End":"06:53.175","Text":"I mean, it\u0027s like they touch tip to tip."},{"Start":"06:53.175 ","End":"06:56.670","Text":"You might think that z has to be positive or"},{"Start":"06:56.670 ","End":"07:00.330","Text":"non-negative because r is non-negative. Not really."},{"Start":"07:00.330 ","End":"07:02.985","Text":"Remember we agreed that r can be negative also,"},{"Start":"07:02.985 ","End":"07:07.090","Text":"if r is negative, we just take the point on the opposite side."},{"Start":"07:07.430 ","End":"07:11.685","Text":"Extend this line in the same distance in the other direction."},{"Start":"07:11.685 ","End":"07:15.315","Text":"Really we get the full cone with both bits."},{"Start":"07:15.315 ","End":"07:18.060","Text":"Anyway, that\u0027s too philosophical for now."},{"Start":"07:18.060 ","End":"07:23.370","Text":"It\u0027s a circular cone and I\u0027ll settle for these 3 examples and that\u0027s it."},{"Start":"07:23.370 ","End":"07:25.920","Text":"Meanwhile, for cylindrical coordinates,"},{"Start":"07:25.920 ","End":"07:27.675","Text":"you might see them again later."},{"Start":"07:27.675 ","End":"07:29.100","Text":"As I said in the next clip,"},{"Start":"07:29.100 ","End":"07:32.320","Text":"it will be spherical coordinates."}],"ID":9728},{"Watched":false,"Name":"The 3D Coordinate System - Spherical","Duration":"31m 7s","ChapterTopicVideoID":9876,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9876.jpeg","UploadDate":"2020-01-16T07:12:25.7470000","DurationForVideoObject":"PT31M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"Hi, and no, you\u0027re not in the wrong clip,"},{"Start":"00:02.430 ","End":"00:05.670","Text":"I just wanted to remind you of what we did previously."},{"Start":"00:05.670 ","End":"00:08.470","Text":"We finished with cartesians then we did cylindrical."},{"Start":"00:08.470 ","End":"00:10.605","Text":"I told you the next would be spherical."},{"Start":"00:10.605 ","End":"00:13.275","Text":"Now is the time for spherical."},{"Start":"00:13.275 ","End":"00:17.430","Text":"I cleaned the board, but I just kept the picture for"},{"Start":"00:17.430 ","End":"00:21.555","Text":"the cylindrical and we can compare it with the spherical."},{"Start":"00:21.555 ","End":"00:26.665","Text":"Now here\u0027s a picture for the spherical."},{"Start":"00:26.665 ","End":"00:30.620","Text":"We\u0027ll go into it in detail right away."},{"Start":"00:30.620 ","End":"00:34.820","Text":"I didn\u0027t draw just the regular cartesian,"},{"Start":"00:34.820 ","End":"00:36.990","Text":"we know that too well."},{"Start":"00:37.760 ","End":"00:41.565","Text":"This time we have 3 Greek letters."},{"Start":"00:41.565 ","End":"00:45.250","Text":"I\u0027ll write them larger. We have the letter Rho"},{"Start":"00:45.250 ","End":"00:51.695","Text":"and we have the letter Theta and we have the letter Phi."},{"Start":"00:51.695 ","End":"00:54.710","Text":"The way they do it as a capital Phi really."},{"Start":"00:54.710 ","End":"00:57.795","Text":"Anyway, Rho, Theta, Phi."},{"Start":"00:57.795 ","End":"01:01.895","Text":"Rho is the distance of the point from the origin,"},{"Start":"01:01.895 ","End":"01:03.469","Text":"as opposed to cylindrical,"},{"Start":"01:03.469 ","End":"01:06.680","Text":"where it\u0027s the distance to the z-axis."},{"Start":"01:06.680 ","End":"01:10.830","Text":"Now, if Rho is constant,"},{"Start":"01:10.830 ","End":"01:16.680","Text":"the points with equal Rho form a sphere of distance Rho from the origin."},{"Start":"01:16.680 ","End":"01:19.535","Text":"That\u0027s why it\u0027s called spherical coordinates."},{"Start":"01:19.535 ","End":"01:24.430","Text":"The other 2 are angles and they\u0027re a bit like latitude and longitude."},{"Start":"01:24.430 ","End":"01:27.085","Text":"If you\u0027re given a sphere and you know its radius,"},{"Start":"01:27.085 ","End":"01:30.650","Text":"everything can be determined by 2 angles."},{"Start":"01:30.870 ","End":"01:34.270","Text":"It\u0027s similar and different to latitude and longitude."},{"Start":"01:34.270 ","End":"01:37.390","Text":"For one thing, we use radians and not degrees."},{"Start":"01:37.390 ","End":"01:40.765","Text":"The second thing is where we start from."},{"Start":"01:40.765 ","End":"01:47.680","Text":"If you took the x-axis or the place it hits the sphere"},{"Start":"01:47.680 ","End":"01:56.750","Text":"as your Greenwich or call that longitude 0."},{"Start":"01:56.850 ","End":"02:06.005","Text":"Also, in geography you measure 0 degrees is the equator and the North Pole is 90 degrees."},{"Start":"02:06.005 ","End":"02:08.890","Text":"But in mathematics they have to do things differently."},{"Start":"02:08.890 ","End":"02:13.024","Text":"The North Pole, the z-axis is 0 degrees,"},{"Start":"02:13.024 ","End":"02:15.980","Text":"and that\u0027s this angle Phi."},{"Start":"02:15.980 ","End":"02:20.990","Text":"Basically Phi has to be between 0 and 180 degrees."},{"Start":"02:20.990 ","End":"02:25.795","Text":"180 degrees will take you to the South Pole or the negative z-axis."},{"Start":"02:25.795 ","End":"02:31.470","Text":"I\u0027ll just write that. Phi has to be between, well,"},{"Start":"02:31.470 ","End":"02:33.905","Text":"in degrees it\u0027s 0 degrees,"},{"Start":"02:33.905 ","End":"02:36.155","Text":"90 degrees at the equator,"},{"Start":"02:36.155 ","End":"02:39.125","Text":"and 180 degrees at the South Pole."},{"Start":"02:39.125 ","End":"02:42.740","Text":"But in radians, 180 degrees is Pi,"},{"Start":"02:42.740 ","End":"02:45.650","Text":"and this is what we restrict Phi to."},{"Start":"02:45.650 ","End":"02:50.100","Text":"Rho is the distance from the origin,"},{"Start":"02:50.100 ","End":"02:55.440","Text":"so that\u0027s restricted naturally by being bigger or equal to 0."},{"Start":"02:55.810 ","End":"03:02.270","Text":"In cylindrical, we sometimes take r to be negative also meaning the other side."},{"Start":"03:02.270 ","End":"03:04.940","Text":"I don\u0027t think this is very commonly done at the spherical,"},{"Start":"03:04.940 ","End":"03:06.995","Text":"so we\u0027ll keep it at that."},{"Start":"03:06.995 ","End":"03:11.880","Text":"Actually Theta will be unrestricted."},{"Start":"03:12.550 ","End":"03:17.030","Text":"Even though you could restrict it to be"},{"Start":"03:17.030 ","End":"03:24.260","Text":"between 0 and Pi and that would give you one full circle, 0 and 2Pi."},{"Start":"03:24.260 ","End":"03:29.600","Text":"That would take us from the x-axis all the way around."},{"Start":"03:29.600 ","End":"03:35.610","Text":"But there\u0027s no reason to restrict it so actually I\u0027m going to erase this."},{"Start":"03:35.690 ","End":"03:40.475","Text":"But you just have to think of it, normally it would be between 0 and 2Pi,"},{"Start":"03:40.475 ","End":"03:44.585","Text":"but you let it go round and round as many times as you want."},{"Start":"03:44.585 ","End":"03:47.450","Text":"Now, the first thing I want to do is show you how to"},{"Start":"03:47.450 ","End":"03:50.330","Text":"convert from spherical to cylindrical."},{"Start":"03:50.330 ","End":"03:55.339","Text":"In principle, there\u0027s actually 6 possible conversions because we have cartesian,"},{"Start":"03:55.339 ","End":"03:57.020","Text":"spherical, and cylindrical and"},{"Start":"03:57.020 ","End":"04:00.290","Text":"all the combinations from one to the other are 6 combinations."},{"Start":"04:00.290 ","End":"04:02.765","Text":"Let\u0027s start with spherical to cylindrical."},{"Start":"04:02.765 ","End":"04:05.625","Text":"What I want to know is,"},{"Start":"04:05.625 ","End":"04:10.695","Text":"what are the 3 quantities?"},{"Start":"04:10.695 ","End":"04:14.580","Text":"What is, let\u0027s say,"},{"Start":"04:14.580 ","End":"04:17.880","Text":"r? What is Theta?"},{"Start":"04:17.880 ","End":"04:22.265","Text":"What is z given Rho, Theta, and Phi?"},{"Start":"04:22.265 ","End":"04:25.970","Text":"Well, the easiest one is that Theta equals Theta."},{"Start":"04:25.970 ","End":"04:27.965","Text":"That\u0027s actually the same in both."},{"Start":"04:27.965 ","End":"04:30.920","Text":"Just like with cylindrical and cartesian,"},{"Start":"04:30.920 ","End":"04:33.685","Text":"we had z being the same."},{"Start":"04:33.685 ","End":"04:38.470","Text":"Now, what we\u0027re going to do is just some simple trigonometry here."},{"Start":"04:38.470 ","End":"04:42.320","Text":"If we look at this triangle,"},{"Start":"04:42.710 ","End":"04:46.125","Text":"this triangle is a right angled triangle."},{"Start":"04:46.125 ","End":"04:50.425","Text":"I guess I should emphasize that this point here"},{"Start":"04:50.425 ","End":"04:57.680","Text":"is what you get dropping a perpendicular from the point to the x y plane."},{"Start":"04:59.270 ","End":"05:03.950","Text":"Because it\u0027s perpendicular, this would be a right angle."},{"Start":"05:04.160 ","End":"05:13.240","Text":"Let\u0027s see, here and here looks like a right angle."},{"Start":"05:13.240 ","End":"05:18.390","Text":"This height is our z."},{"Start":"05:18.470 ","End":"05:20.810","Text":"From here to here,"},{"Start":"05:20.810 ","End":"05:26.915","Text":"this would be r. It\u0027s the same as the distance from this point to the z-axis."},{"Start":"05:26.915 ","End":"05:32.490","Text":"That\u0027s in cylindrical what we called r, and here\u0027s Theta."},{"Start":"05:32.490 ","End":"05:38.585","Text":"Just doing a bit of trigonometry and noting that this angle is also Phi,"},{"Start":"05:38.585 ","End":"05:42.450","Text":"is what you call alternating angles."},{"Start":"05:42.620 ","End":"05:47.630","Text":"This r is the opposite side and z is the adjacent side,"},{"Start":"05:47.630 ","End":"05:49.460","Text":"and Rho is the hypotenuse."},{"Start":"05:49.460 ","End":"05:57.300","Text":"We get from this triangle that r is equal to Rho sine Phi,"},{"Start":"05:59.720 ","End":"06:05.410","Text":"and z is Rho cosine Phi."},{"Start":"06:06.260 ","End":"06:15.300","Text":"This will give us our conversion from spherical to cylindrical."},{"Start":"06:15.300 ","End":"06:21.685","Text":"We could actually continue and go from spherical to cartesian."},{"Start":"06:21.685 ","End":"06:27.360","Text":"Because we know how to get from cylindrical to cartesian."},{"Start":"06:28.430 ","End":"06:34.360","Text":"Here what we get is that z stays the same."},{"Start":"06:34.360 ","End":"06:37.640","Text":"But the z is the same here,"},{"Start":"06:37.640 ","End":"06:42.930","Text":"but it\u0027s equal to Rho cosine Phi."},{"Start":"06:42.930 ","End":"06:44.505","Text":"I just write that."},{"Start":"06:44.505 ","End":"06:46.730","Text":"Then if you recall,"},{"Start":"06:46.730 ","End":"06:51.095","Text":"we have the x and the y was r cosine Theta and r sine Theta."},{"Start":"06:51.095 ","End":"06:54.720","Text":"I\u0027m going to write r and r. Here I\u0027m going to write"},{"Start":"06:54.720 ","End":"06:59.950","Text":"cosine and here I\u0027m going to write sine."},{"Start":"07:03.310 ","End":"07:13.730","Text":"But I want to replace the r with what is here because I need it in terms of Rho,"},{"Start":"07:13.730 ","End":"07:18.445","Text":"so take 2 on the last bit."},{"Start":"07:18.445 ","End":"07:28.180","Text":"Now, we know from cylindrical to cartesian that x is equal to r cosine Theta."},{"Start":"07:28.180 ","End":"07:30.720","Text":"I\u0027ll leave the r for the moment."},{"Start":"07:30.720 ","End":"07:36.380","Text":"Cosine Theta and y equals r sine Theta."},{"Start":"07:36.380 ","End":"07:41.930","Text":"But we can\u0027t use r because we want to use the coordinates from the spherical."},{"Start":"07:41.930 ","End":"07:46.490","Text":"But we look here and see that r is Rho sine Phi."},{"Start":"07:46.490 ","End":"07:51.360","Text":"Here I have Rho sine Phi."},{"Start":"07:51.360 ","End":"07:54.285","Text":"A bit crowded there it\u0027s okay."},{"Start":"07:54.285 ","End":"08:03.450","Text":"Y is the same Rho sine Phi,"},{"Start":"08:03.450 ","End":"08:07.350","Text":"which is r, times sine Theta."},{"Start":"08:07.350 ","End":"08:12.155","Text":"That\u0027s it. We haven\u0027t done everything."},{"Start":"08:12.155 ","End":"08:14.180","Text":"I haven\u0027t given you the formula back"},{"Start":"08:14.180 ","End":"08:24.350","Text":"from cartesian to spherical,"},{"Start":"08:24.350 ","End":"08:29.720","Text":"but you don\u0027t have to remember all the formulas."},{"Start":"08:29.720 ","End":"08:33.290","Text":"I\u0027ll show you the example and you\u0027ll see how we can work"},{"Start":"08:33.290 ","End":"08:36.855","Text":"with the same formulas backwards."},{"Start":"08:36.855 ","End":"08:43.195","Text":"Change your mind. Let\u0027s continue with conversion formulas."},{"Start":"08:43.195 ","End":"08:48.309","Text":"Let\u0027s do from cylindrical to spherical."},{"Start":"08:48.309 ","End":"08:51.430","Text":"Suppose I have r, Theta, and z."},{"Start":"08:51.430 ","End":"08:53.560","Text":"If I have r, Theta, and z,"},{"Start":"08:53.560 ","End":"09:01.540","Text":"what I want is Rho, Theta, and Phi."},{"Start":"09:01.540 ","End":"09:04.180","Text":"Now if I have, as I said,"},{"Start":"09:04.180 ","End":"09:13.555","Text":"well I just write these letters that this is based on r, Theta, and z."},{"Start":"09:13.555 ","End":"09:18.400","Text":"Then just like in this formula and the reverse formula,"},{"Start":"09:18.400 ","End":"09:23.289","Text":"Theta equals Theta, that\u0027s the same in the cylindrical and the spherical."},{"Start":"09:23.289 ","End":"09:27.400","Text":"Now, what we can say also looking for this triangle,"},{"Start":"09:27.400 ","End":"09:29.275","Text":"which is a right-angled triangle,"},{"Start":"09:29.275 ","End":"09:33.520","Text":"is that Rho squared equals z squared plus r squared."},{"Start":"09:33.520 ","End":"09:35.500","Text":"That\u0027s Pythagoras\u0027 theorem."},{"Start":"09:35.500 ","End":"09:37.570","Text":"Just taking the square root,"},{"Start":"09:37.570 ","End":"09:44.125","Text":"Rho is the square root of r squared plus z squared,"},{"Start":"09:44.125 ","End":"09:46.570","Text":"and as for Phi,"},{"Start":"09:46.570 ","End":"09:48.835","Text":"there\u0027s more than 1 way of doing it."},{"Start":"09:48.835 ","End":"09:54.715","Text":"Now, we could actually on this triangle use the Sine, Cosine, or Tangent."},{"Start":"09:54.715 ","End":"10:00.745","Text":"But I prefer the Cosine because when you look up our Cosine on the calculator,"},{"Start":"10:00.745 ","End":"10:05.590","Text":"it automatically gives you between 0 and Pi or 0 and 180 degrees,"},{"Start":"10:05.590 ","End":"10:07.105","Text":"and that\u0027s good for us."},{"Start":"10:07.105 ","End":"10:10.360","Text":"So if we look at what Cosine of Phi is,"},{"Start":"10:10.360 ","End":"10:12.610","Text":"it\u0027s just z over Rho."},{"Start":"10:12.610 ","End":"10:19.899","Text":"What we want is the arc Cosine of z over."},{"Start":"10:19.899 ","End":"10:26.110","Text":"Now, I could say Rho but Rho is given here as"},{"Start":"10:26.110 ","End":"10:32.810","Text":"the square root of r squared plus z squared."},{"Start":"10:32.970 ","End":"10:35.650","Text":"Now, some books use the tangent,"},{"Start":"10:35.650 ","End":"10:40.374","Text":"and then it\u0027s a bit simpler because then it\u0027s just r over z,"},{"Start":"10:40.374 ","End":"10:43.930","Text":"but I prefer it this way because other people do it on"},{"Start":"10:43.930 ","End":"10:49.270","Text":"the calculator and then they forget to adjust it, and so on."},{"Start":"10:49.270 ","End":"10:51.655","Text":"That\u0027s another conversion formula."},{"Start":"10:51.655 ","End":"10:58.010","Text":"I guess you\u0027ve pretty much got them all except we don\u0027t have from Cartesian to spherical."},{"Start":"10:59.220 ","End":"11:02.365","Text":"The cylindrical to Cartesian and back,"},{"Start":"11:02.365 ","End":"11:04.840","Text":"that was done in the section on cylindrical."},{"Start":"11:04.840 ","End":"11:06.160","Text":"Really we\u0027ve got 1,"},{"Start":"11:06.160 ","End":"11:07.960","Text":"2, 3, we need another one,"},{"Start":"11:07.960 ","End":"11:12.590","Text":"which is how to find Rho,"},{"Start":"11:12.720 ","End":"11:20.690","Text":"Theta, and Phi if we have x, y, and z."},{"Start":"11:21.390 ","End":"11:24.310","Text":"Let\u0027s take Rho first."},{"Start":"11:24.310 ","End":"11:30.340","Text":"Now notice this side that Rho"},{"Start":"11:30.340 ","End":"11:37.420","Text":"squared from Pythagoras is r squared plus z squared,"},{"Start":"11:37.420 ","End":"11:40.870","Text":"but r squared is x squared plus y squared."},{"Start":"11:40.870 ","End":"11:47.230","Text":"So we get x squared plus y squared plus z squared just like the equation of"},{"Start":"11:47.230 ","End":"11:54.669","Text":"a sphere in 3D or the distance formula in the extended Pythagoras to 3D."},{"Start":"11:54.669 ","End":"11:59.710","Text":"In any event, we can write Rho as the square root of,"},{"Start":"11:59.710 ","End":"12:01.780","Text":"because we know it\u0027s bigger or equal to 0,"},{"Start":"12:01.780 ","End":"12:07.945","Text":"of x squared plus y squared plus z squared."},{"Start":"12:07.945 ","End":"12:13.270","Text":"Now, Theta. For this one,"},{"Start":"12:13.270 ","End":"12:16.660","Text":"we can just use the polar coordinate. We don\u0027t need z."},{"Start":"12:16.660 ","End":"12:22.059","Text":"We can get the arc tangent of y over x,"},{"Start":"12:22.059 ","End":"12:23.680","Text":"just like with polar,"},{"Start":"12:23.680 ","End":"12:27.790","Text":"as they say, and as far as Phi goes, again,"},{"Start":"12:27.790 ","End":"12:33.760","Text":"there\u0027s more than 1 possibility to use either the tangent or the cosine."},{"Start":"12:33.760 ","End":"12:36.295","Text":"Again, I\u0027m going to use the cosine."},{"Start":"12:36.295 ","End":"12:43.510","Text":"Cosine Phi is equal"},{"Start":"12:43.510 ","End":"12:47.200","Text":"to z over Rho."},{"Start":"12:47.200 ","End":"12:53.589","Text":"So this is the arc cosine of z over Rho,"},{"Start":"12:53.589 ","End":"12:56.515","Text":"but I won\u0027t write Rho."},{"Start":"12:56.515 ","End":"12:58.960","Text":"I could, but then we have to find this 1 first."},{"Start":"12:58.960 ","End":"13:01.030","Text":"But in case you wanted to go straight here,"},{"Start":"13:01.030 ","End":"13:03.715","Text":"and I won\u0027t write Rho, I\u0027ll write what it\u0027s equal to,"},{"Start":"13:03.715 ","End":"13:12.085","Text":"which is the square root of x squared plus y squared plus z squared."},{"Start":"13:12.085 ","End":"13:17.510","Text":"Scattered around here we have all the conversion formulas we need."},{"Start":"13:20.610 ","End":"13:23.665","Text":"1 more thing before the example."},{"Start":"13:23.665 ","End":"13:27.710","Text":"Just want to make a remark on this condition here."},{"Start":"13:27.710 ","End":"13:31.525","Text":"It\u0027s not a universal."},{"Start":"13:31.525 ","End":"13:37.840","Text":"Some allow Rho to be negative just like we allowed r to be negative in polar coordinates,"},{"Start":"13:37.840 ","End":"13:40.675","Text":"and we sort of said that cylindrical like polar."},{"Start":"13:40.675 ","End":"13:43.120","Text":"I guess r can be negative here too."},{"Start":"13:43.120 ","End":"13:47.875","Text":"What it means for negative Rho is we would take the same position as positive Rho,"},{"Start":"13:47.875 ","End":"13:51.230","Text":"but we would then go on the other side,"},{"Start":"13:51.230 ","End":"13:53.820","Text":"like the mirror image through the origin,"},{"Start":"13:53.820 ","End":"13:56.655","Text":"continue this line in the same distance on the other side."},{"Start":"13:56.655 ","End":"14:01.005","Text":"Mathematically what this means is that if we have a positive number Rho,"},{"Start":"14:01.005 ","End":"14:04.650","Text":"then the point minus Rho, Theta,"},{"Start":"14:04.650 ","End":"14:10.255","Text":"Phi would be equivalent to positive Rho,"},{"Start":"14:10.255 ","End":"14:15.265","Text":"but add 180 degrees or Pi radians to Theta to the other side,"},{"Start":"14:15.265 ","End":"14:17.575","Text":"and then again the same Phi,"},{"Start":"14:17.575 ","End":"14:19.870","Text":"so that these would be equivalent."},{"Start":"14:19.870 ","End":"14:21.715","Text":"Now, this is not standard,"},{"Start":"14:21.715 ","End":"14:23.320","Text":"and even with cylindrical,"},{"Start":"14:23.320 ","End":"14:27.220","Text":"whether r can be positive and negative, which is positive."},{"Start":"14:27.220 ","End":"14:29.725","Text":"I\u0027m leaving it a bit vague."},{"Start":"14:29.725 ","End":"14:32.620","Text":"Now on to the examples,"},{"Start":"14:32.620 ","End":"14:35.829","Text":"and we\u0027ll take some conversion examples."},{"Start":"14:35.829 ","End":"14:39.295","Text":"Let me just make sure that I keep the formulas."},{"Start":"14:39.295 ","End":"14:48.325","Text":"The first example I\u0027ll take will be cartesian to spherical,"},{"Start":"14:48.325 ","End":"14:51.955","Text":"and I\u0027ll give you a point, an x, y, z."},{"Start":"14:51.955 ","End":"15:01.990","Text":"Let\u0027s see, root 3, 1 minus 2."},{"Start":"15:01.990 ","End":"15:05.410","Text":"Let\u0027s convert this to spherical."},{"Start":"15:05.410 ","End":"15:08.710","Text":"I\u0027ll find the right formulas,"},{"Start":"15:08.710 ","End":"15:14.470","Text":"and it looks like these are the set of formulas that we need because it\u0027s an x,"},{"Start":"15:14.470 ","End":"15:17.485","Text":"y, z, and it gives us Rho, Theta, Phi."},{"Start":"15:17.485 ","End":"15:19.719","Text":"So 1 at a time,"},{"Start":"15:19.719 ","End":"15:23.230","Text":"we\u0027ll get that Rho is equal to"},{"Start":"15:23.230 ","End":"15:30.890","Text":"the square root of x squared plus y squared plus z squared."},{"Start":"15:31.050 ","End":"15:40.075","Text":"Let\u0027s see, 3 plus 1 plus 4."},{"Start":"15:40.075 ","End":"15:46.435","Text":"In other words, Rho is equal to square root of 8,"},{"Start":"15:46.435 ","End":"15:50.545","Text":"which we can also write as twice root 2."},{"Start":"15:50.545 ","End":"15:52.270","Text":"We\u0027ve done this sort of thing before."},{"Start":"15:52.270 ","End":"15:54.100","Text":"Root of 8 is root 4 root 2,"},{"Start":"15:54.100 ","End":"15:57.505","Text":"which is 2 root 2. That\u0027s Rho."},{"Start":"15:57.505 ","End":"16:05.710","Text":"Now, Theta is going to equal the arc tangent"},{"Start":"16:05.710 ","End":"16:09.655","Text":"of and this is just like"},{"Start":"16:09.655 ","End":"16:16.525","Text":"in converting from cartesian to polar in 2D,"},{"Start":"16:16.525 ","End":"16:20.170","Text":"the arc tangent of y over x."},{"Start":"16:20.170 ","End":"16:23.300","Text":"But we have to watch out for the quadrant."},{"Start":"16:23.460 ","End":"16:29.020","Text":"Y over x is 1 over root 3,"},{"Start":"16:29.020 ","End":"16:35.425","Text":"and we know that the arc tangent of 1 over root 3 is one of those famous ones,"},{"Start":"16:35.425 ","End":"16:39.865","Text":"is 30 degrees or Pi over 6."},{"Start":"16:39.865 ","End":"16:48.760","Text":"But it could also be this thing plus Pi or 30 plus 180 degrees."},{"Start":"16:48.760 ","End":"16:51.085","Text":"But since the first 2 coordinates,"},{"Start":"16:51.085 ","End":"16:53.995","Text":"the x and the y are both positive,"},{"Start":"16:53.995 ","End":"16:56.754","Text":"the thing is in the first quadrant,"},{"Start":"16:56.754 ","End":"16:58.540","Text":"so this is the correct answer,"},{"Start":"16:58.540 ","End":"17:00.460","Text":"but it might not have been."},{"Start":"17:00.460 ","End":"17:05.020","Text":"Both of these were negative and still get the same 1 over root 3,"},{"Start":"17:05.020 ","End":"17:07.880","Text":"but then I would have to add Pi."},{"Start":"17:07.980 ","End":"17:13.495","Text":"Then the third quantity is Phi,"},{"Start":"17:13.495 ","End":"17:19.690","Text":"and we get this by taking the arc cosine of,"},{"Start":"17:19.690 ","End":"17:22.270","Text":"basically, it\u0027s z over Rho."},{"Start":"17:22.270 ","End":"17:23.770","Text":"This thing is Rho."},{"Start":"17:23.770 ","End":"17:32.740","Text":"So z is minus 2 over Rho,"},{"Start":"17:32.740 ","End":"17:38.329","Text":"we already found is 2 root 2."},{"Start":"17:39.630 ","End":"17:48.220","Text":"The 2s cancel and 1 over root 2 is the famous angle and it\u0027s the cosine of 45."},{"Start":"17:48.220 ","End":"17:50.830","Text":"But here I have minus 1 over root 2."},{"Start":"17:50.830 ","End":"17:57.070","Text":"All I have to do is subtract 180 degrees minus the 45."},{"Start":"17:57.070 ","End":"17:59.395","Text":"I\u0027ve got 135 degrees,"},{"Start":"17:59.395 ","End":"18:01.405","Text":"but we\u0027re working in radians,"},{"Start":"18:01.405 ","End":"18:08.670","Text":"so this is equal to 3 Pi over 4."},{"Start":"18:08.670 ","End":"18:10.910","Text":"The easiest thing to do would be to do it on"},{"Start":"18:10.910 ","End":"18:18.140","Text":"your calculator to figure out what is minus 1 over root 2 and take the arc cosine,"},{"Start":"18:18.140 ","End":"18:20.300","Text":"and if you set it for degrees,"},{"Start":"18:20.300 ","End":"18:22.280","Text":"you\u0027ll get 135 degrees,"},{"Start":"18:22.280 ","End":"18:23.600","Text":"and if it\u0027s radians,"},{"Start":"18:23.600 ","End":"18:30.115","Text":"you\u0027ll get some number that is actually equal to 3 Pi over 4."},{"Start":"18:30.115 ","End":"18:33.280","Text":"So yeah, sure at the end,"},{"Start":"18:33.280 ","End":"18:38.885","Text":"we finally write the answer as this cartesian and then spherical,"},{"Start":"18:38.885 ","End":"18:42.920","Text":"it would be 2 root 2,"},{"Start":"18:42.920 ","End":"18:52.700","Text":"Pi over 6, and 3 Pi over 4."},{"Start":"18:52.700 ","End":"18:56.870","Text":"This has to come out between 0 and Pi, and it is."},{"Start":"18:56.870 ","End":"19:00.165","Text":"That\u0027s the cartesian to spherical."},{"Start":"19:00.165 ","End":"19:02.215","Text":"Let me just erase this,"},{"Start":"19:02.215 ","End":"19:05.470","Text":"the next example will be cylindrical to spherical,"},{"Start":"19:05.470 ","End":"19:06.850","Text":"and as an example,"},{"Start":"19:06.850 ","End":"19:10.245","Text":"I\u0027ll take, let\u0027s see,"},{"Start":"19:10.245 ","End":"19:18.310","Text":"root 3 Pi over 2, 1,"},{"Start":"19:18.310 ","End":"19:23.950","Text":"and this would be the r, Theta,"},{"Start":"19:23.950 ","End":"19:28.720","Text":"and z. I guess I should have written here like x, y,"},{"Start":"19:28.720 ","End":"19:32.035","Text":"z, and here r,"},{"Start":"19:32.035 ","End":"19:35.980","Text":"Theta, z because that\u0027s the cylindrical."},{"Start":"19:35.980 ","End":"19:38.095","Text":"Now I want the spherical."},{"Start":"19:38.095 ","End":"19:40.640","Text":"Let\u0027s see, 1 at a time,"},{"Start":"19:40.640 ","End":"19:45.040","Text":"Rho is equal to, and let\u0027s look."},{"Start":"19:45.040 ","End":"19:47.060","Text":"Here\u0027s the formulas."},{"Start":"19:47.060 ","End":"19:51.904","Text":"Rho is equal to the square root of r squared plus z squared."},{"Start":"19:51.904 ","End":"19:55.545","Text":"I need r squared is 3,"},{"Start":"19:55.545 ","End":"19:59.980","Text":"z squared is 1, 3 plus 1 is 4."},{"Start":"19:59.980 ","End":"20:03.590","Text":"So Rho is 2, came out nice."},{"Start":"20:04.590 ","End":"20:10.840","Text":"Theta is even easier because Theta is just the same."},{"Start":"20:10.840 ","End":"20:14.529","Text":"So it\u0027s going to be Pi over 2,"},{"Start":"20:14.529 ","End":"20:25.675","Text":"and all we\u0027re left with is Phi and this is the arc cosine of."},{"Start":"20:25.675 ","End":"20:29.260","Text":"Now, this thing in the denominator is just Rho,"},{"Start":"20:29.260 ","End":"20:30.955","Text":"so that thing is 2,"},{"Start":"20:30.955 ","End":"20:33.355","Text":"and z is 1."},{"Start":"20:33.355 ","End":"20:39.505","Text":"So it\u0027s arc cosine of 1 over 2."},{"Start":"20:39.505 ","End":"20:42.355","Text":"You could do it in the calculator."},{"Start":"20:42.355 ","End":"20:45.040","Text":"But which 1 of the special angles?"},{"Start":"20:45.040 ","End":"20:51.805","Text":"Anyway, if you remember that the cosine of 60 is a half and 60 is between 0 and 180,"},{"Start":"20:51.805 ","End":"20:56.454","Text":"then we get that Phi is equal to,"},{"Start":"20:56.454 ","End":"21:01.495","Text":"I won\u0027t write 60 degrees because we work in radians, Pi over 3."},{"Start":"21:01.495 ","End":"21:06.805","Text":"So the answer is that when we convert this from cylindrical to spherical,"},{"Start":"21:06.805 ","End":"21:09.085","Text":"the answer is 2,"},{"Start":"21:09.085 ","End":"21:15.085","Text":"Pi over 2, Pi over 3."},{"Start":"21:15.085 ","End":"21:20.080","Text":"I suppose I should really also above just so it\u0027s really clear to"},{"Start":"21:20.080 ","End":"21:25.810","Text":"write that this is Rho, Theta, Phi."},{"Start":"21:25.810 ","End":"21:30.895","Text":"The same here, maybe Rho, Theta, Phi."},{"Start":"21:30.895 ","End":"21:34.090","Text":"This point perhaps I\u0027ll mention that the physicists tend"},{"Start":"21:34.090 ","End":"21:40.524","Text":"to get Theta and Phi in the other order, they\u0027re reversed."},{"Start":"21:40.524 ","End":"21:42.880","Text":"So if you\u0027re studying physics,"},{"Start":"21:42.880 ","End":"21:47.260","Text":"then this might look backwards anyway."},{"Start":"21:47.260 ","End":"21:50.080","Text":"So that\u0027s the examples of conversions."},{"Start":"21:50.080 ","End":"21:56.080","Text":"Next, I\u0027ll do some examples of equations of surfaces."},{"Start":"21:56.080 ","End":"22:03.565","Text":"I\u0027ll write a spherical coordinate equation and we\u0027ll find out what the surface is."},{"Start":"22:03.565 ","End":"22:06.295","Text":"Let\u0027s start it on a new page."},{"Start":"22:06.295 ","End":"22:09.415","Text":"I copied the picture and the formulas."},{"Start":"22:09.415 ","End":"22:12.295","Text":"Now, the idea of these exercises,"},{"Start":"22:12.295 ","End":"22:16.105","Text":"I\u0027m going to do something similar to what we saw with cylindrical."},{"Start":"22:16.105 ","End":"22:21.490","Text":"Remember with cylindrical, we showed what it means for a constant r,"},{"Start":"22:21.490 ","End":"22:22.720","Text":"or a constant Theta,"},{"Start":"22:22.720 ","End":"22:26.800","Text":"or a constant z. I want to do the same thing with the"},{"Start":"22:26.800 ","End":"22:32.080","Text":"spherical and with more numerical examples to see what it means for constant Rho,"},{"Start":"22:32.080 ","End":"22:35.120","Text":"constant Theta, and constant Phi."},{"Start":"22:35.120 ","End":"22:39.270","Text":"My first example will be the equation"},{"Start":"22:39.270 ","End":"22:44.235","Text":"Rho equals 4 to see what it means for a constant Rho."},{"Start":"22:44.235 ","End":"22:47.060","Text":"I just took a numerical example of 4."},{"Start":"22:47.060 ","End":"22:54.655","Text":"Now, Rho equals 4 means that if I look at it in the Cartesian,"},{"Start":"22:54.655 ","End":"22:59.515","Text":"it means that the square root of x squared plus y squared plus z squared is 4."},{"Start":"22:59.515 ","End":"23:06.040","Text":"This gives us that x squared plus y squared plus z squared equals 16,"},{"Start":"23:06.040 ","End":"23:08.170","Text":"which I\u0027ll write as 4 squared."},{"Start":"23:08.170 ","End":"23:13.105","Text":"We immediately recognize this from the chapter on quadric surfaces."},{"Start":"23:13.105 ","End":"23:15.380","Text":"This is a sphere,"},{"Start":"23:15.750 ","End":"23:19.060","Text":"and its radius is 4."},{"Start":"23:19.060 ","End":"23:22.525","Text":"That\u0027s 1 of the reasons these are spherical coordinates."},{"Start":"23:22.525 ","End":"23:27.820","Text":"Now, let\u0027s take example 2 where we\u0027ll take a constant Theta."},{"Start":"23:27.820 ","End":"23:31.315","Text":"Let\u0027s take Theta equals,"},{"Start":"23:31.315 ","End":"23:34.525","Text":"let\u0027s say Pi over 4."},{"Start":"23:34.525 ","End":"23:36.685","Text":"Let\u0027s try this equation."},{"Start":"23:36.685 ","End":"23:39.700","Text":"Theta is arc tangent of y over x."},{"Start":"23:39.700 ","End":"23:45.430","Text":"So if I take the tangent of this equation on both sides,"},{"Start":"23:45.430 ","End":"23:50.170","Text":"I get the tangent Thetas y over x. Tangent"},{"Start":"23:50.170 ","End":"23:55.915","Text":"Theta is y over x. I should really write this over here."},{"Start":"23:55.915 ","End":"24:01.915","Text":"Here, I want to actually substitute Theta equals Pi over 4,"},{"Start":"24:01.915 ","End":"24:04.795","Text":"and tangent Pi over 4 is 1."},{"Start":"24:04.795 ","End":"24:09.684","Text":"So this gives me that y over x equals 1,"},{"Start":"24:09.684 ","End":"24:12.114","Text":"tangent of 45 degrees."},{"Start":"24:12.114 ","End":"24:16.900","Text":"So we get that y equals x,"},{"Start":"24:16.900 ","End":"24:18.910","Text":"or if we like,"},{"Start":"24:18.910 ","End":"24:21.430","Text":"I\u0027d rather write it as, say,"},{"Start":"24:21.430 ","End":"24:28.420","Text":"x equals y or even x minus y plus 0, z equals 0."},{"Start":"24:28.420 ","End":"24:31.000","Text":"This is actually a plane because in general,"},{"Start":"24:31.000 ","End":"24:39.815","Text":"when we have Ax plus By plus cz equals d, it\u0027s a plane."},{"Start":"24:39.815 ","End":"24:42.464","Text":"Since z doesn\u0027t appear,"},{"Start":"24:42.464 ","End":"24:44.699","Text":"it\u0027s going to be a vertical plane."},{"Start":"24:44.699 ","End":"24:47.250","Text":"It\u0027s just like the line x equals y,"},{"Start":"24:47.250 ","End":"24:53.770","Text":"which is the 45-degree line in the plane and then raised upwards."},{"Start":"24:53.770 ","End":"24:58.160","Text":"In any event, it\u0027s the equation of a plane."},{"Start":"24:59.100 ","End":"25:02.125","Text":"That\u0027s a sphere, that\u0027s a plane."},{"Start":"25:02.125 ","End":"25:07.645","Text":"What do we get if we let Phi equals a constant?"},{"Start":"25:07.645 ","End":"25:09.790","Text":"Let\u0027s say 30 degrees,"},{"Start":"25:09.790 ","End":"25:13.790","Text":"which I\u0027ll write as Pi over 6."},{"Start":"25:14.670 ","End":"25:22.900","Text":"With the equations, it might be more difficult although we could try."},{"Start":"25:22.900 ","End":"25:25.150","Text":"If we were going to get an equation,"},{"Start":"25:25.150 ","End":"25:30.240","Text":"what I would suggest would be to do it here and to say that take the cosine of"},{"Start":"25:30.240 ","End":"25:37.110","Text":"both sides and then get that z over the square root is equal to cosine of Pi over 6."},{"Start":"25:37.110 ","End":"25:38.850","Text":"If you play around with it a bit,"},{"Start":"25:38.850 ","End":"25:41.265","Text":"you\u0027ll see that we would get a cone."},{"Start":"25:41.265 ","End":"25:43.720","Text":"But I don\u0027t want to do it with equations,"},{"Start":"25:43.720 ","End":"25:45.295","Text":"I want to do it with the picture."},{"Start":"25:45.295 ","End":"25:47.440","Text":"I mean, think about this."},{"Start":"25:47.440 ","End":"25:54.925","Text":"Theta is this angle that the line makes with the vertical line."},{"Start":"25:54.925 ","End":"25:58.540","Text":"If we fix this angle,"},{"Start":"25:58.540 ","End":"26:01.540","Text":"what we could get is we could rotate"},{"Start":"26:01.540 ","End":"26:09.250","Text":"this line around the z-axis and we can get a cone, basically."},{"Start":"26:09.250 ","End":"26:14.420","Text":"If I draw the thing on the other side and then I rotate it,"},{"Start":"26:14.820 ","End":"26:21.444","Text":"what we\u0027d basically get is half a cone."},{"Start":"26:21.444 ","End":"26:24.760","Text":"Although if we let Rho be negative,"},{"Start":"26:24.760 ","End":"26:26.230","Text":"we would get the full cone,"},{"Start":"26:26.230 ","End":"26:30.925","Text":"which we get when we continue over here."},{"Start":"26:30.925 ","End":"26:34.540","Text":"So it all depends whether we get a half cone or a whole,"},{"Start":"26:34.540 ","End":"26:36.835","Text":"which I\u0027ll just call a double cone."},{"Start":"26:36.835 ","End":"26:39.310","Text":"As a last example,"},{"Start":"26:39.310 ","End":"26:48.190","Text":"we\u0027ll take Rho sine Phi equals 3."},{"Start":"26:48.190 ","End":"26:51.220","Text":"I\u0027m going to do it in a couple of ways."},{"Start":"26:51.220 ","End":"26:55.720","Text":"1 way, which many people just try to do automatically,"},{"Start":"26:55.720 ","End":"26:58.585","Text":"is to convert the Cartesian coordinates."},{"Start":"26:58.585 ","End":"27:03.025","Text":"Let\u0027s try that first and then we\u0027ll do another way which will turn out to be shorter."},{"Start":"27:03.025 ","End":"27:11.030","Text":"If we try for the Cartesian and then we\u0027ll be going for these formulas,"},{"Start":"27:11.850 ","End":"27:16.250","Text":"just get us some more space here."},{"Start":"27:16.560 ","End":"27:18.985","Text":"Now, here\u0027s the trick."},{"Start":"27:18.985 ","End":"27:28.675","Text":"Let\u0027s square both sides and get Rho squared sine squared Phi equals 9."},{"Start":"27:28.675 ","End":"27:31.180","Text":"Now, there\u0027s a trick I want to use."},{"Start":"27:31.180 ","End":"27:34.990","Text":"I\u0027m going to make use of the fact that sine squared plus cosine squared is 1."},{"Start":"27:34.990 ","End":"27:36.550","Text":"That\u0027s what I have in mind."},{"Start":"27:36.550 ","End":"27:40.660","Text":"So I\u0027m going to add the same thing to both sides."},{"Start":"27:40.660 ","End":"27:42.625","Text":"I\u0027m going to add Rho squared,"},{"Start":"27:42.625 ","End":"27:47.020","Text":"cosine squared Phi to both sides."},{"Start":"27:47.020 ","End":"27:48.700","Text":"I can see that here afterwards,"},{"Start":"27:48.700 ","End":"27:50.320","Text":"I\u0027ll take the Rho squared out."},{"Start":"27:50.320 ","End":"27:55.915","Text":"So I\u0027ve got 9 plus cosine squared Phi."},{"Start":"27:55.915 ","End":"28:02.230","Text":"Now, as I said, the trick here is that cosine squared plus sine squared is 1."},{"Start":"28:02.230 ","End":"28:03.955","Text":"On the left-hand side,"},{"Start":"28:03.955 ","End":"28:06.895","Text":"I\u0027m just left with Rho squared,"},{"Start":"28:06.895 ","End":"28:10.465","Text":"and on the right-hand side,"},{"Start":"28:10.465 ","End":"28:12.655","Text":"I forgot the Rho squared,"},{"Start":"28:12.655 ","End":"28:15.250","Text":"so let me add it, Rho squared."},{"Start":"28:15.250 ","End":"28:22.210","Text":"Now, notice that Rho cosine Phi is z."},{"Start":"28:22.210 ","End":"28:32.860","Text":"So I\u0027ve got that Rho squared equals 9 plus z squared."},{"Start":"28:32.860 ","End":"28:37.900","Text":"But Rho squared, this is equal to x"},{"Start":"28:37.900 ","End":"28:44.580","Text":"squared plus y squared plus z squared,"},{"Start":"28:44.580 ","End":"28:51.700","Text":"this, I had it written earlier but it disappeared."},{"Start":"28:51.700 ","End":"28:55.360","Text":"But remember that this is what Rho squared equals."},{"Start":"28:55.360 ","End":"29:01.225","Text":"So this is now equal to 9 plus z squared."},{"Start":"29:01.225 ","End":"29:11.665","Text":"So finally, if I just subtract z squared from both sides,"},{"Start":"29:11.665 ","End":"29:14.650","Text":"then I end up with the equation,"},{"Start":"29:14.650 ","End":"29:20.110","Text":"x squared plus y squared equals 9."},{"Start":"29:20.110 ","End":"29:23.394","Text":"We discussed this kind of equation earlier."},{"Start":"29:23.394 ","End":"29:25.300","Text":"If it was in 2-dimensions,"},{"Start":"29:25.300 ","End":"29:28.120","Text":"it would be a circle of radius 3."},{"Start":"29:28.120 ","End":"29:29.710","Text":"This is 3 squared."},{"Start":"29:29.710 ","End":"29:32.005","Text":"But when we add a third dimension,"},{"Start":"29:32.005 ","End":"29:33.130","Text":"because z is missing,"},{"Start":"29:33.130 ","End":"29:40.640","Text":"it becomes a cylinder centered around the z-axis with radius 3."},{"Start":"29:41.030 ","End":"29:45.765","Text":"That\u0027s what we get. Now, I mentioned there was an easier way."},{"Start":"29:45.765 ","End":"29:47.985","Text":"To do it the other way,"},{"Start":"29:47.985 ","End":"29:51.190","Text":"let me scroll back up."},{"Start":"29:53.280 ","End":"29:59.275","Text":"Turns out that Cartesian was not the easiest."},{"Start":"29:59.275 ","End":"30:01.825","Text":"Actually, cylindrical was the easiest."},{"Start":"30:01.825 ","End":"30:05.005","Text":"If we convert this to cylindrical,"},{"Start":"30:05.005 ","End":"30:07.720","Text":"then I can immediately,"},{"Start":"30:07.720 ","End":"30:14.690","Text":"from here, go to say that r equals 3."},{"Start":"30:14.730 ","End":"30:19.515","Text":"Because Rho sine Phi is r,"},{"Start":"30:19.515 ","End":"30:22.185","Text":"so I have r equals 3."},{"Start":"30:22.185 ","End":"30:25.845","Text":"Remember, we talked about constants in cylindrical coordinates."},{"Start":"30:25.845 ","End":"30:29.175","Text":"The same r means the same distance from the z-axis."},{"Start":"30:29.175 ","End":"30:31.775","Text":"So this also gives us a cylinder."},{"Start":"30:31.775 ","End":"30:33.745","Text":"So in both cases,"},{"Start":"30:33.745 ","End":"30:37.750","Text":"whether we look at it in cylindrical or whether we"},{"Start":"30:37.750 ","End":"30:41.950","Text":"look at it in Cartesian, in both cases,"},{"Start":"30:41.950 ","End":"30:50.830","Text":"it\u0027s a cylinder with radius 3,"},{"Start":"30:50.830 ","End":"30:54.475","Text":"and it\u0027s centered around the z-axis."},{"Start":"30:54.475 ","End":"30:58.795","Text":"That\u0027s the last example for spherical coordinates."},{"Start":"30:58.795 ","End":"31:03.340","Text":"Spherical coordinates is the last section in the 3D"},{"Start":"31:03.340 ","End":"31:08.600","Text":"coordinate system and 3D space. We are done."}],"ID":9729},{"Watched":false,"Name":"Exercise 1","Duration":"5m ","ChapterTopicVideoID":9856,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9856.jpeg","UploadDate":"2017-08-06T05:15:22.9400000","DurationForVideoObject":"PT5M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.135","Text":"In this exercise, we have to convert from Cartesian to cylindrical coordinates,"},{"Start":"00:06.135 ","End":"00:09.720","Text":"and I\u0027ve provided the formulas that we\u0027re going to use."},{"Start":"00:09.720 ","End":"00:13.395","Text":"Let\u0027s tackle the first one, a."},{"Start":"00:13.395 ","End":"00:15.135","Text":"Note that the first two,"},{"Start":"00:15.135 ","End":"00:17.160","Text":"it\u0027s like polar coordinates."},{"Start":"00:17.160 ","End":"00:19.350","Text":"We get r Theta from x and y,"},{"Start":"00:19.350 ","End":"00:21.915","Text":"and then z just stays the same."},{"Start":"00:21.915 ","End":"00:26.895","Text":"We have that r is equal to"},{"Start":"00:26.895 ","End":"00:33.870","Text":"the square root of 3 squared plus, minus 7 squared,"},{"Start":"00:33.870 ","End":"00:41.985","Text":"and that\u0027s just 9 plus 49 is 58,"},{"Start":"00:41.985 ","End":"00:45.825","Text":"so this is a square root of 58."},{"Start":"00:45.825 ","End":"00:48.795","Text":"Now, Theta."},{"Start":"00:48.795 ","End":"00:53.820","Text":"Theta is sometimes called the arctangent,"},{"Start":"00:53.820 ","End":"00:59.825","Text":"I\u0027ll write it as arctangent of y over x,"},{"Start":"00:59.825 ","End":"01:02.300","Text":"except that we have to afterwards check that we\u0027re in"},{"Start":"01:02.300 ","End":"01:04.895","Text":"the right quadrant with the arctangent."},{"Start":"01:04.895 ","End":"01:09.010","Text":"We know that we want to point,"},{"Start":"01:09.010 ","End":"01:11.965","Text":"well, when I say quadrant, I mean just the x, y."},{"Start":"01:11.965 ","End":"01:14.365","Text":"If we\u0027re just looking at the x, y part,"},{"Start":"01:14.365 ","End":"01:18.185","Text":"then x positive y negative is the fourth quadrant."},{"Start":"01:18.185 ","End":"01:26.869","Text":"We want either a negative angle or something between 270 and 360 degrees."},{"Start":"01:28.200 ","End":"01:34.375","Text":"Let\u0027s see, this is equal to the arctangent"},{"Start":"01:34.375 ","End":"01:41.000","Text":"of minus 7 over 3."},{"Start":"01:41.000 ","End":"01:43.795","Text":"We\u0027ll need a calculator for this."},{"Start":"01:43.795 ","End":"01:50.200","Text":"The arctangent you do by shift tangent, or inverse tangent, or something like that,"},{"Start":"01:50.200 ","End":"01:57.680","Text":"I make that approximately equal to 66.8 degrees,"},{"Start":"01:57.680 ","End":"02:06.180","Text":"but we usually want that in radians, so minus 1.166."},{"Start":"02:06.180 ","End":"02:08.235","Text":"Sometimes with radians, you add a little c,"},{"Start":"02:08.235 ","End":"02:12.660","Text":"circular measure and it\u0027s in the right quadrant."},{"Start":"02:12.660 ","End":"02:15.245","Text":"I forgot the minus here, silly me."},{"Start":"02:15.245 ","End":"02:21.770","Text":"Yes, it is the right quadrant between 0 and minus 90, so that\u0027s fine."},{"Start":"02:21.770 ","End":"02:24.800","Text":"Z is just z."},{"Start":"02:24.800 ","End":"02:31.320","Text":"Z is just equal to 4, doesn\u0027t change."},{"Start":"02:31.320 ","End":"02:34.580","Text":"If I want to write the answer in radians,"},{"Start":"02:34.580 ","End":"02:40.045","Text":"then I would write the answer as square root of 58,"},{"Start":"02:40.045 ","End":"02:47.500","Text":"minus 1.166, assuming we want it in radians, and 4."},{"Start":"02:47.690 ","End":"02:50.300","Text":"Now for part b,"},{"Start":"02:50.300 ","End":"02:51.965","Text":"I\u0027ll do it on the same page."},{"Start":"02:51.965 ","End":"02:56.270","Text":"We want r is equal to the square root"},{"Start":"02:56.270 ","End":"03:02.600","Text":"of minus 3 squared plus 10 squared,"},{"Start":"03:02.600 ","End":"03:07.440","Text":"and this is 109 square root of."},{"Start":"03:07.750 ","End":"03:11.430","Text":"Then we want Theta."},{"Start":"03:14.830 ","End":"03:18.070","Text":"The x and y have to be in the second quadrant,"},{"Start":"03:18.070 ","End":"03:19.990","Text":"x is negative and y is positive,"},{"Start":"03:19.990 ","End":"03:21.730","Text":"so we have to make sure of that."},{"Start":"03:21.730 ","End":"03:26.350","Text":"We have the arctangent"},{"Start":"03:26.350 ","End":"03:33.540","Text":"of 10 over minus 3 or minus 10 over 3."},{"Start":"03:33.540 ","End":"03:37.135","Text":"I\u0027ll just do it straight away in radians."},{"Start":"03:37.135 ","End":"03:43.030","Text":"I make it roughly minus 1.28 radians."},{"Start":"03:43.030 ","End":"03:46.010","Text":"I think I\u0027ll do it in degrees also."},{"Start":"03:46.010 ","End":"03:52.690","Text":"It comes out minus 77.3 degrees also approximately."},{"Start":"03:52.690 ","End":"03:58.340","Text":"Now, this is in the wrong quadrant."},{"Start":"03:58.340 ","End":"04:03.080","Text":"This would be a fourth quadrant angle and we want a second quadrant,"},{"Start":"04:03.080 ","End":"04:07.530","Text":"so what we do is we add a 180 degrees."},{"Start":"04:08.000 ","End":"04:10.275","Text":"If we do that,"},{"Start":"04:10.275 ","End":"04:14.255","Text":"then we get 106.7 degrees."},{"Start":"04:14.255 ","End":"04:16.490","Text":"But if we want it in radians,"},{"Start":"04:16.490 ","End":"04:19.535","Text":"see, I did it in degrees, because it\u0027s easier to see where you are."},{"Start":"04:19.535 ","End":"04:21.775","Text":"Then I have to add Pi,"},{"Start":"04:21.775 ","End":"04:24.170","Text":"and again, on the calculator,"},{"Start":"04:24.170 ","End":"04:29.460","Text":"it comes out to about 1.86."},{"Start":"04:30.270 ","End":"04:33.790","Text":"The z stays the same,"},{"Start":"04:33.790 ","End":"04:41.744","Text":"which is 4, z equals 4."},{"Start":"04:41.744 ","End":"04:47.930","Text":"Now, I can write the point as square root of 109,"},{"Start":"04:47.930 ","End":"04:49.340","Text":"doing it in radians,"},{"Start":"04:49.340 ","End":"04:54.905","Text":"1.86 radians, and 4,"},{"Start":"04:54.905 ","End":"04:57.200","Text":"and that\u0027s the answer to part b."},{"Start":"04:57.200 ","End":"05:00.150","Text":"So we\u0027re done, a and b."}],"ID":9730},{"Watched":false,"Name":"Exercise 2","Duration":"2m 31s","ChapterTopicVideoID":9857,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9857.jpeg","UploadDate":"2017-08-06T05:15:28.6970000","DurationForVideoObject":"PT2M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise where we were given a Cartesian equation in x,"},{"Start":"00:06.000 ","End":"00:12.120","Text":"y, z, and we want to convert it to cylindrical coordinates in r Theta z."},{"Start":"00:12.120 ","End":"00:20.940","Text":"Here\u0027s the conversion formula and that\u0027s just substitute."},{"Start":"00:20.940 ","End":"00:22.305","Text":"It\u0027s just straightforward."},{"Start":"00:22.305 ","End":"00:26.530","Text":"We have 2, y is r sine Theta."},{"Start":"00:27.350 ","End":"00:37.680","Text":"Want that cubed minus 3 and here we have r cosine sine Theta squared,"},{"Start":"00:37.680 ","End":"00:45.434","Text":"z is the same for z and here we have plus 3y"},{"Start":"00:45.434 ","End":"00:55.320","Text":"squared is r sine Theta squared."},{"Start":"00:55.320 ","End":"00:58.920","Text":"Now, we could have actually,"},{"Start":"00:58.920 ","End":"01:02.445","Text":"if we\u0027d seen it, done a bit of a shortcut."},{"Start":"01:02.445 ","End":"01:06.029","Text":"If I would have put the 3x squared over to the other side,"},{"Start":"01:06.029 ","End":"01:10.740","Text":"I would have got 3y-squared plus 3x squared and there\u0027s"},{"Start":"01:10.740 ","End":"01:16.215","Text":"also a formula that x squared plus y squared equals r-squared."},{"Start":"01:16.215 ","End":"01:17.730","Text":"We could have used that."},{"Start":"01:17.730 ","End":"01:20.030","Text":"But it doesn\u0027t matter because when we bring this over,"},{"Start":"01:20.030 ","End":"01:22.160","Text":"we\u0027ll still get the same thing."},{"Start":"01:22.160 ","End":"01:25.845","Text":"Just a bit lengthier."},{"Start":"01:25.845 ","End":"01:27.350","Text":"Let me do that."},{"Start":"01:27.350 ","End":"01:30.200","Text":"Let me bring this to the other side."},{"Start":"01:30.200 ","End":"01:31.880","Text":"Meanwhile, here I have"},{"Start":"01:31.880 ","End":"01:36.480","Text":"2 cubed sine cubed"},{"Start":"01:36.480 ","End":"01:42.855","Text":"Theta plus 4z equals."},{"Start":"01:42.855 ","End":"01:50.820","Text":"Here I have from here 3 squared cosine"},{"Start":"01:50.820 ","End":"01:58.890","Text":"squared Theta plus 3r squared sine squared Theta."},{"Start":"01:58.890 ","End":"02:04.040","Text":"Since cosine squared plus sine squared is 1 and the 3 r-squared is common."},{"Start":"02:04.040 ","End":"02:13.665","Text":"We can just write this side as 3r squared and maybe we should copy this side."},{"Start":"02:13.665 ","End":"02:15.810","Text":"Now I did a copy-paste."},{"Start":"02:15.810 ","End":"02:20.710","Text":"This would be the answer or you might want to bring everything to 1 side."},{"Start":"02:20.710 ","End":"02:26.449","Text":"I could make this into a minus and I could put equals 0."},{"Start":"02:26.449 ","End":"02:27.770","Text":"I don\u0027t know if it\u0027s any better."},{"Start":"02:27.770 ","End":"02:32.130","Text":"Anyway, that\u0027s the answer and we\u0027re done."}],"ID":9731},{"Watched":false,"Name":"Exercise 3","Duration":"2m 14s","ChapterTopicVideoID":9858,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9858.jpeg","UploadDate":"2017-08-06T05:15:34.4630000","DurationForVideoObject":"PT2M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.185","Text":"In this exercise, we have to convert the equations from cylindrical to Cartesian."},{"Start":"00:07.185 ","End":"00:11.235","Text":"I brought both sets of conversions."},{"Start":"00:11.235 ","End":"00:12.960","Text":"This 1 gives r, Theta,"},{"Start":"00:12.960 ","End":"00:14.720","Text":"and z in terms of x, y, z,"},{"Start":"00:14.720 ","End":"00:17.205","Text":"and this is the other way round."},{"Start":"00:17.205 ","End":"00:20.760","Text":"In principle, you could just blindly substitute r,"},{"Start":"00:20.760 ","End":"00:23.680","Text":"Theta, and z into the equation."},{"Start":"00:23.680 ","End":"00:25.895","Text":"But it would be messy because,"},{"Start":"00:25.895 ","End":"00:29.690","Text":"for example when you had cosine Theta,"},{"Start":"00:29.690 ","End":"00:33.130","Text":"you\u0027d write cosine of arc tangent of y over,"},{"Start":"00:33.130 ","End":"00:37.280","Text":"so you have to be a bit smarter and use both sets."},{"Start":"00:37.280 ","End":"00:42.920","Text":"For example, I see r cosine Theta and I write x. I see r squared,"},{"Start":"00:42.920 ","End":"00:44.690","Text":"and that\u0027s x squared plus y squared."},{"Start":"00:44.690 ","End":"00:47.555","Text":"As for the rest of it, go by the formula."},{"Start":"00:47.555 ","End":"00:51.359","Text":"In a we have 3 minus,"},{"Start":"00:51.359 ","End":"01:00.330","Text":"now r squared is x squared plus y squared equals z is unchanged in cylindrical,"},{"Start":"01:00.330 ","End":"01:04.190","Text":"and r we just have to use what is here."},{"Start":"01:04.190 ","End":"01:07.670","Text":"Square root of x squared plus y squared,"},{"Start":"01:07.670 ","End":"01:11.960","Text":"and then plus r cosine Theta is x."},{"Start":"01:11.960 ","End":"01:14.285","Text":"Then we could tidy it up a bit,"},{"Start":"01:14.285 ","End":"01:17.940","Text":"but it\u0027s just fine to leave it as is."},{"Start":"01:17.940 ","End":"01:23.100","Text":"In part b, let\u0027s see,"},{"Start":"01:23.100 ","End":"01:26.395","Text":"we have cosine Theta and sine Theta."},{"Start":"01:26.395 ","End":"01:28.970","Text":"If we had r cosine Theta like before,"},{"Start":"01:28.970 ","End":"01:30.545","Text":"that would be nice."},{"Start":"01:30.545 ","End":"01:34.925","Text":"Why don\u0027t we multiply this whole equation by r?"},{"Start":"01:34.925 ","End":"01:36.935","Text":"This is a common trick."},{"Start":"01:36.935 ","End":"01:42.380","Text":"We\u0027ll get 3r cosine Theta plus"},{"Start":"01:42.380 ","End":"01:48.065","Text":"5r sine Theta equals r squared z."},{"Start":"01:48.065 ","End":"01:51.605","Text":"This really helps us because we have r cosine Theta is x,"},{"Start":"01:51.605 ","End":"01:53.495","Text":"r sine Theta is y,"},{"Start":"01:53.495 ","End":"01:56.060","Text":"and even r squared, we don\u0027t need a square root."},{"Start":"01:56.060 ","End":"02:00.679","Text":"What we get is 3 times x,"},{"Start":"02:00.679 ","End":"02:05.780","Text":"which is r cosine Theta, plus 5r sine Theta is y"},{"Start":"02:05.780 ","End":"02:11.750","Text":"equals r squared is x squared plus y squared times z."},{"Start":"02:11.750 ","End":"02:14.490","Text":"That\u0027s all there is to it."}],"ID":9732},{"Watched":false,"Name":"Exercise 4","Duration":"3m 3s","ChapterTopicVideoID":9859,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9859.jpeg","UploadDate":"2017-08-06T05:15:41.7970000","DurationForVideoObject":"PT3M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we have to identify the surface generated"},{"Start":"00:04.170 ","End":"00:09.190","Text":"by the following cylindrical coordinate equation."},{"Start":"00:09.620 ","End":"00:16.960","Text":"The idea is to first convert it to Cartesian coordinates and it\u0027ll be easier to identify."},{"Start":"00:18.680 ","End":"00:23.715","Text":"I\u0027m assuming at this point you remember all the formulas or you have them handy."},{"Start":"00:23.715 ","End":"00:30.000","Text":"R squared is x squared plus y squared."},{"Start":"00:30.000 ","End":"00:34.680","Text":"Then r sine Theta is y,"},{"Start":"00:34.680 ","End":"00:41.100","Text":"so I have plus 6y equals 13."},{"Start":"00:41.100 ","End":"00:51.439","Text":"Now, what we want to do here to do a completing the square."},{"Start":"00:51.439 ","End":"00:54.980","Text":"It looks like it\u0027s going to be a circle from the x squared plus y squared."},{"Start":"00:54.980 ","End":"01:02.210","Text":"What we do is we say x squared plus y squared plus 6y."},{"Start":"01:02.210 ","End":"01:07.340","Text":"Now we want to add something here to make it a perfect square,"},{"Start":"01:07.340 ","End":"01:09.785","Text":"then we\u0027ll add the same thing to the other side."},{"Start":"01:09.785 ","End":"01:14.735","Text":"Now this looks like it\u0027s going to be y plus"},{"Start":"01:14.735 ","End":"01:20.685","Text":"3 squared because it has to be twice y times 3 is 6."},{"Start":"01:20.685 ","End":"01:24.220","Text":"We need here 3 squared, which is 9."},{"Start":"01:24.220 ","End":"01:26.680","Text":"Now if I\u0027ve added 9 to the left-hand side,"},{"Start":"01:26.680 ","End":"01:29.620","Text":"I need to add 9 here."},{"Start":"01:29.620 ","End":"01:37.385","Text":"So I get 22, so 22,"},{"Start":"01:37.385 ","End":"01:41.680","Text":"and then we can rewrite it as x"},{"Start":"01:41.680 ","End":"01:49.360","Text":"squared plus y plus 3 squared equals,"},{"Start":"01:49.360 ","End":"01:51.640","Text":"I want to put it as something squared,"},{"Start":"01:51.640 ","End":"01:55.135","Text":"so I\u0027ll write it as square root of 22 squared."},{"Start":"01:55.135 ","End":"01:59.990","Text":"This way I can see that if it was in the plane,"},{"Start":"01:59.990 ","End":"02:06.735","Text":"then I would say this was a circle and it would be centered."},{"Start":"02:06.735 ","End":"02:10.530","Text":"The center would be 0,"},{"Start":"02:10.530 ","End":"02:16.750","Text":"minus 3 because this is x minus 0 squared plus y minus minus 3 squared."},{"Start":"02:16.750 ","End":"02:23.385","Text":"Center and radius would be equal to root 22."},{"Start":"02:23.385 ","End":"02:25.440","Text":"Only we\u0027re not in the plane,"},{"Start":"02:25.440 ","End":"02:29.035","Text":"we\u0027re in 3D and z doesn\u0027t appear here."},{"Start":"02:29.035 ","End":"02:31.510","Text":"So z is anything,"},{"Start":"02:31.510 ","End":"02:36.325","Text":"which means that we take our circle and extend it infinitely up and down"},{"Start":"02:36.325 ","End":"02:42.500","Text":"so the circle actually becomes a cylinder,"},{"Start":"02:43.610 ","End":"02:48.695","Text":"which is the axis is parallel to the z-axis."},{"Start":"02:48.695 ","End":"02:55.265","Text":"If you look down from the z-axis at where it cuts the x-y plane and it would be this."},{"Start":"02:55.265 ","End":"02:57.950","Text":"But it extends infinitely up and down,"},{"Start":"02:57.950 ","End":"03:03.000","Text":"so it\u0027s a cylinder, not a circle. That\u0027s it."}],"ID":9733},{"Watched":false,"Name":"Exercise 5","Duration":"2m 25s","ChapterTopicVideoID":9860,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9860.jpeg","UploadDate":"2017-08-06T05:15:47.8230000","DurationForVideoObject":"PT2M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"Here we have to identify the surface generated by this equation which is in"},{"Start":"00:05.100 ","End":"00:10.500","Text":"cylindrical coordinates and we\u0027re going to convert it first to Cartesian,"},{"Start":"00:10.500 ","End":"00:11.985","Text":"then it will be easier."},{"Start":"00:11.985 ","End":"00:15.045","Text":"What we\u0027ll get if we convert to Cartesian,"},{"Start":"00:15.045 ","End":"00:19.875","Text":"z is just z equals 3 minus 2."},{"Start":"00:19.875 ","End":"00:28.360","Text":"We know that r squared is x squared plus y squared."},{"Start":"00:28.430 ","End":"00:35.964","Text":"I\u0027ll just rewrite it as 3 minus 2x squared minus 2y squared."},{"Start":"00:35.964 ","End":"00:40.320","Text":"Now I claim that this is a paraboloid."},{"Start":"00:40.450 ","End":"00:44.510","Text":"If we didn\u0027t have it like this,"},{"Start":"00:44.510 ","End":"00:50.674","Text":"if we had z equals 2x squared plus 2y squared,"},{"Start":"00:50.674 ","End":"00:53.299","Text":"you\u0027d see immediately that it\u0027s a paraboloid,"},{"Start":"00:53.299 ","End":"00:57.185","Text":"a circular paraboloid because the coefficients of x and y are equal."},{"Start":"00:57.185 ","End":"00:59.420","Text":"In general, we have 2 different numbers here."},{"Start":"00:59.420 ","End":"01:00.980","Text":"It\u0027s an elliptical paraboloid."},{"Start":"01:00.980 ","End":"01:03.750","Text":"So this is a circular paraboloid."},{"Start":"01:03.880 ","End":"01:06.890","Text":"I can\u0027t really sketch it in 3D,"},{"Start":"01:06.890 ","End":"01:09.290","Text":"but I can give you an idea of what it looks like."},{"Start":"01:09.290 ","End":"01:15.475","Text":"If this is the z-axis and this is either the x or the y-axis."},{"Start":"01:15.475 ","End":"01:17.715","Text":"It could be x or y looking at it,"},{"Start":"01:17.715 ","End":"01:19.605","Text":"or even the xy-plane,"},{"Start":"01:19.605 ","End":"01:23.100","Text":"then it looks like if y is 0,"},{"Start":"01:23.100 ","End":"01:27.009","Text":"x is 0, you can see it\u0027s a parabola facing upwards."},{"Start":"01:27.009 ","End":"01:33.440","Text":"Now, what happens if I was to take 3 minus this?"},{"Start":"01:33.440 ","End":"01:41.135","Text":"It just means the minus means it\u0027s upside down and 3 means I start 3 units up."},{"Start":"01:41.135 ","End":"01:46.895","Text":"So again, just looking at the side where this is the z-axis,"},{"Start":"01:46.895 ","End":"01:51.950","Text":"and this is either the x-axis or the y-axis or the xy-plane."},{"Start":"01:51.950 ","End":"01:54.245","Text":"Instead of this, I\u0027d start at the point,"},{"Start":"01:54.245 ","End":"01:57.140","Text":"let\u0027s say this is 3 and then it would just be"},{"Start":"01:57.140 ","End":"02:00.590","Text":"the same thing but facing down and from here."},{"Start":"02:00.590 ","End":"02:10.830","Text":"You would write something like a circular paraboloid,"},{"Start":"02:10.830 ","End":"02:14.255","Text":"then I might say it was centered on the z-axis."},{"Start":"02:14.255 ","End":"02:22.080","Text":"I might also say that the apex is where z is 3 and of course x and y are both 0."},{"Start":"02:22.080 ","End":"02:25.700","Text":"Then this should do it, so we\u0027re done."}],"ID":9734},{"Watched":false,"Name":"Exercise 6","Duration":"6m 48s","ChapterTopicVideoID":9861,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9861.jpeg","UploadDate":"2017-08-06T05:16:04.4500000","DurationForVideoObject":"PT6M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.450","Text":"In this exercise, we have to convert from Cartesian to Spherical coordinates,"},{"Start":"00:06.450 ","End":"00:07.860","Text":"and it\u0027s 2 in 1."},{"Start":"00:07.860 ","End":"00:09.685","Text":"We have 2 separate problems."},{"Start":"00:09.685 ","End":"00:13.300","Text":"I copied the formula from the tutorial."},{"Start":"00:13.300 ","End":"00:14.905","Text":"I just did a copy paste."},{"Start":"00:14.905 ","End":"00:16.540","Text":"This should suffice."},{"Start":"00:16.540 ","End":"00:19.210","Text":"Let\u0027s start with Part A."},{"Start":"00:19.210 ","End":"00:21.625","Text":"These are x, y, and z,"},{"Start":"00:21.625 ","End":"00:24.830","Text":"and we need Rho, Theta, Phi."},{"Start":"00:27.210 ","End":"00:34.555","Text":"Rho is equal to the square root of x squared plus y squared plus z squared,"},{"Start":"00:34.555 ","End":"00:40.715","Text":"which is minus 3 squared plus 9 squared plus 4 squared."},{"Start":"00:40.715 ","End":"00:42.570","Text":"That is equal to,"},{"Start":"00:42.570 ","End":"00:44.025","Text":"let\u0027s see if we can do it in our heads,"},{"Start":"00:44.025 ","End":"00:51.150","Text":"9 plus 81 is 90 plus 16 is 106,"},{"Start":"00:51.150 ","End":"00:53.825","Text":"so we have the square root of 106."},{"Start":"00:53.825 ","End":"00:58.930","Text":"We\u0027ll leave it like that. Next, we need Theta."},{"Start":"00:58.930 ","End":"01:07.954","Text":"Theta is the arctangent of y over x,"},{"Start":"01:07.954 ","End":"01:14.945","Text":"except that we have to make sure that we\u0027re in the right quadrant in the xy plane."},{"Start":"01:14.945 ","End":"01:20.230","Text":"In this case, minus 3,9 is going to be in the second quadrant,"},{"Start":"01:20.230 ","End":"01:24.470","Text":"so the answer from the calculator might need adjusting."},{"Start":"01:24.470 ","End":"01:28.830","Text":"Sorry, I meant to plug in the actual numbers already."},{"Start":"01:29.360 ","End":"01:36.020","Text":"We have 9 over minus 3,"},{"Start":"01:36.020 ","End":"01:40.145","Text":"which is the arctangent of minus 3."},{"Start":"01:40.145 ","End":"01:47.030","Text":"On the calculator, I like to do it in both degrees and radians."},{"Start":"01:47.030 ","End":"01:53.120","Text":"In degrees, I got minus 71 point something degrees,"},{"Start":"01:53.120 ","End":"02:01.370","Text":"and in radians, I got minus 1.249 something."},{"Start":"02:01.370 ","End":"02:07.410","Text":"Sorry, radian, sometimes we write little c there for radians."},{"Start":"02:07.410 ","End":"02:09.770","Text":"Now, we\u0027re in the wrong quadrant because"},{"Start":"02:09.770 ","End":"02:14.390","Text":"minus 71 degrees is in the fourth quadrant and we want to be in the second quadrant."},{"Start":"02:14.390 ","End":"02:21.410","Text":"What we have to do is add a 180 degrees or in this case, add Pi."},{"Start":"02:21.410 ","End":"02:23.330","Text":"Now, I just want it in radians."},{"Start":"02:23.330 ","End":"02:26.735","Text":"The degrees is just so I can see clearly which quadrant it\u0027s in."},{"Start":"02:26.735 ","End":"02:30.200","Text":"What I\u0027m going to do is take this and add Pi,"},{"Start":"02:30.200 ","End":"02:40.520","Text":"and this gives me 1.892 something radians."},{"Start":"02:40.520 ","End":"02:45.845","Text":"Now for Phi, Phi is"},{"Start":"02:45.845 ","End":"02:52.665","Text":"equal to the arccosine of z,"},{"Start":"02:52.665 ","End":"02:57.980","Text":"which is 4 divided by, well,"},{"Start":"02:57.980 ","End":"03:00.620","Text":"this thing in the denominator is just Rho,"},{"Start":"03:00.620 ","End":"03:02.510","Text":"and we\u0027ve computed it already,"},{"Start":"03:02.510 ","End":"03:08.165","Text":"so we don\u0027t need to compute it again, root 106."},{"Start":"03:08.165 ","End":"03:14.810","Text":"I make it 1.171 something in radians."},{"Start":"03:14.810 ","End":"03:16.520","Text":"Just to give you an idea,"},{"Start":"03:16.520 ","End":"03:21.210","Text":"it\u0027s 67 point something in degrees."},{"Start":"03:21.210 ","End":"03:23.430","Text":"Look at this, this is a check."},{"Start":"03:23.430 ","End":"03:28.550","Text":"I know that the z is positive and if I\u0027m above the xy plane,"},{"Start":"03:28.550 ","End":"03:31.430","Text":"then the Phi is going to be between 0 and 90,"},{"Start":"03:31.430 ","End":"03:34.650","Text":"which it is, and that\u0027s just a check."},{"Start":"03:34.900 ","End":"03:40.025","Text":"Last thing you do is just to write the answer neatly,"},{"Start":"03:40.025 ","End":"03:44.855","Text":"106 root then the Phi,"},{"Start":"03:44.855 ","End":"03:47.365","Text":"which we\u0027ll do in radians,"},{"Start":"03:47.365 ","End":"03:49.665","Text":"we\u0027ll write the approximate,"},{"Start":"03:49.665 ","End":"03:55.020","Text":"and then Phi, 1.171."},{"Start":"03:55.020 ","End":"03:56.985","Text":"That\u0027s the answer to Part A."},{"Start":"03:56.985 ","End":"03:59.950","Text":"Now, let\u0027s go and do Part B."},{"Start":"04:00.250 ","End":"04:05.015","Text":"Let\u0027s see if I can do it on the same page."},{"Start":"04:05.015 ","End":"04:07.360","Text":"I\u0027ll start anyway."},{"Start":"04:07.360 ","End":"04:11.960","Text":"B, and why don\u0027t I just copy it and then I don\u0027t have to worry if it\u0027s"},{"Start":"04:11.960 ","End":"04:16.920","Text":"scrolls out of sight and then we get plenty of space?"},{"Start":"04:17.350 ","End":"04:21.715","Text":"Using these formulae again,"},{"Start":"04:21.715 ","End":"04:26.480","Text":"we get that Rho is equal to"},{"Start":"04:26.480 ","End":"04:35.130","Text":"the square root of 3 squared plus 4 squared plus minus 5 squared."},{"Start":"04:35.130 ","End":"04:37.230","Text":"Now, 3 squared and 4 squared,"},{"Start":"04:37.230 ","End":"04:40.425","Text":"9 and 16 is 25 plus 25 is 50,"},{"Start":"04:40.425 ","End":"04:44.560","Text":"so this is root 50."},{"Start":"04:45.020 ","End":"04:48.025","Text":"Next is Theta."},{"Start":"04:48.025 ","End":"04:55.520","Text":"Theta is equal to the arctangent of y over x."},{"Start":"04:55.520 ","End":"04:58.280","Text":"Now, these are both positive first quadrant,"},{"Start":"04:58.280 ","End":"04:59.915","Text":"so everything will be okay."},{"Start":"04:59.915 ","End":"05:02.670","Text":"Arctangent of 4 over 3,"},{"Start":"05:02.670 ","End":"05:05.210","Text":"which on the calculator,"},{"Start":"05:05.210 ","End":"05:12.710","Text":"I\u0027ll just do it in radians and we can see it\u0027s supposed to be between 0 and 0.5 Pi,"},{"Start":"05:12.710 ","End":"05:18.080","Text":"which is between 0 and 1.5 something."},{"Start":"05:18.080 ","End":"05:20.340","Text":"Let me check that,"},{"Start":"05:20.780 ","End":"05:26.870","Text":"0.927 something in radians."},{"Start":"05:26.870 ","End":"05:32.210","Text":"If you\u0027re curious, it\u0027s 53 point something degrees,"},{"Start":"05:32.210 ","End":"05:35.265","Text":"but we want in radians."},{"Start":"05:35.265 ","End":"05:42.300","Text":"Lastly, Phi, which is the arccosine of z,"},{"Start":"05:42.300 ","End":"05:46.440","Text":"which is minus 5 over,"},{"Start":"05:46.440 ","End":"05:53.255","Text":"this we don\u0027t compute again because we computed it in the first part with Rho,"},{"Start":"05:53.255 ","End":"06:00.920","Text":"is root 50, and this is equal"},{"Start":"06:00.920 ","End":"06:09.165","Text":"to 2.356 approximately in radians."},{"Start":"06:09.165 ","End":"06:10.640","Text":"By the way, in degrees,"},{"Start":"06:10.640 ","End":"06:12.260","Text":"it comes out a nice number."},{"Start":"06:12.260 ","End":"06:16.190","Text":"It comes out exactly 135 degrees,"},{"Start":"06:16.190 ","End":"06:21.094","Text":"and it\u0027s more than 90 and less than 180 because the z is negative,"},{"Start":"06:21.094 ","End":"06:23.910","Text":"so we\u0027re in the lower hemisphere."},{"Start":"06:24.020 ","End":"06:34.515","Text":"Now, we just have to write the answer that Rho is root 50, Theta,"},{"Start":"06:34.515 ","End":"06:39.695","Text":"we\u0027ll do everything in radians here, that\u0027s approximately this,"},{"Start":"06:39.695 ","End":"06:44.900","Text":"and Phi is approximately 2.356,"},{"Start":"06:44.900 ","End":"06:48.150","Text":"and that\u0027s the answer."}],"ID":9735},{"Watched":false,"Name":"Exercise 7","Duration":"1m 29s","ChapterTopicVideoID":9862,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9862.jpeg","UploadDate":"2017-08-06T05:16:58.2270000","DurationForVideoObject":"PT1M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.395","Text":"Here we have to convert from spherical to Cartesian coordinates."},{"Start":"00:04.395 ","End":"00:05.669","Text":"If these are spherical,"},{"Start":"00:05.669 ","End":"00:07.020","Text":"the order is first of all,"},{"Start":"00:07.020 ","End":"00:09.855","Text":"it\u0027s Rho, then it\u0027s Theta,"},{"Start":"00:09.855 ","End":"00:11.790","Text":"and then it\u0027s Phi."},{"Start":"00:11.790 ","End":"00:13.290","Text":"Now I brought the equations."},{"Start":"00:13.290 ","End":"00:15.660","Text":"I copy pasted them from the tutorial."},{"Start":"00:15.660 ","End":"00:17.535","Text":"Let\u0027s get started."},{"Start":"00:17.535 ","End":"00:23.445","Text":"X is going to equal Rho which is 5, sine Phi,"},{"Start":"00:23.445 ","End":"00:26.549","Text":"which Phi is equal to Pi,"},{"Start":"00:26.549 ","End":"00:30.360","Text":"and then cosine of Theta."},{"Start":"00:30.360 ","End":"00:32.955","Text":"Theta is 0."},{"Start":"00:32.955 ","End":"00:36.045","Text":"I\u0027ll compute it in a moment. Let\u0027s just write it."},{"Start":"00:36.045 ","End":"00:39.540","Text":"Y equals same thing as this,"},{"Start":"00:39.540 ","End":"00:41.775","Text":"except the sine at the end,"},{"Start":"00:41.775 ","End":"00:43.550","Text":"so I just copied this,"},{"Start":"00:43.550 ","End":"00:45.755","Text":"but now I\u0027m going to erase the cosine,"},{"Start":"00:45.755 ","End":"00:48.530","Text":"and write sine instead."},{"Start":"00:48.530 ","End":"00:52.574","Text":"The last 1, z is Rho 5,"},{"Start":"00:52.574 ","End":"00:56.200","Text":"cosine of this angle, which is Pi."},{"Start":"00:56.200 ","End":"01:01.980","Text":"Let\u0027s see what these are actually equal to numerically."},{"Start":"01:02.990 ","End":"01:06.390","Text":"Sine of Pi is 0,"},{"Start":"01:06.390 ","End":"01:08.580","Text":"so that makes this 0,"},{"Start":"01:08.580 ","End":"01:10.590","Text":"and it makes this 0,"},{"Start":"01:10.590 ","End":"01:13.650","Text":"because sine of 180 degrees is 0."},{"Start":"01:13.650 ","End":"01:17.505","Text":"Cosine of 180 degrees is minus 1,"},{"Start":"01:17.505 ","End":"01:20.009","Text":"so this gives us minus 5,"},{"Start":"01:20.009 ","End":"01:26.380","Text":"so our answer is that we have the point 0, 0, -"}],"ID":9736},{"Watched":false,"Name":"Exercise 8","Duration":"2m 14s","ChapterTopicVideoID":9851,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9851.jpeg","UploadDate":"2017-08-06T05:14:52.5570000","DurationForVideoObject":"PT2M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, we have to convert from cylindrical to"},{"Start":"00:04.620 ","End":"00:10.260","Text":"spherical and I\u0027ve got all the formulas, I need."},{"Start":"00:10.260 ","End":"00:13.005","Text":"I copy-pasted them from the tutorial."},{"Start":"00:13.005 ","End":"00:17.160","Text":"Notice that the order, this is r,"},{"Start":"00:17.160 ","End":"00:23.130","Text":"this is Theta, and this is z."},{"Start":"00:23.130 ","End":"00:25.995","Text":"Now I\u0027m going to apply these."},{"Start":"00:25.995 ","End":"00:33.015","Text":"I get Rho is equal square root of r squared plus z squared is the square root of"},{"Start":"00:33.015 ","End":"00:40.750","Text":"2 squared plus z squared is 3."},{"Start":"00:43.370 ","End":"00:47.590","Text":"This is root 7."},{"Start":"00:47.590 ","End":"00:50.195","Text":"Next is the easy part."},{"Start":"00:50.195 ","End":"00:53.330","Text":"Theta is the same in cylindrical and spherical,"},{"Start":"00:53.330 ","End":"00:59.315","Text":"so I just have to copy it 1.23 assuming this is radians."},{"Start":"00:59.315 ","End":"01:03.020","Text":"Sometimes you write a little c to indicate radians."},{"Start":"01:03.020 ","End":"01:07.490","Text":"Lastly, we have the angle Phi,"},{"Start":"01:07.490 ","End":"01:13.310","Text":"which is equal to the arc cosine of z,"},{"Start":"01:13.310 ","End":"01:17.855","Text":"which is root 3 over."},{"Start":"01:17.855 ","End":"01:21.620","Text":"Now, we don\u0027t have to compute the denominator again"},{"Start":"01:21.620 ","End":"01:25.085","Text":"because we have it already over here, which is this."},{"Start":"01:25.085 ","End":"01:33.050","Text":"It\u0027s root 7 and this is root of 3 over 7."},{"Start":"01:33.050 ","End":"01:34.640","Text":"That\u0027s how we would do it on the calculator,"},{"Start":"01:34.640 ","End":"01:36.230","Text":"I do 3 divided by 7,"},{"Start":"01:36.230 ","End":"01:39.600","Text":"take the square root then the inverse cosine."},{"Start":"01:39.730 ","End":"01:47.000","Text":"I make it 0.428 something, in radians."},{"Start":"01:47.000 ","End":"01:49.070","Text":"If you want it in degrees,"},{"Start":"01:49.070 ","End":"01:52.440","Text":"I\u0027ll give it to you, I\u0027ll make it roughly, I don\u0027t know,"},{"Start":"01:52.440 ","End":"01:55.370","Text":"64.6 something in degrees and even,"},{"Start":"01:55.370 ","End":"01:59.930","Text":"we\u0027ll use radians and we\u0027ll write our answer in the correct order Rho,"},{"Start":"01:59.930 ","End":"02:01.895","Text":"then Theta then Phi."},{"Start":"02:01.895 ","End":"02:07.325","Text":"Root 7, then 1.23,"},{"Start":"02:07.325 ","End":"02:14.370","Text":"and then 0.428. That\u0027s it."}],"ID":9737},{"Watched":false,"Name":"Exercise 9","Duration":"1m 28s","ChapterTopicVideoID":9852,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9852.jpeg","UploadDate":"2017-08-06T05:14:56.4330000","DurationForVideoObject":"PT1M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we have to convert the following equation,"},{"Start":"00:03.660 ","End":"00:09.600","Text":"which is in Cartesian coordinates to an equation in spherical coordinates."},{"Start":"00:09.600 ","End":"00:14.640","Text":"Now, we can just blindly do it using the formula x equals y equals,"},{"Start":"00:14.640 ","End":"00:19.335","Text":"and z equals, but we could see if there\u0027s anything shorter."},{"Start":"00:19.335 ","End":"00:24.225","Text":"We know that x squared plus y squared plus z squared is Rho squared,"},{"Start":"00:24.225 ","End":"00:32.760","Text":"so I\u0027m going to write this as x squared plus y squared plus z squared,"},{"Start":"00:32.760 ","End":"00:38.760","Text":"and then I\u0027m going to write minus 2z squared equals 0,"},{"Start":"00:38.760 ","End":"00:41.535","Text":"and then I haven\u0027t changed anything,"},{"Start":"00:41.535 ","End":"00:45.555","Text":"but these now together, Rho squared."},{"Start":"00:45.555 ","End":"00:48.105","Text":"So I have Rho squared."},{"Start":"00:48.105 ","End":"00:57.770","Text":"Also, I mean z"},{"Start":"00:57.770 ","End":"01:01.475","Text":"is Rho cosine Phi,"},{"Start":"01:01.475 ","End":"01:03.155","Text":"one of the formulas,"},{"Start":"01:03.155 ","End":"01:05.210","Text":"so this is what we have,"},{"Start":"01:05.210 ","End":"01:09.120","Text":"and we can simplify it a bit,"},{"Start":"01:09.120 ","End":"01:10.685","Text":"if you open the brackets,"},{"Start":"01:10.685 ","End":"01:13.490","Text":"you could take Rho squared outside,"},{"Start":"01:13.490 ","End":"01:19.610","Text":"and then you\u0027d get 1 minus 2 cosine squared"},{"Start":"01:19.610 ","End":"01:28.350","Text":"Phi equals 0, and that\u0027s it."}],"ID":9738},{"Watched":false,"Name":"Exercise 10","Duration":"1m 28s","ChapterTopicVideoID":9853,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9853.jpeg","UploadDate":"2017-08-06T05:15:00.2370000","DurationForVideoObject":"PT1M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have an equation in"},{"Start":"00:03.360 ","End":"00:08.085","Text":"Spherical coordinates and we want to convert it to Cartesian."},{"Start":"00:08.085 ","End":"00:13.335","Text":"Let\u0027s look at it for a moment and it should be ringing some bells."},{"Start":"00:13.335 ","End":"00:17.490","Text":"For example, if there wasn\u0027t the squareds here,"},{"Start":"00:17.490 ","End":"00:23.050","Text":"this is exactly the formula for y."},{"Start":"00:25.250 ","End":"00:34.595","Text":"Let me just write it as Rho sine Phi, sine Theta squared."},{"Start":"00:34.595 ","End":"00:39.365","Text":"The other 1 is also a formula for Rho cosine Phi."},{"Start":"00:39.365 ","End":"00:43.410","Text":"Remember that\u0027s equal to z."},{"Start":"00:44.690 ","End":"00:56.150","Text":"This is y, so I get y squared plus z squared equals 16,"},{"Start":"00:56.150 ","End":"00:58.880","Text":"which I\u0027m going to write as 4 squared."},{"Start":"00:58.880 ","End":"01:00.170","Text":"We can stop here."},{"Start":"01:00.170 ","End":"01:05.899","Text":"But I just wanted to give you an idea of what this equation is."},{"Start":"01:05.899 ","End":"01:13.050","Text":"It\u0027s a cylinder and if I take the y, z,"},{"Start":"01:13.050 ","End":"01:15.090","Text":"this is y and this is z,"},{"Start":"01:15.090 ","End":"01:18.650","Text":"it\u0027s a circle of radius 2,"},{"Start":"01:18.650 ","End":"01:19.700","Text":"but it\u0027s not a circle,"},{"Start":"01:19.700 ","End":"01:22.250","Text":"it\u0027s a cylinder because x doesn\u0027t appear here."},{"Start":"01:22.250 ","End":"01:24.170","Text":"It extends indefinitely up and down,"},{"Start":"01:24.170 ","End":"01:25.445","Text":"so it\u0027s a cylinder."},{"Start":"01:25.445 ","End":"01:28.380","Text":"Anyway, this is the answer."}],"ID":9739},{"Watched":false,"Name":"Exercise 11","Duration":"1m 48s","ChapterTopicVideoID":9854,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9854.jpeg","UploadDate":"2017-08-06T05:15:04.5930000","DurationForVideoObject":"PT1M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.955","Text":"In this exercise, we have to identify the surface generated by the spherical equation."},{"Start":"00:05.955 ","End":"00:08.525","Text":"Pi equals 3Pi over 4."},{"Start":"00:08.525 ","End":"00:18.060","Text":"Notice that Rho and Theta are not in the equation."},{"Start":"00:18.060 ","End":"00:20.400","Text":"They can be anything we want."},{"Start":"00:20.400 ","End":"00:23.910","Text":"Now, I\u0027ll try and do a sketch for this, although it\u0027s 3D,"},{"Start":"00:23.910 ","End":"00:31.850","Text":"I\u0027ll just take a 2D sketch where this is the z and this would be the x-y plane."},{"Start":"00:31.850 ","End":"00:35.465","Text":"Either, you can think of it as the x-axis or the y-axis."},{"Start":"00:35.465 ","End":"00:38.450","Text":"Now, we have an angle of 3Pi over 4,"},{"Start":"00:38.450 ","End":"00:40.040","Text":"with the positive z-axis."},{"Start":"00:40.040 ","End":"00:42.400","Text":"We need to take an angle of 3Pi over 4."},{"Start":"00:42.400 ","End":"00:46.185","Text":"This is 135 degrees."},{"Start":"00:46.185 ","End":"00:50.590","Text":"If we rotate 135 degrees,"},{"Start":"00:50.590 ","End":"00:56.130","Text":"we\u0027ll get to here."},{"Start":"00:56.130 ","End":"00:57.530","Text":"Now, Rho can be anything,"},{"Start":"00:57.530 ","End":"01:01.010","Text":"so all of these points are okay."},{"Start":"01:01.010 ","End":"01:02.945","Text":"Theta can be anything."},{"Start":"01:02.945 ","End":"01:05.600","Text":"I\u0027m going to try and draw a little bit of a 3D thing."},{"Start":"01:05.600 ","End":"01:09.030","Text":"I\u0027ll draw the opposite 1 on the other side."},{"Start":"01:10.100 ","End":"01:16.520","Text":"If you think about it, if we rotate this line or this ray,"},{"Start":"01:16.520 ","End":"01:21.230","Text":"that it always makes a 135 degrees with the positive z-axis,"},{"Start":"01:21.230 ","End":"01:24.095","Text":"then what we\u0027ll get is a cone with"},{"Start":"01:24.095 ","End":"01:30.540","Text":"the vertex at the origin and facing downwards, opens downwards."},{"Start":"01:30.700 ","End":"01:37.370","Text":"Let us say that it\u0027s a cone facing down with"},{"Start":"01:37.370 ","End":"01:43.985","Text":"the vertex at the origin and centered along the z-axis."},{"Start":"01:43.985 ","End":"01:48.240","Text":"I\u0027m not going to put that in words. Okay, we\u0027re done."}],"ID":9740},{"Watched":false,"Name":"Exercise 12","Duration":"2m 21s","ChapterTopicVideoID":9855,"CourseChapterTopicPlaylistID":57485,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9855.jpeg","UploadDate":"2017-08-06T05:15:10.2900000","DurationForVideoObject":"PT2M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.030 ","End":"00:06.190","Text":"In this exercise, we have to identify the surface generated by the spherical equation,"},{"Start":"00:06.190 ","End":"00:09.570","Text":"Rho equals 4 sine Phi sine Theta."},{"Start":"00:09.570 ","End":"00:11.825","Text":"After staring at it a bit,"},{"Start":"00:11.825 ","End":"00:15.820","Text":"we realized the thing to do is to multiply both sides by Rho."},{"Start":"00:15.820 ","End":"00:20.560","Text":"The reason is we have a formula for Rho sine Phi sine Theta as just y,"},{"Start":"00:20.560 ","End":"00:23.480","Text":"and we also have an expression for Rho squared,"},{"Start":"00:23.480 ","End":"00:25.710","Text":"which is, I\u0027ll write it."},{"Start":"00:25.710 ","End":"00:29.635","Text":"Let\u0027s write it as Rho squared equals 4"},{"Start":"00:29.635 ","End":"00:36.870","Text":"Rho sine Phi sine Theta."},{"Start":"00:36.870 ","End":"00:39.625","Text":"Now, if you look at the formulas,"},{"Start":"00:39.625 ","End":"00:44.240","Text":"you\u0027ll see that Rho squared is x squared plus y"},{"Start":"00:44.240 ","End":"00:49.010","Text":"squared plus z squared in Cartesian coordinates."},{"Start":"00:49.010 ","End":"00:52.910","Text":"I\u0027m converting this to Cartesian by the way, I should have said."},{"Start":"00:52.910 ","End":"00:56.180","Text":"It\u0027ll identify it more readily."},{"Start":"00:56.180 ","End":"00:58.730","Text":"Then on the right-hand side,"},{"Start":"00:58.730 ","End":"01:03.390","Text":"we have 4, and this is exactly y."},{"Start":"01:04.750 ","End":"01:08.135","Text":"Don\u0027t immediately see what surface this is,"},{"Start":"01:08.135 ","End":"01:11.555","Text":"but I\u0027m going to show you that it\u0027s a sphere."},{"Start":"01:11.555 ","End":"01:14.090","Text":"Let\u0027s just collect the y together."},{"Start":"01:14.090 ","End":"01:19.545","Text":"We have x squared plus y squared minus 4y,"},{"Start":"01:19.545 ","End":"01:22.520","Text":"and I\u0027m going to leave a gap for completing the square,"},{"Start":"01:22.520 ","End":"01:25.475","Text":"plus z squared equals."},{"Start":"01:25.475 ","End":"01:27.925","Text":"Now, this would be 0,"},{"Start":"01:27.925 ","End":"01:30.680","Text":"but I\u0027m going to add something to both sides."},{"Start":"01:30.680 ","End":"01:32.975","Text":"To make this a perfect square,"},{"Start":"01:32.975 ","End":"01:34.690","Text":"I would add 4."},{"Start":"01:34.690 ","End":"01:37.200","Text":"Then it\u0027s going to be y minus 2 squared."},{"Start":"01:37.200 ","End":"01:40.755","Text":"I have to add 4 to the right-hand side also."},{"Start":"01:40.755 ","End":"01:42.660","Text":"Now I\u0027ve got x,"},{"Start":"01:42.660 ","End":"01:45.960","Text":"and just for emphasis I\u0027m going to show you this is a sphere."},{"Start":"01:45.960 ","End":"01:48.975","Text":"I\u0027ll write it as x minus 0 squared."},{"Start":"01:48.975 ","End":"01:53.040","Text":"Here I\u0027ve got y minus 2 squared and z"},{"Start":"01:53.040 ","End":"01:57.440","Text":"squared also just to emphasize I\u0027ll write it as z minus 0 squared,"},{"Start":"01:57.440 ","End":"02:00.230","Text":"and the 4 I\u0027ll write as 2 squared."},{"Start":"02:00.230 ","End":"02:09.769","Text":"What we\u0027ve got is a sphere and the radius is 2,"},{"Start":"02:09.769 ","End":"02:18.870","Text":"and the center is 0, 2, 0."},{"Start":"02:18.870 ","End":"02:21.630","Text":"We\u0027ve identified it and we\u0027re done."}],"ID":9741}],"Thumbnail":null,"ID":57485},{"Name":"Integrals of Vector Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"10m 16s","ChapterTopicVideoID":10192,"CourseChapterTopicPlaylistID":57486,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10192.jpeg","UploadDate":"2017-08-22T04:12:12.4300000","DurationForVideoObject":"PT10M16S","Description":null,"VideoComments":[],"Subtitles":[],"ID":10520}],"Thumbnail":null,"ID":57486}]

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