Introduction to 2D and 3D Vectors
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Vector Arithmetic
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Vectors Dot Product
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Vectors Cross Product
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The 3D Coordinates System
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Equations of Lines
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Equations of Planes
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Quadric Surfaces
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[{"Name":"Introduction to 2D and 3D Vectors","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors (mostly 2D and 3D)","Duration":"27m 41s","ChapterTopicVideoID":10295,"CourseChapterTopicPlaylistID":12288,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Starting a new topic,"},{"Start":"00:01.830 ","End":"00:05.235","Text":"the 1 of the concept vectors,"},{"Start":"00:05.235 ","End":"00:08.745","Text":"which are mostly applied in physics,"},{"Start":"00:08.745 ","End":"00:13.260","Text":"but they\u0027re used also abstractly in mathematics and they are used to"},{"Start":"00:13.260 ","End":"00:18.270","Text":"represent quantities that have both a magnitude,"},{"Start":"00:18.270 ","End":"00:26.070","Text":"magnitude is the size and direction,"},{"Start":"00:26.070 ","End":"00:28.455","Text":"both of these things."},{"Start":"00:28.455 ","End":"00:33.435","Text":"Examples which I will bring from physics,"},{"Start":"00:33.435 ","End":"00:36.210","Text":"1 example is force,"},{"Start":"00:36.210 ","End":"00:40.350","Text":"which doesn\u0027t just have a magnitude measured in newtons,"},{"Start":"00:40.350 ","End":"00:42.145","Text":"but it also has a direction."},{"Start":"00:42.145 ","End":"00:46.730","Text":"The other example would be velocity."},{"Start":"00:46.730 ","End":"00:55.205","Text":"Velocity, as opposed to speed, has a direction."},{"Start":"00:55.205 ","End":"01:02.045","Text":"Speed is just how many miles per hour or kilometers per hour you\u0027re going,"},{"Start":"01:02.045 ","End":"01:05.925","Text":"say 60 kilometers an hour,"},{"Start":"01:05.925 ","End":"01:07.415","Text":"the velocity would be"},{"Start":"01:07.415 ","End":"01:12.110","Text":"maybe 60 kilometers an hour going North and the same thing with force."},{"Start":"01:12.110 ","End":"01:15.020","Text":"Let me show you a diagram."},{"Start":"01:15.020 ","End":"01:21.080","Text":"Typically a vector is represented with an arrow of a certain length and going in"},{"Start":"01:21.080 ","End":"01:28.175","Text":"a certain direction and all these vectors would be considered to be the same vector."},{"Start":"01:28.175 ","End":"01:32.110","Text":"Each of them, if you look, would go,"},{"Start":"01:32.110 ","End":"01:40.325","Text":"that\u0027s 2 units to the left and 5 units up the same thing."},{"Start":"01:40.325 ","End":"01:44.210","Text":"They\u0027re all parallel and they all have the same size."},{"Start":"01:44.210 ","End":"01:45.650","Text":"These are all considered to be"},{"Start":"01:45.650 ","End":"01:50.150","Text":"the same vector and the position at which they are applied,"},{"Start":"01:50.150 ","End":"01:52.625","Text":"like here or here,"},{"Start":"01:52.625 ","End":"01:56.405","Text":"or here, or here, is not significant."},{"Start":"01:56.405 ","End":"01:59.180","Text":"If it\u0027s a force, it\u0027s just the size of the force."},{"Start":"01:59.180 ","End":"02:02.465","Text":"It might be 5 newtons in this direction."},{"Start":"02:02.465 ","End":"02:04.010","Text":"If it\u0027s a velocity,"},{"Start":"02:04.010 ","End":"02:09.020","Text":"it could be 80 kilometers an hour in this direction,"},{"Start":"02:09.020 ","End":"02:11.000","Text":"doesn\u0027t matter where the car is."},{"Start":"02:11.000 ","End":"02:16.655","Text":"The only thing that matters is the speed and direction,"},{"Start":"02:16.655 ","End":"02:19.550","Text":"which is called LSA velocity."},{"Start":"02:19.550 ","End":"02:25.115","Text":"This is an example of vectors and this is in 2-dimensions."},{"Start":"02:25.115 ","End":"02:29.230","Text":"There is also such a concept in 3-dimensions,"},{"Start":"02:29.230 ","End":"02:32.900","Text":"It\u0027s exactly the same magnitude and direction,"},{"Start":"02:32.900 ","End":"02:35.405","Text":"but in 3D, but we often use 2D,"},{"Start":"02:35.405 ","End":"02:36.760","Text":"especially with drawings,"},{"Start":"02:36.760 ","End":"02:40.910","Text":"it\u0027s easier to draw diagrams in 2-dimensions."},{"Start":"02:40.910 ","End":"02:48.250","Text":"Graphically, vectors are represented as what we call a directed line segment."},{"Start":"02:48.250 ","End":"02:50.345","Text":"When we have a picture like this,"},{"Start":"02:50.345 ","End":"02:55.025","Text":"each of these is called a representation of the vector."},{"Start":"02:55.025 ","End":"02:56.404","Text":"They\u0027re all the same."},{"Start":"02:56.404 ","End":"03:01.310","Text":"Each 1 of them could be a representation of the same vector."},{"Start":"03:01.310 ","End":"03:06.245","Text":"The way we write the vector numerically not graphically,"},{"Start":"03:06.245 ","End":"03:10.185","Text":"the notation is something like,"},{"Start":"03:10.185 ","End":"03:12.400","Text":"you take a left up, often V,"},{"Start":"03:12.400 ","End":"03:16.160","Text":"and put a little arrow above it and when you see the arrow above it,"},{"Start":"03:16.160 ","End":"03:18.980","Text":"it means it\u0027s a vector, not a number."},{"Start":"03:18.980 ","End":"03:23.385","Text":"Numbers are also called scalars. We\u0027ll get to that."},{"Start":"03:23.385 ","End":"03:27.165","Text":"A vector, this might be a vector V,"},{"Start":"03:27.165 ","End":"03:29.760","Text":"and this would be the same V,"},{"Start":"03:29.760 ","End":"03:33.820","Text":"they\u0027re all the same vector."},{"Start":"03:35.810 ","End":"03:40.415","Text":"The way we specifically say what this vector is,"},{"Start":"03:40.415 ","End":"03:45.920","Text":"is we do it in like an x,"},{"Start":"03:45.920 ","End":"03:51.364","Text":"y coordinate system, but the number of units"},{"Start":"03:51.364 ","End":"03:57.290","Text":"we go in the x-direction and the x-direction positively goes here,"},{"Start":"03:57.290 ","End":"04:01.170","Text":"and the y-direction positively goes here."},{"Start":"04:02.180 ","End":"04:04.845","Text":"The x and y,"},{"Start":"04:04.845 ","End":"04:08.745","Text":"we say this 1 is negative 2,"},{"Start":"04:08.745 ","End":"04:11.310","Text":"they all go 2 to the left,"},{"Start":"04:11.310 ","End":"04:12.840","Text":"which means negative 2,"},{"Start":"04:12.840 ","End":"04:14.490","Text":"and they go up 5,"},{"Start":"04:14.490 ","End":"04:16.230","Text":"which is a positive 5,"},{"Start":"04:16.230 ","End":"04:19.760","Text":"and we use angular brackets to distinguish"},{"Start":"04:19.760 ","End":"04:27.115","Text":"from the point minus 2, 5."},{"Start":"04:27.115 ","End":"04:30.830","Text":"Now it turns out that if you start"},{"Start":"04:30.830 ","End":"04:35.610","Text":"this vector and place it so that 1 end is at the origin,"},{"Start":"04:35.930 ","End":"04:44.945","Text":"then the other end is going to be the point minus 2, 5."},{"Start":"04:44.945 ","End":"04:48.265","Text":"But there\u0027s a difference between the point."},{"Start":"04:48.265 ","End":"04:52.970","Text":"This is a vector and if I just say minus 2,"},{"Start":"04:52.970 ","End":"04:55.790","Text":"5, then it\u0027s a point."},{"Start":"04:55.790 ","End":"05:00.830","Text":"But this is called the position vector for this point."},{"Start":"05:00.830 ","End":"05:04.790","Text":"When you join the origin to a point,"},{"Start":"05:04.790 ","End":"05:08.000","Text":"that vector is the position vector of the point."},{"Start":"05:08.000 ","End":"05:10.790","Text":"It basically tells you how to get there,"},{"Start":"05:10.790 ","End":"05:15.545","Text":"what magnitude and direction to get from the origin to that point."},{"Start":"05:15.545 ","End":"05:18.560","Text":"There is a relation between the 2."},{"Start":"05:18.560 ","End":"05:26.015","Text":"The same numbers appear in both the point and the position vector for the point,"},{"Start":"05:26.015 ","End":"05:32.600","Text":"would probably put that under examples of a vector as something called a position vector."},{"Start":"05:32.600 ","End":"05:37.530","Text":"Just want to have it written so the concept has been introduced."},{"Start":"05:37.810 ","End":"05:41.330","Text":"Also note that when you take a representation of a vector,"},{"Start":"05:41.330 ","End":"05:44.270","Text":"let\u0027s choose this 1 and you take the 2 endpoints."},{"Start":"05:44.270 ","End":"05:48.385","Text":"Let\u0027s say this is point A and this is point B,"},{"Start":"05:48.385 ","End":"05:51.975","Text":"A would be the point in this case."},{"Start":"05:51.975 ","End":"05:58.535","Text":"Let\u0027s see, the x of it would be minus 3 and the why of it would be minus 4."},{"Start":"05:58.535 ","End":"06:06.360","Text":"B would be the point minus 5, 1."},{"Start":"06:06.360 ","End":"06:09.995","Text":"I\u0027d like to point out the relationship between"},{"Start":"06:09.995 ","End":"06:14.705","Text":"the head of the vector and the vector itself."},{"Start":"06:14.705 ","End":"06:21.565","Text":"The vector we said was minus 2, 5."},{"Start":"06:21.565 ","End":"06:26.345","Text":"Note that if I take my minus 3,"},{"Start":"06:26.345 ","End":"06:32.070","Text":"minus 4 and I add the minus 2,"},{"Start":"06:32.070 ","End":"06:34.410","Text":"5 to it, that minus 2,"},{"Start":"06:34.410 ","End":"06:36.195","Text":"5, I\u0027m just going to write that here,"},{"Start":"06:36.195 ","End":"06:38.350","Text":"minus 2, 5."},{"Start":"06:38.480 ","End":"06:44.660","Text":"Coordinate-wise, what I\u0027ll get is minus 3, minus 2,"},{"Start":"06:44.660 ","End":"06:52.440","Text":"I\u0027m adding negative 2 and then minus 4 plus 5,"},{"Start":"06:52.520 ","End":"06:56.370","Text":"which is just equal to,"},{"Start":"06:56.370 ","End":"07:02.010","Text":"minus 3 minus 2 is minus 5, 1."},{"Start":"07:02.010 ","End":"07:07.370","Text":"In other words, if I have point A at the tail of the vector and point B at the head of"},{"Start":"07:07.370 ","End":"07:13.680","Text":"the vector and this is the vector that goes from here to here,"},{"Start":"07:13.680 ","End":"07:19.009","Text":"we can get the coordinates of the tail from the head."},{"Start":"07:19.009 ","End":"07:22.935","Text":"We could get any 1 of these 3 from the other 2."},{"Start":"07:22.935 ","End":"07:24.875","Text":"If we\u0027re given the tail and the head,"},{"Start":"07:24.875 ","End":"07:26.600","Text":"we could subtract minus 5,"},{"Start":"07:26.600 ","End":"07:28.715","Text":"takeaway minus 3 is minus 2."},{"Start":"07:28.715 ","End":"07:34.540","Text":"In some ways, the tail plus the vector is the head as far as numbers go,"},{"Start":"07:34.540 ","End":"07:37.190","Text":"they\u0027re coordinates, but they are different entities,"},{"Start":"07:37.190 ","End":"07:39.680","Text":"these 2 are points and this is a vector."},{"Start":"07:39.680 ","End":"07:41.590","Text":"But the math works that way."},{"Start":"07:41.590 ","End":"07:48.470","Text":"In general, if I have a vector that goes from the point x,"},{"Start":"07:48.470 ","End":"07:50.780","Text":"y, and the vector,"},{"Start":"07:50.780 ","End":"07:52.520","Text":"this is 1 point,"},{"Start":"07:52.520 ","End":"07:55.195","Text":"let\u0027s say this is my A,"},{"Start":"07:55.195 ","End":"08:02.090","Text":"and I want to go to a point B and the vector that takes me there is,"},{"Start":"08:02.090 ","End":"08:05.540","Text":"let\u0027s say a_1, a_2."},{"Start":"08:05.540 ","End":"08:11.235","Text":"Then the B will be x"},{"Start":"08:11.235 ","End":"08:17.220","Text":"plus a_1 and y plus a_2."},{"Start":"08:17.220 ","End":"08:20.150","Text":"I should really put a bar at least or the whole vector,"},{"Start":"08:20.150 ","End":"08:24.350","Text":"sometimes just the bar sometimes you forget the thing altogether."},{"Start":"08:24.350 ","End":"08:29.225","Text":"Now so far we\u0027ve just talked about 2D, 2-dimensional."},{"Start":"08:29.225 ","End":"08:32.180","Text":"But the same thing works in 3D."},{"Start":"08:32.180 ","End":"08:39.465","Text":"In 3D, we would have a point would be 3 coordinates,"},{"Start":"08:39.465 ","End":"08:43.910","Text":"x, y, and z and even if you hadn\u0027t studied this,"},{"Start":"08:43.910 ","End":"08:46.790","Text":"you can imagine just generalizing from x and y."},{"Start":"08:46.790 ","End":"08:50.330","Text":"We also have a height and we have"},{"Start":"08:50.330 ","End":"09:01.300","Text":"a vector V and this time the vector V will have 3 components,"},{"Start":"09:01.580 ","End":"09:05.310","Text":"a_1, a_2, a_3,"},{"Start":"09:05.310 ","End":"09:10.039","Text":"and this will take us from point A to point B,"},{"Start":"09:10.039 ","End":"09:13.700","Text":"which will be x plus first coordinate,"},{"Start":"09:13.700 ","End":"09:17.365","Text":"y plus the second coordinate,"},{"Start":"09:17.365 ","End":"09:27.240","Text":"and I\u0027m running out of space here, z plus a_3."},{"Start":"09:28.050 ","End":"09:31.390","Text":"This concept works the other way round."},{"Start":"09:31.390 ","End":"09:33.565","Text":"Also given 2 points,"},{"Start":"09:33.565 ","End":"09:37.420","Text":"I can find the vector that takes me from 1 to the other."},{"Start":"09:37.420 ","End":"09:42.910","Text":"Suppose I had a point A which was,"},{"Start":"09:42.910 ","End":"09:45.940","Text":"let\u0027s say, I\u0027ll call it a1,"},{"Start":"09:45.940 ","End":"09:50.650","Text":"a2, a3 and I have a point B,"},{"Start":"09:50.650 ","End":"09:55.645","Text":"which is call it say b1, b2, b3."},{"Start":"09:55.645 ","End":"09:58.855","Text":"I want to know the vector that takes me from here to here."},{"Start":"09:58.855 ","End":"10:01.600","Text":"First of all, the name for that vector,"},{"Start":"10:01.600 ","End":"10:05.455","Text":"it\u0027s called AB with an arrow on top."},{"Start":"10:05.455 ","End":"10:08.230","Text":"The order is important as we\u0027ll soon see."},{"Start":"10:08.230 ","End":"10:14.395","Text":"This will equal the second coordinate minus the first coordinate."},{"Start":"10:14.395 ","End":"10:21.955","Text":"It will be this minus this and then this minus this,"},{"Start":"10:21.955 ","End":"10:23.680","Text":"b2 minus a2,"},{"Start":"10:23.680 ","End":"10:27.745","Text":"and finally b3 minus a3."},{"Start":"10:27.745 ","End":"10:32.665","Text":"If we went the other direction from B to A,"},{"Start":"10:32.665 ","End":"10:36.830","Text":"let\u0027s suppose I want to know what is BA,"},{"Start":"10:36.830 ","End":"10:38.820","Text":"give it another name,"},{"Start":"10:38.820 ","End":"10:43.425","Text":"say w, this would be the other way."},{"Start":"10:43.425 ","End":"10:48.330","Text":"This minus this, this is actually going to be a1 minus b1,"},{"Start":"10:48.330 ","End":"10:55.960","Text":"a2 minus b2, and a3 minus b3."},{"Start":"10:55.960 ","End":"10:58.330","Text":"This is actually the negatives of this."},{"Start":"10:58.330 ","End":"10:59.710","Text":"For example in here,"},{"Start":"10:59.710 ","End":"11:03.550","Text":"if I took in 2-dimensions let\u0027s say,"},{"Start":"11:03.550 ","End":"11:06.850","Text":"w was the opposite of this,"},{"Start":"11:06.850 ","End":"11:11.215","Text":"plus 2 and minus 5,"},{"Start":"11:11.215 ","End":"11:17.395","Text":"then it would take me the other way from B to A."},{"Start":"11:17.395 ","End":"11:21.580","Text":"You can check minus 3 takeaway minus 5 is 2,"},{"Start":"11:21.580 ","End":"11:23.905","Text":"minus 4 minus 1,"},{"Start":"11:23.905 ","End":"11:29.755","Text":"minus 5 or you could say that minus 5 plus 2 is minus 3 and so on."},{"Start":"11:29.755 ","End":"11:32.335","Text":"You get the opposite signs."},{"Start":"11:32.335 ","End":"11:34.615","Text":"It\u0027s important the order,"},{"Start":"11:34.615 ","End":"11:38.245","Text":"the vector from A to B opposite from the vector from B to A."},{"Start":"11:38.245 ","End":"11:41.890","Text":"It\u0027s the second minus the first when you do the coordinates,"},{"Start":"11:41.890 ","End":"11:44.260","Text":"so don\u0027t get it backwards."},{"Start":"11:44.260 ","End":"11:48.745","Text":"Also, note that this works fine with the position vector."},{"Start":"11:48.745 ","End":"11:50.110","Text":"Like in this example,"},{"Start":"11:50.110 ","End":"11:53.365","Text":"position vector is the vector from the origin to the point."},{"Start":"11:53.365 ","End":"11:56.230","Text":"If I take 0,"},{"Start":"11:56.230 ","End":"11:58.450","Text":"0 and add minus 2,"},{"Start":"11:58.450 ","End":"12:01.465","Text":"5, I just get minus 2, 5."},{"Start":"12:01.465 ","End":"12:05.370","Text":"For position vectors, the vector and"},{"Start":"12:05.370 ","End":"12:09.300","Text":"the point at the head of the vector actually have the same coordinates,"},{"Start":"12:09.300 ","End":"12:10.965","Text":"but ones with a round brackets,"},{"Start":"12:10.965 ","End":"12:12.300","Text":"ones with angular brackets,"},{"Start":"12:12.300 ","End":"12:13.830","Text":"and the different entities."},{"Start":"12:13.830 ","End":"12:16.615","Text":"Let\u0027s do just some exercises."},{"Start":"12:16.615 ","End":"12:21.835","Text":"I\u0027ll show you the exercises you might get at this very basic level."},{"Start":"12:21.835 ","End":"12:24.130","Text":"In this example exercise,"},{"Start":"12:24.130 ","End":"12:31.280","Text":"I\u0027m going to ask you to find the following vectors."},{"Start":"12:31.770 ","End":"12:35.845","Text":"Let\u0027s do 3 examples that give you a, b,"},{"Start":"12:35.845 ","End":"12:39.580","Text":"and c. In a,"},{"Start":"12:39.580 ","End":"12:45.355","Text":"I\u0027ll ask you to find the vector that takes me from the point,"},{"Start":"12:45.355 ","End":"12:47.650","Text":"just make up some numbers, 1, 2,"},{"Start":"12:47.650 ","End":"12:50.830","Text":"3 to the point,"},{"Start":"12:50.830 ","End":"12:56.720","Text":"minus 7, 4, 5."},{"Start":"12:57.510 ","End":"13:05.485","Text":"In part B, I\u0027ll give a 2-dimensional example."},{"Start":"13:05.485 ","End":"13:10.165","Text":"The vector from minus 2, 7,"},{"Start":"13:10.165 ","End":"13:16.010","Text":"2, 1, minus 5."},{"Start":"13:16.050 ","End":"13:21.895","Text":"In the last 1, I want you to give me a position vector"},{"Start":"13:21.895 ","End":"13:32.420","Text":"for the 3-dimensional point minus 5, 9, 20."},{"Start":"13:34.890 ","End":"13:39.775","Text":"I\u0027ll solve them. When we take a vector from here to here,"},{"Start":"13:39.775 ","End":"13:42.850","Text":"remember we do the second minus the first."},{"Start":"13:42.850 ","End":"13:46.075","Text":"What we do is minus 7,"},{"Start":"13:46.075 ","End":"13:47.950","Text":"take away 1,"},{"Start":"13:47.950 ","End":"13:55.675","Text":"then second coordinate 4 takeaway 2 and 5 takeaway 3."},{"Start":"13:55.675 ","End":"13:58.105","Text":"In angular brackets,"},{"Start":"13:58.105 ","End":"13:59.710","Text":"of course, you do the calculation,"},{"Start":"13:59.710 ","End":"14:07.455","Text":"so it comes out to be minus 8, 2, 2."},{"Start":"14:07.455 ","End":"14:09.150","Text":"That would be the answer."},{"Start":"14:09.150 ","End":"14:11.850","Text":"The same thing in 2-dimensions."},{"Start":"14:11.850 ","End":"14:15.825","Text":"We\u0027ll take the second minus the first."},{"Start":"14:15.825 ","End":"14:24.580","Text":"1 takeaway, minus 2 and minus 5, takeaway 7."},{"Start":"14:24.580 ","End":"14:27.955","Text":"The answer is the 2-dimensional vector."},{"Start":"14:27.955 ","End":"14:33.700","Text":"3, minus 5,"},{"Start":"14:33.700 ","End":"14:36.695","Text":"minus 7 is minus 12."},{"Start":"14:36.695 ","End":"14:38.940","Text":"The last 1 is the easiest."},{"Start":"14:38.940 ","End":"14:42.270","Text":"The position vector for a point is from the origin to the point."},{"Start":"14:42.270 ","End":"14:44.640","Text":"We take away 0, 0, 0."},{"Start":"14:44.640 ","End":"14:48.960","Text":"Basically, what you do is you just get the vector with the same numbers."},{"Start":"14:48.960 ","End":"14:50.930","Text":"We saw this before."},{"Start":"14:50.930 ","End":"14:53.890","Text":"The same numbers but a different interpretation."},{"Start":"14:53.890 ","End":"14:59.080","Text":"This is a point, and this is the vector that takes me from the origin to this point."},{"Start":"14:59.080 ","End":"15:01.375","Text":"That\u0027s a distinction."},{"Start":"15:01.375 ","End":"15:04.090","Text":"Let\u0027s move on a bit."},{"Start":"15:04.090 ","End":"15:07.030","Text":"I\u0027m going to talk about the magnitude of a vector."},{"Start":"15:07.030 ","End":"15:10.090","Text":"We\u0027ve seen various examples of vectors."},{"Start":"15:10.090 ","End":"15:12.895","Text":"Remember the beginning and I\u0027m going to go back up."},{"Start":"15:12.895 ","End":"15:16.585","Text":"We said that the vector has magnitude and direction."},{"Start":"15:16.585 ","End":"15:21.910","Text":"Well, I want to focus on the magnitude of a vector."},{"Start":"15:21.910 ","End":"15:24.100","Text":"I\u0027m going to erase what I don\u0027t need."},{"Start":"15:24.100 ","End":"15:29.980","Text":"I\u0027m going to introduce a notation and a formula to compute the magnitude of a vector."},{"Start":"15:29.980 ","End":"15:33.160","Text":"For example, in a 2D case,"},{"Start":"15:33.160 ","End":"15:36.355","Text":"I might have a vector minus 3,"},{"Start":"15:36.355 ","End":"15:39.715","Text":"4, and I want to know what its magnitude is."},{"Start":"15:39.715 ","End":"15:42.010","Text":"Or in 3-dimensions,"},{"Start":"15:42.010 ","End":"15:45.010","Text":"I might have a vector,"},{"Start":"15:45.010 ","End":"15:51.860","Text":"let\u0027s say 3, 4, 12."},{"Start":"15:51.870 ","End":"15:56.270","Text":"I want to know what it\u0027s magnitude is."},{"Start":"15:56.730 ","End":"16:04.735","Text":"I\u0027m going to show you something about a generalized Pythagoras theorem,"},{"Start":"16:04.735 ","End":"16:07.480","Text":"which will help us with all this."},{"Start":"16:07.480 ","End":"16:10.105","Text":"Let\u0027s just leave the vectors for a moment."},{"Start":"16:10.105 ","End":"16:14.110","Text":"Now, Pythagoras\u0027 theorem in 2-dimensions,"},{"Start":"16:14.110 ","End":"16:17.500","Text":"if we just look at the base of this box,"},{"Start":"16:17.500 ","End":"16:20.185","Text":"this is 90 degrees,"},{"Start":"16:20.185 ","End":"16:26.095","Text":"then what Pythagoras\u0027 theorem says is that the hypotenuse in this case c,"},{"Start":"16:26.095 ","End":"16:30.940","Text":"will be given by the formula that c squared is x squared plus y squared."},{"Start":"16:30.940 ","End":"16:34.765","Text":"The hypotenuse squared is the sum of the squares on the other 2 sides."},{"Start":"16:34.765 ","End":"16:36.460","Text":"Now, suppose I have something."},{"Start":"16:36.460 ","End":"16:42.325","Text":"In 3-dimensions, I have this line here inside a box."},{"Start":"16:42.325 ","End":"16:45.144","Text":"It\u0027s called an x, a y,"},{"Start":"16:45.144 ","End":"16:47.560","Text":"and it\u0027s z up."},{"Start":"16:47.560 ","End":"16:50.560","Text":"Turns out that you can just extend"},{"Start":"16:50.560 ","End":"16:53.920","Text":"this formula and say that the square of this is the sums of"},{"Start":"16:53.920 ","End":"17:00.010","Text":"the squares of all the 3 sides but the dimensions."},{"Start":"17:00.010 ","End":"17:01.960","Text":"The reason for this is,"},{"Start":"17:01.960 ","End":"17:04.165","Text":"well, they\u0027ve proved it here basically,"},{"Start":"17:04.165 ","End":"17:09.670","Text":"is that you can use Pythagoras on this triangle here and get the z squared"},{"Start":"17:09.670 ","End":"17:15.010","Text":"plus c squared is s squared but c squared from here is x squared plus y squared."},{"Start":"17:15.010 ","End":"17:19.420","Text":"So that proves it. That\u0027s a generalization of Pythagoras theorem."},{"Start":"17:19.420 ","End":"17:24.040","Text":"Instead of 2-dimensions having just x-squared plus y-squared,"},{"Start":"17:24.040 ","End":"17:27.220","Text":"in 3-dimensions, it\u0027s x-squared plus y-squared plus z-squared."},{"Start":"17:27.220 ","End":"17:30.340","Text":"Now let\u0027s see how this relates to vectors."},{"Start":"17:30.340 ","End":"17:33.680","Text":"This could be a vector V."},{"Start":"17:35.400 ","End":"17:38.395","Text":"In general, in 2-dimensions,"},{"Start":"17:38.395 ","End":"17:39.850","Text":"vector v would be,"},{"Start":"17:39.850 ","End":"17:46.630","Text":"let\u0027s say a_1, a_2."},{"Start":"17:46.630 ","End":"17:48.324","Text":"In our particular case,"},{"Start":"17:48.324 ","End":"17:56.340","Text":"we could draw a vector goes 3 units this way and 4 units this way."},{"Start":"17:56.340 ","End":"17:59.700","Text":"This would be 3 in this direction,"},{"Start":"17:59.700 ","End":"18:06.940","Text":"and I go up 4 and then I get the vector and this is the tail of the vector,"},{"Start":"18:06.940 ","End":"18:08.365","Text":"the head of the vector,"},{"Start":"18:08.365 ","End":"18:10.480","Text":"and it goes from here to here,"},{"Start":"18:10.480 ","End":"18:12.800","Text":"and we called it v,"},{"Start":"18:12.800 ","End":"18:15.090","Text":"and the question is, what is the magnitude?"},{"Start":"18:15.090 ","End":"18:16.740","Text":"What is the size of v?"},{"Start":"18:16.740 ","End":"18:18.630","Text":"We just use Pythagoras."},{"Start":"18:18.630 ","End":"18:26.950","Text":"We would say the square root of 3 squared plus 4 squared."},{"Start":"18:26.950 ","End":"18:30.820","Text":"Notice that if I\u0027d actually put the minus 3 squared here,"},{"Start":"18:30.820 ","End":"18:33.220","Text":"it wouldn\u0027t have made any difference."},{"Start":"18:33.220 ","End":"18:36.520","Text":"In other words, the magnitude of"},{"Start":"18:36.520 ","End":"18:39.610","Text":"a vector in 2 dimensions is the square root of the sum of the squares."},{"Start":"18:39.610 ","End":"18:41.425","Text":"Let\u0027s just write this in general."},{"Start":"18:41.425 ","End":"18:43.780","Text":"What I would say was,"},{"Start":"18:43.780 ","End":"18:47.530","Text":"we use the notation a double bar around"},{"Start":"18:47.530 ","End":"18:52.720","Text":"the vector another double vertical bar like the absolute value,"},{"Start":"18:52.720 ","End":"18:54.865","Text":"but just doubled up."},{"Start":"18:54.865 ","End":"18:58.210","Text":"This is called the magnitude of the vector V,"},{"Start":"18:58.210 ","End":"19:00.190","Text":"and it\u0027s given by the formula,"},{"Start":"19:00.190 ","End":"19:05.590","Text":"the square root of a_1 squared plus a_2 squared,"},{"Start":"19:05.590 ","End":"19:08.320","Text":"and this is using Pythagoras."},{"Start":"19:08.320 ","End":"19:13.120","Text":"Now, let\u0027s just extend the same thing to 2-dimensions."},{"Start":"19:13.120 ","End":"19:15.550","Text":"If I have a vector, well,"},{"Start":"19:15.550 ","End":"19:17.950","Text":"I\u0027ll use the same letter again, it doesn\u0027t matter,"},{"Start":"19:17.950 ","End":"19:21.160","Text":"but I use in different contexts,"},{"Start":"19:21.160 ","End":"19:22.660","Text":"maybe a different color."},{"Start":"19:22.660 ","End":"19:26.005","Text":"Yeah, that\u0027ll do it. I have in 2-dimensions,"},{"Start":"19:26.005 ","End":"19:28.150","Text":"let\u0027s take our example here."},{"Start":"19:28.150 ","End":"19:33.760","Text":"Suppose I took 3, 4, 12."},{"Start":"19:33.760 ","End":"19:36.085","Text":"I didn\u0027t actually finish the exercise here."},{"Start":"19:36.085 ","End":"19:41.365","Text":"Just to complete it, it\u0027s the square root of 9 plus 16,"},{"Start":"19:41.365 ","End":"19:46.480","Text":"which is the square root of 25,"},{"Start":"19:46.480 ","End":"19:48.250","Text":"which is 5,"},{"Start":"19:48.250 ","End":"19:53.035","Text":"and so the magnitude of V is 5."},{"Start":"19:53.035 ","End":"19:55.209","Text":"Now Iet\u0027s take the other example."},{"Start":"19:55.209 ","End":"20:01.855","Text":"In general, we will have V as being a_1, a_2,"},{"Start":"20:01.855 ","End":"20:07.165","Text":"a_3, and the magnitude of a vector,"},{"Start":"20:07.165 ","End":"20:11.095","Text":"even if it\u0027s 3_d, is just the analogy of this."},{"Start":"20:11.095 ","End":"20:14.845","Text":"I won\u0027t draw the sketch because we have a sketch here,"},{"Start":"20:14.845 ","End":"20:17.455","Text":"but this might be a_1,"},{"Start":"20:17.455 ","End":"20:19.300","Text":"this would be a_2,"},{"Start":"20:19.300 ","End":"20:21.220","Text":"this would be a_3,"},{"Start":"20:21.220 ","End":"20:28.990","Text":"and this line here would be the vector V. So what we would get,"},{"Start":"20:28.990 ","End":"20:30.775","Text":"the length of this,"},{"Start":"20:30.775 ","End":"20:32.500","Text":"the magnitude is the length,"},{"Start":"20:32.500 ","End":"20:39.504","Text":"is the square root of a_1 squared plus a_2 squared plus a_3 squared."},{"Start":"20:39.504 ","End":"20:41.320","Text":"Again, by Pythagoras."},{"Start":"20:41.320 ","End":"20:44.740","Text":"Notice that minuses don\u0027t make any difference because everything\u0027s"},{"Start":"20:44.740 ","End":"20:48.865","Text":"being squared anyway so we don\u0027t have to worry about that."},{"Start":"20:48.865 ","End":"20:54.590","Text":"In our case, let\u0027s just see what we get."},{"Start":"20:56.550 ","End":"20:59.140","Text":"Magnitude in our case,"},{"Start":"20:59.140 ","End":"21:03.610","Text":"we will get the square root of"},{"Start":"21:03.610 ","End":"21:09.250","Text":"3 squared plus 4 squared plus 12 squared,"},{"Start":"21:09.250 ","End":"21:15.340","Text":"that\u0027s 9 plus 16 is 25 plus 144,"},{"Start":"21:15.340 ","End":"21:22.850","Text":"that\u0027s 169, the square root of 169 is 13 in this case."},{"Start":"21:24.120 ","End":"21:29.305","Text":"Now suppose V was the vector 0,"},{"Start":"21:29.305 ","End":"21:32.320","Text":"0 in 2 dimensions and"},{"Start":"21:32.320 ","End":"21:36.925","Text":"the magnitude of"},{"Start":"21:36.925 ","End":"21:43.900","Text":"V is equal to,"},{"Start":"21:43.900 ","End":"21:49.840","Text":"by the formula, the square root of 0 squared plus 0 squared, which is just 0."},{"Start":"21:49.840 ","End":"21:52.119","Text":"This is a very special vector."},{"Start":"21:52.119 ","End":"21:54.054","Text":"It\u0027s the 0 vector,"},{"Start":"21:54.054 ","End":"21:58.315","Text":"and sometimes we just write it as a 0 with an arrow above it,"},{"Start":"21:58.315 ","End":"22:01.210","Text":"not the number 0, but the vector 0."},{"Start":"22:01.210 ","End":"22:08.785","Text":"Similarly in 3D, we could take a vector V in 3D to be 0,"},{"Start":"22:08.785 ","End":"22:10.840","Text":"0, 0,"},{"Start":"22:10.840 ","End":"22:17.710","Text":"and then the magnitude of V would be the square root of 0 squared,"},{"Start":"22:17.710 ","End":"22:19.120","Text":"plus 0 squared,"},{"Start":"22:19.120 ","End":"22:21.445","Text":"plus 0 squared,"},{"Start":"22:21.445 ","End":"22:27.460","Text":"and this would also be denoted as 0 with an arrow."},{"Start":"22:27.460 ","End":"22:29.200","Text":"But, you know, from the context,"},{"Start":"22:29.200 ","End":"22:32.740","Text":"whether it\u0027s the 2D or 3D or whatever,"},{"Start":"22:32.740 ","End":"22:36.820","Text":"depends on the context but that\u0027s the special name."},{"Start":"22:36.820 ","End":"22:40.255","Text":"Besides the 0 vector, which is special,"},{"Start":"22:40.255 ","End":"22:44.515","Text":"another concept that\u0027s important is called the Unit Vector."},{"Start":"22:44.515 ","End":"22:48.984","Text":"Let me just write down."},{"Start":"22:48.984 ","End":"22:50.860","Text":"That\u0027s all that we\u0027ve learned,"},{"Start":"22:50.860 ","End":"22:54.160","Text":"something called the 0 vector."},{"Start":"22:54.160 ","End":"23:00.200","Text":"Now I want to write down, the next thing I\u0027m gonna do is something called a Unit Vector."},{"Start":"23:00.900 ","End":"23:07.240","Text":"Well, a 0 vector is a vector whose magnitude is a 0,"},{"Start":"23:07.240 ","End":"23:10.420","Text":"equal 0, and that\u0027s what makes it the 0 vector,"},{"Start":"23:10.420 ","End":"23:12.490","Text":"the magnitude 0, same as here."},{"Start":"23:12.490 ","End":"23:18.190","Text":"The unit vector would be a vector with magnitude 1."},{"Start":"23:18.190 ","End":"23:21.025","Text":"I\u0027ll give an example first of all in 2D."},{"Start":"23:21.025 ","End":"23:24.280","Text":"In fact, we can just put a magnitude on a vector."},{"Start":"23:24.280 ","End":"23:26.365","Text":"We don\u0027t have to give it a name with a letter,"},{"Start":"23:26.365 ","End":"23:35.080","Text":"I could say minus 1 over square root of 5,"},{"Start":"23:35.080 ","End":"23:38.380","Text":"2 over square root of 5."},{"Start":"23:38.380 ","End":"23:41.125","Text":"Let\u0027s see what the magnitude of this vector is."},{"Start":"23:41.125 ","End":"23:44.035","Text":"It\u0027s the square root of this squared."},{"Start":"23:44.035 ","End":"23:47.155","Text":"This squared is 1/5,"},{"Start":"23:47.155 ","End":"23:49.960","Text":"this squared is 4/5,"},{"Start":"23:49.960 ","End":"23:53.080","Text":"2 squared is 4, and root 5 squared is 5,"},{"Start":"23:53.080 ","End":"23:55.105","Text":"which is the square root of,"},{"Start":"23:55.105 ","End":"23:58.240","Text":"1/5 plus 4/5 is 1,"},{"Start":"23:58.240 ","End":"24:00.265","Text":"so this is equal to 1,"},{"Start":"24:00.265 ","End":"24:04.390","Text":"so this vector here is a unit factor,"},{"Start":"24:04.390 ","End":"24:07.030","Text":"and I\u0027ll give an example in 3D."},{"Start":"24:07.030 ","End":"24:15.505","Text":"Let\u0027s see what is the magnitude of the vector 1, 0, 0."},{"Start":"24:15.505 ","End":"24:20.170","Text":"This is equal to the square root of 1 squared,"},{"Start":"24:20.170 ","End":"24:24.114","Text":"plus 0 squared, plus 0 squared."},{"Start":"24:24.114 ","End":"24:26.710","Text":"1 plus 0 plus 0 is 1,"},{"Start":"24:26.710 ","End":"24:28.225","Text":"square root of 1 is 1,"},{"Start":"24:28.225 ","End":"24:31.070","Text":"so this is also equal to 1."},{"Start":"24:31.740 ","End":"24:36.549","Text":"This is an example of a unit vector in 2D,"},{"Start":"24:36.549 ","End":"24:40.120","Text":"and this is an example of a unit vector in 3D."},{"Start":"24:40.120 ","End":"24:45.700","Text":"Now, some of the unit vectors have special names."},{"Start":"24:45.700 ","End":"24:48.760","Text":"This particular vector, 1, 0,"},{"Start":"24:48.760 ","End":"24:52.045","Text":"0 is called i,"},{"Start":"24:52.045 ","End":"24:54.460","Text":"I\u0027m not sure if it has the dot or not,"},{"Start":"24:54.460 ","End":"24:58.255","Text":"and that\u0027s the 1, 0, 0."},{"Start":"24:58.255 ","End":"25:02.710","Text":"There is also a special unit vector called j,"},{"Start":"25:02.710 ","End":"25:07.270","Text":"and that is 0, 1, 0."},{"Start":"25:07.270 ","End":"25:12.295","Text":"It\u0027s also a unit vector of course because 0 squared plus 1 squared plus 0 squared is 1."},{"Start":"25:12.295 ","End":"25:16.179","Text":"We also have a third unit vector in 3D,"},{"Start":"25:16.179 ","End":"25:17.875","Text":"it\u0027s called k,"},{"Start":"25:17.875 ","End":"25:20.440","Text":"and this is equal to, you might guess,"},{"Start":"25:20.440 ","End":"25:23.335","Text":"0, 0, 1."},{"Start":"25:23.335 ","End":"25:27.235","Text":"In 2D, there are also unit vectors."},{"Start":"25:27.235 ","End":"25:29.050","Text":"We only have 2 of them."},{"Start":"25:29.050 ","End":"25:30.760","Text":"We have i,"},{"Start":"25:30.760 ","End":"25:33.655","Text":"which is 1, 0,"},{"Start":"25:33.655 ","End":"25:40.240","Text":"and we have a j which is 0, 1."},{"Start":"25:40.240 ","End":"25:47.275","Text":"There\u0027s actually a special name for these vectors with single 1 and the rest zeros,"},{"Start":"25:47.275 ","End":"25:54.235","Text":"they\u0027re actually called Standard Basis Vectors,"},{"Start":"25:54.235 ","End":"25:57.640","Text":"and it will always be clear in which context of talking about;"},{"Start":"25:57.640 ","End":"26:00.310","Text":"the 2D or 3D case."},{"Start":"26:00.310 ","End":"26:06.130","Text":"I just want to mention that although we\u0027ve talked about 2D and 3D vectors,"},{"Start":"26:06.130 ","End":"26:11.875","Text":"in general, there are n-dimensional vectors,"},{"Start":"26:11.875 ","End":"26:15.580","Text":"and it\u0027s not restricted to 2 or 3."},{"Start":"26:15.580 ","End":"26:16.840","Text":"There\u0027s a fourth, fifth, sixth,"},{"Start":"26:16.840 ","End":"26:19.990","Text":"any number of dimensions at least in Abstract Mathematics."},{"Start":"26:19.990 ","End":"26:23.905","Text":"It doesn\u0027t necessarily mean that we can imagine it in physical space,"},{"Start":"26:23.905 ","End":"26:30.800","Text":"but an n-dimensional vector V would be something like a_1,"},{"Start":"26:30.800 ","End":"26:34.105","Text":"a_2, and depending on n and so on,"},{"Start":"26:34.105 ","End":"26:38.725","Text":"up to an and could be 4 or 5, 6 whatever."},{"Start":"26:38.725 ","End":"26:43.540","Text":"Most of the formulas are just straightforward extensions."},{"Start":"26:43.540 ","End":"26:47.800","Text":"The magnitude of a vector instead of a_1 squared plus"},{"Start":"26:47.800 ","End":"26:51.805","Text":"a_2 squared plus a_3 squared would be the sum of the squares of all of them,"},{"Start":"26:51.805 ","End":"26:53.695","Text":"and you take the square root,"},{"Start":"26:53.695 ","End":"26:58.135","Text":"and each of these would be called the ai\u0027s,"},{"Start":"26:58.135 ","End":"27:01.910","Text":"a_1 and a_2 are called components of the vector."},{"Start":"27:02.520 ","End":"27:06.160","Text":"Sometimes we say the x component, the y component,"},{"Start":"27:06.160 ","End":"27:07.840","Text":"the z component or the first, second,"},{"Start":"27:07.840 ","End":"27:09.985","Text":"and third component and so on."},{"Start":"27:09.985 ","End":"27:13.150","Text":"Just mentioning it, we\u0027re mostly going to deal with"},{"Start":"27:13.150 ","End":"27:18.160","Text":"just 2 and 3 dimensions and the formulas will be almost the same,"},{"Start":"27:18.160 ","End":"27:24.205","Text":"just like in 2-dimensions,"},{"Start":"27:24.205 ","End":"27:27.220","Text":"we had a_1 squared plus a_2 squared,"},{"Start":"27:27.220 ","End":"27:28.240","Text":"and in 3-dimensions,"},{"Start":"27:28.240 ","End":"27:30.700","Text":"a_1 squared plus a_2 squared, a_3 squared."},{"Start":"27:30.700 ","End":"27:32.230","Text":"All the formulas are analogous,"},{"Start":"27:32.230 ","End":"27:34.045","Text":"pretty much 2 or 3 dimensions,"},{"Start":"27:34.045 ","End":"27:37.010","Text":"and that\u0027s what we\u0027ll be sticking with mostly."},{"Start":"27:37.380 ","End":"27:41.330","Text":"We\u0027re done for now."}],"ID":10644},{"Watched":false,"Name":"Exercise 1","Duration":"6m 13s","ChapterTopicVideoID":10296,"CourseChapterTopicPlaylistID":12288,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"This exercise has several parts,"},{"Start":"00:01.800 ","End":"00:07.720","Text":"and it\u0027s to do with the basics of vectors in 2D and 3D."},{"Start":"00:07.940 ","End":"00:10.740","Text":"In the first set,"},{"Start":"00:10.740 ","End":"00:13.500","Text":"we have to, in each of a, b, c, d,"},{"Start":"00:13.500 ","End":"00:19.530","Text":"find the vector and then its magnitude and say whether or not it\u0027s a unit vector."},{"Start":"00:19.530 ","End":"00:21.270","Text":"In these 2,"},{"Start":"00:21.270 ","End":"00:23.370","Text":"we have what is called a displacement vector."},{"Start":"00:23.370 ","End":"00:25.530","Text":"I\u0027m not sure if I mentioned the term displacement."},{"Start":"00:25.530 ","End":"00:27.555","Text":"When you have 2 points,"},{"Start":"00:27.555 ","End":"00:32.400","Text":"then the displacement vector is the vector that"},{"Start":"00:32.400 ","End":"00:39.900","Text":"goes from 1 to the other displacement vector."},{"Start":"00:39.900 ","End":"00:43.000","Text":"What we do is we take the coordinates of"},{"Start":"00:43.000 ","End":"00:49.040","Text":"the first point from the coordinates of the second point."},{"Start":"00:49.250 ","End":"00:52.660","Text":"In part A,"},{"Start":"00:53.030 ","End":"00:56.400","Text":"I use the angular brackets."},{"Start":"00:56.400 ","End":"01:01.510","Text":"From here to here is 5 minus minus 8."},{"Start":"01:01.970 ","End":"01:09.030","Text":"The second 1 would be minus 2 minus 3,"},{"Start":"01:09.030 ","End":"01:13.540","Text":"which gives us 13,"},{"Start":"01:14.230 ","End":"01:18.305","Text":"minus 2 minus 3 is minus 5,"},{"Start":"01:18.305 ","End":"01:23.000","Text":"and I\u0027ll be using both notations interchangeably."},{"Start":"01:23.000 ","End":"01:29.485","Text":"We could write it as 13i minus 5j,"},{"Start":"01:29.485 ","End":"01:31.170","Text":"either 1 of these."},{"Start":"01:31.170 ","End":"01:34.305","Text":"In part B, we have a 3D vector."},{"Start":"01:34.305 ","End":"01:37.020","Text":"Once again, we do the same thing."},{"Start":"01:37.020 ","End":"01:43.300","Text":"We have 2 minus 2."},{"Start":"01:43.310 ","End":"01:46.500","Text":"I\u0027ll just write it already here, is 0."},{"Start":"01:46.500 ","End":"01:49.830","Text":"4 minus 3 is 1."},{"Start":"01:49.830 ","End":"01:54.010","Text":"4 minus 4 is 0."},{"Start":"01:54.200 ","End":"01:59.410","Text":"I didn\u0027t relate to the magnitude."},{"Start":"02:00.020 ","End":"02:07.205","Text":"The magnitude, I\u0027ll stick to the angular brackets."},{"Start":"02:07.205 ","End":"02:09.050","Text":"I\u0027ll give it some space."},{"Start":"02:09.050 ","End":"02:13.160","Text":"Magnitude of 13,"},{"Start":"02:13.160 ","End":"02:18.020","Text":"minus 5 is just the square root"},{"Start":"02:18.020 ","End":"02:24.775","Text":"of 13 squared plus negative 5 squared,"},{"Start":"02:24.775 ","End":"02:27.000","Text":"which comes out to 169,"},{"Start":"02:27.000 ","End":"02:31.860","Text":"and 25 is 194."},{"Start":"02:31.860 ","End":"02:34.580","Text":"That\u0027s the magnitude. In any event,"},{"Start":"02:34.580 ","End":"02:37.370","Text":"the magnitude is not equal to 1,"},{"Start":"02:37.370 ","End":"02:41.885","Text":"so the answer is that it\u0027s not a unit vector."},{"Start":"02:41.885 ","End":"02:42.980","Text":"I\u0027ll just write the word,"},{"Start":"02:42.980 ","End":"02:45.095","Text":"not a unit vector."},{"Start":"02:45.095 ","End":"02:47.030","Text":"In the next 1,"},{"Start":"02:47.030 ","End":"02:52.500","Text":"we have the vector and now its magnitude,"},{"Start":"02:53.360 ","End":"02:55.890","Text":"put it in bars,"},{"Start":"02:55.890 ","End":"03:00.130","Text":"and what it means is the square root of this squared,"},{"Start":"03:00.130 ","End":"03:04.760","Text":"0 squared plus 1 squared plus 0 squared,"},{"Start":"03:04.760 ","End":"03:07.385","Text":"which is the square root of 1, which is 1."},{"Start":"03:07.385 ","End":"03:13.155","Text":"Yes, this is a unit vector."},{"Start":"03:13.155 ","End":"03:16.845","Text":"Now, this one\u0027s in 3D, this one\u0027s in 2D."},{"Start":"03:16.845 ","End":"03:19.500","Text":"Now in part C and D,"},{"Start":"03:19.500 ","End":"03:21.795","Text":"we are talking about the position vector."},{"Start":"03:21.795 ","End":"03:25.055","Text":"When we have the position vector of a point,"},{"Start":"03:25.055 ","End":"03:31.705","Text":"is actually a displacement vector from the origin to the point."},{"Start":"03:31.705 ","End":"03:35.960","Text":"What it is is just the origin is 0,0."},{"Start":"03:35.960 ","End":"03:37.925","Text":"We just subtract 0s from everything."},{"Start":"03:37.925 ","End":"03:42.405","Text":"In other words, just change the brackets to angular brackets."},{"Start":"03:42.405 ","End":"03:49.510","Text":"In Part C, I can straight away write the answer as 1/2 root 3/2,"},{"Start":"03:49.510 ","End":"03:58.555","Text":"and in part D, the answer will be minus 8,3,5."},{"Start":"03:58.555 ","End":"04:01.640","Text":"For those who like the other notation,"},{"Start":"04:01.640 ","End":"04:06.570","Text":"1/2i plus root 3/2j."},{"Start":"04:10.250 ","End":"04:13.660","Text":"By the way, this happens to be a unit vector."},{"Start":"04:13.660 ","End":"04:17.110","Text":"I\u0027m familiar with it, but that wasn\u0027t what was asked for here."},{"Start":"04:17.110 ","End":"04:18.790","Text":"If you take this squared, it\u0027s 1/4."},{"Start":"04:18.790 ","End":"04:20.440","Text":"This squared is 3/4."},{"Start":"04:20.440 ","End":"04:23.605","Text":"This happens to be a unit vector, just saying."},{"Start":"04:23.605 ","End":"04:26.800","Text":"This 1, the other notation,"},{"Start":"04:26.800 ","End":"04:34.230","Text":"minus 8i plus 3j plus 5k, wasn\u0027t asked for."},{"Start":"04:34.230 ","End":"04:36.820","Text":"You could use either form, whatever you prefer."},{"Start":"04:36.820 ","End":"04:39.510","Text":"I\u0027m going to be using both interchangeably."},{"Start":"04:39.510 ","End":"04:41.845","Text":"That\u0027s the part 1."},{"Start":"04:41.845 ","End":"04:45.480","Text":"In part 2,"},{"Start":"04:45.480 ","End":"04:50.595","Text":"I didn\u0027t write that these are all exercise 1,"},{"Start":"04:50.595 ","End":"04:53.385","Text":"it\u0027s a slightly different setup,"},{"Start":"04:53.385 ","End":"04:57.885","Text":"we\u0027re given the point,"},{"Start":"04:57.885 ","End":"05:02.580","Text":"and we\u0027re given a vector,"},{"Start":"05:02.580 ","End":"05:05.880","Text":"that\u0027s the tail, and the head of the vector."},{"Start":"05:05.880 ","End":"05:11.110","Text":"This is given, this is given, the vector,"},{"Start":"05:11.110 ","End":"05:12.800","Text":"and this is what we have to find,"},{"Start":"05:12.800 ","End":"05:17.615","Text":"is what\u0027s the head of the vector, where it ends."},{"Start":"05:17.615 ","End":"05:24.795","Text":"All we do is we take for this the coordinates of P,"},{"Start":"05:24.795 ","End":"05:33.330","Text":"and we add the vector V. The answer\u0027s going to be a point."},{"Start":"05:33.330 ","End":"05:37.055","Text":"It\u0027s going to be minus 3 plus 7,"},{"Start":"05:37.055 ","End":"05:43.355","Text":"and then 4 plus negative 3,"},{"Start":"05:43.355 ","End":"05:46.640","Text":"and then minus 1 plus 0."},{"Start":"05:46.640 ","End":"05:53.000","Text":"That\u0027s the endpoint because if this vector is the endpoint minus the start point,"},{"Start":"05:53.000 ","End":"05:55.760","Text":"the start point plus the vector is the endpoint."},{"Start":"05:55.760 ","End":"05:59.160","Text":"When I say plus, I mean coordinate-wise."},{"Start":"05:59.420 ","End":"06:01.545","Text":"This will be,"},{"Start":"06:01.545 ","End":"06:03.300","Text":"let\u0027s see, 4,"},{"Start":"06:03.300 ","End":"06:06.540","Text":"this comes out to be 1,"},{"Start":"06:06.540 ","End":"06:09.405","Text":"this comes out to be minus 1,"},{"Start":"06:09.405 ","End":"06:11.780","Text":"and that\u0027s it,"},{"Start":"06:11.780 ","End":"06:14.039","Text":"we\u0027re done for this exercise."}],"ID":10645}],"Thumbnail":null,"ID":12288},{"Name":"Vector Arithmetic","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vector Arithmetic","Duration":"23m 37s","ChapterTopicVideoID":10300,"CourseChapterTopicPlaylistID":12289,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"Continuing with vectors, we\u0027re going to be talking about vector arithmetic."},{"Start":"00:04.860 ","End":"00:08.460","Text":"This will apply to all dimensions,"},{"Start":"00:08.460 ","End":"00:12.165","Text":"2D, 3D, higher dimensions."},{"Start":"00:12.165 ","End":"00:15.330","Text":"But mostly I\u0027ll be using the 3D as"},{"Start":"00:15.330 ","End":"00:19.935","Text":"an example and the same principles will work for other dimensions."},{"Start":"00:19.935 ","End":"00:24.285","Text":"Let\u0027s start with the simplest which is addition,"},{"Start":"00:24.285 ","End":"00:29.500","Text":"and then we\u0027ll go on to subtraction and other operations."},{"Start":"00:29.660 ","End":"00:33.509","Text":"Let\u0027s say in 3 dimensions that I have 2 vectors,"},{"Start":"00:33.509 ","End":"00:36.059","Text":"1 of them I\u0027ll call a,"},{"Start":"00:36.059 ","End":"00:41.210","Text":"and this will be a1, a2, a3."},{"Start":"00:41.210 ","End":"00:43.115","Text":"These are the 3 components."},{"Start":"00:43.115 ","End":"00:46.070","Text":"The other vector, we\u0027ll call it b,"},{"Start":"00:46.070 ","End":"00:53.220","Text":"and let this be, b1, b2, b3."},{"Start":"00:53.220 ","End":"01:00.315","Text":"What I\u0027m going to do is define the addition a plus b."},{"Start":"01:00.315 ","End":"01:04.280","Text":"The obvious thing to do is component-wise addition."},{"Start":"01:04.280 ","End":"01:06.350","Text":"In other words, we add the first components,"},{"Start":"01:06.350 ","End":"01:12.980","Text":"so that will be a1 plus b1 and the next 1 would be a2 plus b2,"},{"Start":"01:12.980 ","End":"01:17.375","Text":"and then a3 plus b3."},{"Start":"01:17.375 ","End":"01:21.980","Text":"Of course, the same principle would hold in 2 dimensions."},{"Start":"01:21.980 ","End":"01:28.920","Text":"We just have a1 plus a2 and b1 plus b2 in 2 dimensions."},{"Start":"01:29.990 ","End":"01:33.540","Text":"I\u0027ll just give you a simple example."},{"Start":"01:33.540 ","End":"01:38.465","Text":"I\u0027ll just make up some numbers, 2 minus 4,"},{"Start":"01:38.465 ","End":"01:47.700","Text":"7 plus 3, 8 minus 11."},{"Start":"01:47.700 ","End":"01:49.770","Text":"Then what this equals,"},{"Start":"01:49.770 ","End":"01:51.390","Text":"this will be my a, this will be my b."},{"Start":"01:51.390 ","End":"01:54.510","Text":"We add 2 plus 3 is 5."},{"Start":"01:54.510 ","End":"01:57.675","Text":"Negative 4 and 8 is 4."},{"Start":"01:57.675 ","End":"02:00.930","Text":"7 and minus 11 is minus 4."},{"Start":"02:00.930 ","End":"02:05.535","Text":"That\u0027s all there is to it and similarly for in 2D."},{"Start":"02:05.535 ","End":"02:10.534","Text":"There is an illustration geometrically of what this means."},{"Start":"02:10.534 ","End":"02:13.700","Text":"I\u0027ll illustrate it in 2 dimensions where this is"},{"Start":"02:13.700 ","End":"02:16.795","Text":"the x-direction and this is the y-direction."},{"Start":"02:16.795 ","End":"02:20.155","Text":"Suppose I have 2 vectors, a."},{"Start":"02:20.155 ","End":"02:23.650","Text":"Let\u0027s say that a is a1,"},{"Start":"02:23.650 ","End":"02:29.225","Text":"a2, and b is b1, b2."},{"Start":"02:29.225 ","End":"02:35.990","Text":"Then a plus b is represented by completing the parallelogram,"},{"Start":"02:35.990 ","End":"02:37.610","Text":"it\u0027s the red vector."},{"Start":"02:37.610 ","End":"02:39.980","Text":"In fact, I\u0027ll even show you why."},{"Start":"02:39.980 ","End":"02:47.600","Text":"What we would expect this to be is a1 plus a2, b1 plus b2."},{"Start":"02:47.600 ","End":"02:49.970","Text":"Let me just show you briefly why this is so"},{"Start":"02:49.970 ","End":"02:52.910","Text":"although you could accept it on faith without proof."},{"Start":"02:52.910 ","End":"02:54.920","Text":"To say that this vector is a1,"},{"Start":"02:54.920 ","End":"03:03.900","Text":"a2 means that this part here would be a1,"},{"Start":"03:03.900 ","End":"03:06.305","Text":"and this part here would be a2."},{"Start":"03:06.305 ","End":"03:16.295","Text":"Similarly, I could make a little vertical line here and another line here."},{"Start":"03:16.295 ","End":"03:19.530","Text":"This would be b1,"},{"Start":"03:19.530 ","End":"03:20.790","Text":"just this bit,"},{"Start":"03:20.790 ","End":"03:26.955","Text":"and this part would be b2 from here to here."},{"Start":"03:26.955 ","End":"03:29.680","Text":"If you think about it,"},{"Start":"03:30.390 ","End":"03:34.285","Text":"because this and this are parallel,"},{"Start":"03:34.285 ","End":"03:36.730","Text":"then from here to here,"},{"Start":"03:36.730 ","End":"03:39.860","Text":"it\u0027s also going to be b1."},{"Start":"03:39.860 ","End":"03:42.450","Text":"I meant to draw this over here."},{"Start":"03:42.450 ","End":"03:45.805","Text":"Since this 1 is parallel to this 1,"},{"Start":"03:45.805 ","End":"03:51.600","Text":"this bit here is also going to be a2,"},{"Start":"03:51.600 ","End":"03:53.655","Text":"the same as this."},{"Start":"03:53.655 ","End":"03:57.340","Text":"We can see that the coordinates of this up to here,"},{"Start":"03:57.340 ","End":"03:58.570","Text":"it\u0027s a1 plus b1,"},{"Start":"03:58.570 ","End":"04:03.640","Text":"up to here it\u0027s a2 plus b2 and that just gives you an idea of why this is so,"},{"Start":"04:03.640 ","End":"04:05.665","Text":"but we don\u0027t need the proof,"},{"Start":"04:05.665 ","End":"04:08.290","Text":"and this is the proof in 2D."},{"Start":"04:08.290 ","End":"04:12.280","Text":"Back to vector arithmetic."},{"Start":"04:12.280 ","End":"04:14.165","Text":"That explains addition,"},{"Start":"04:14.165 ","End":"04:18.805","Text":"and the next operation will be, of course, subtraction."},{"Start":"04:18.805 ","End":"04:21.665","Text":"Works very similar to addition."},{"Start":"04:21.665 ","End":"04:25.495","Text":"For subtraction we\u0027ll use the same a and b,"},{"Start":"04:25.495 ","End":"04:28.300","Text":"and I\u0027ll start out by copying this formula."},{"Start":"04:28.300 ","End":"04:33.350","Text":"All we have to do is change the pluses into"},{"Start":"04:33.350 ","End":"04:39.640","Text":"minuses and there we have the formula for subtraction of vectors."},{"Start":"04:39.640 ","End":"04:43.125","Text":"If I take the same 2 as an example."},{"Start":"04:43.125 ","End":"04:45.090","Text":"I copied the addition."},{"Start":"04:45.090 ","End":"04:49.655","Text":"We\u0027ll change the plus to a minus and then the answer will be different."},{"Start":"04:49.655 ","End":"04:52.130","Text":"This time we just subtract component-wise."},{"Start":"04:52.130 ","End":"04:55.605","Text":"2 minus 3 is minus 1."},{"Start":"04:55.605 ","End":"05:01.790","Text":"Minus 4 minus 8 is minus 12,"},{"Start":"05:01.790 ","End":"05:07.610","Text":"and 7 minus minus 11 is 18."},{"Start":"05:07.610 ","End":"05:13.725","Text":"There\u0027s also a diagram pictorially to show what\u0027s going on,"},{"Start":"05:13.725 ","End":"05:18.905","Text":"and here\u0027s the picture for the subtraction."},{"Start":"05:18.905 ","End":"05:21.920","Text":"Not as intuitive and I won\u0027t go into it."},{"Start":"05:21.920 ","End":"05:24.515","Text":"But basically, if you take a and b,"},{"Start":"05:24.515 ","End":"05:31.010","Text":"then the vector a minus b is what you get when you draw a vector from the tip of b,"},{"Start":"05:31.010 ","End":"05:34.175","Text":"the second 1, to the tip of the first 1."},{"Start":"05:34.175 ","End":"05:39.800","Text":"This time I\u0027m not going to prove it like I did with the addition."},{"Start":"05:39.800 ","End":"05:44.330","Text":"By the way, the addition rule is sometimes"},{"Start":"05:44.330 ","End":"05:49.535","Text":"called the parallelogram law for obvious reasons."},{"Start":"05:49.535 ","End":"05:51.620","Text":"But it also has another name."},{"Start":"05:51.620 ","End":"05:56.090","Text":"It\u0027s sometimes called the triangle law."},{"Start":"05:56.090 ","End":"06:01.130","Text":"The reason for that is there\u0027s a slightly different way of showing it."},{"Start":"06:01.130 ","End":"06:03.485","Text":"Instead of putting the vector b here,"},{"Start":"06:03.485 ","End":"06:07.075","Text":"we could put the vector b here,"},{"Start":"06:07.075 ","End":"06:12.520","Text":"and remember what we said about vectors that they have magnitude and direction,"},{"Start":"06:12.520 ","End":"06:15.760","Text":"but it doesn\u0027t matter where you attach the tail to."},{"Start":"06:15.760 ","End":"06:17.440","Text":"So this is vector b,"},{"Start":"06:17.440 ","End":"06:19.480","Text":"this is also vector b."},{"Start":"06:19.480 ","End":"06:24.570","Text":"Then if we place the head of the 1 to the tail of the other, then we get a triangle."},{"Start":"06:24.570 ","End":"06:26.800","Text":"That\u0027s why it\u0027s also called the triangle law."},{"Start":"06:26.800 ","End":"06:28.840","Text":"You could also think of this picture,"},{"Start":"06:28.840 ","End":"06:31.955","Text":"whichever you prefer, the parallelogram or the triangle."},{"Start":"06:31.955 ","End":"06:34.390","Text":"As I said, with the subtraction,"},{"Start":"06:34.390 ","End":"06:37.870","Text":"you just have to remember that when it\u0027s a minus b,"},{"Start":"06:37.870 ","End":"06:40.960","Text":"it goes from the tip of b to the tip of"},{"Start":"06:40.960 ","End":"06:47.140","Text":"a. I want to remind you again that this thing works in 2D or even in 4D."},{"Start":"06:47.140 ","End":"06:48.280","Text":"We just generalize it."},{"Start":"06:48.280 ","End":"06:49.690","Text":"Instead of 1, 2, 3,"},{"Start":"06:49.690 ","End":"06:51.920","Text":"1, 2 up to whatever we need."},{"Start":"06:51.920 ","End":"06:53.300","Text":"If it\u0027s just in 2 dimensions,"},{"Start":"06:53.300 ","End":"06:55.535","Text":"we just take the first 2 components."},{"Start":"06:55.535 ","End":"07:00.290","Text":"The next thing will be scalar multiplication."},{"Start":"07:00.290 ","End":"07:01.850","Text":"First of all, the word scalar,"},{"Start":"07:01.850 ","End":"07:03.140","Text":"I may have mentioned it before,"},{"Start":"07:03.140 ","End":"07:04.520","Text":"but I want to emphasize."},{"Start":"07:04.520 ","End":"07:06.305","Text":"Because we now have vectors,"},{"Start":"07:06.305 ","End":"07:09.925","Text":"a regular number is called a scalar."},{"Start":"07:09.925 ","End":"07:13.790","Text":"Just what we call a real number is now"},{"Start":"07:13.790 ","End":"07:17.645","Text":"called a scalar to distinguish it from a quantity called a vector."},{"Start":"07:17.645 ","End":"07:20.900","Text":"There\u0027s a thing called scalar multiplication,"},{"Start":"07:20.900 ","End":"07:27.710","Text":"which actually means multiplication of a scalar by a vector."},{"Start":"07:27.710 ","End":"07:33.055","Text":"I\u0027ll explain. Suppose we have a vector a,"},{"Start":"07:33.055 ","End":"07:34.935","Text":"same as before,"},{"Start":"07:34.935 ","End":"07:38.130","Text":"take it in 3D, it\u0027s a1, a2,"},{"Start":"07:38.130 ","End":"07:40.305","Text":"a3 or its components,"},{"Start":"07:40.305 ","End":"07:42.940","Text":"and that\u0027s a vector."},{"Start":"07:44.930 ","End":"07:47.950","Text":"It\u0027s a different color c,"},{"Start":"07:47.950 ","End":"07:50.000","Text":"which is a scalar,"},{"Start":"07:50.000 ","End":"07:52.339","Text":"which means a number."},{"Start":"07:52.339 ","End":"07:58.440","Text":"We have that c times a,"},{"Start":"07:58.440 ","End":"08:05.175","Text":"we just write it as ca without anything in between or you could put a dot optionally."},{"Start":"08:05.175 ","End":"08:09.440","Text":"Scalar times a vector is going to give a new vector."},{"Start":"08:09.440 ","End":"08:19.405","Text":"This new vector, we start off by a3, a2, a1,"},{"Start":"08:19.405 ","End":"08:26.180","Text":"and we multiply each of the components by that same c. I think the reason it\u0027s"},{"Start":"08:26.180 ","End":"08:32.900","Text":"called a scalar is that it makes a scaled version larger or smaller of the original."},{"Start":"08:32.900 ","End":"08:36.395","Text":"I\u0027ll give some examples and then we\u0027ll show it on a diagram."},{"Start":"08:36.395 ","End":"08:41.700","Text":"I\u0027m going to give a 2D example so that afterwards I can draw it better."},{"Start":"08:41.700 ","End":"08:45.895","Text":"Let\u0027s take a to be the vector 2,4,"},{"Start":"08:45.895 ","End":"08:49.790","Text":"and let\u0027s take the scalar c. Well,"},{"Start":"08:49.790 ","End":"08:52.070","Text":"I want 3 different examples."},{"Start":"08:52.070 ","End":"08:55.790","Text":"I want to take c first as 3,"},{"Start":"08:55.790 ","End":"08:57.920","Text":"then as 1/2,"},{"Start":"08:57.920 ","End":"09:01.190","Text":"just to show you that we can take fractions that are smaller than 1."},{"Start":"09:01.190 ","End":"09:04.190","Text":"Finally, we\u0027ll take a third example with minus 2."},{"Start":"09:04.190 ","End":"09:06.095","Text":"That\u0027s 3 exercises in 1."},{"Start":"09:06.095 ","End":"09:09.655","Text":"I have to compute c times a for all of these."},{"Start":"09:09.655 ","End":"09:11.730","Text":"If I take the 3,"},{"Start":"09:11.730 ","End":"09:16.430","Text":"then 3a is going to be just 3 times 2 is 6."},{"Start":"09:16.430 ","End":"09:18.925","Text":"3 times 4 is 12."},{"Start":"09:18.925 ","End":"09:23.010","Text":"I didn\u0027t even bother to write that 6 is 3 times 2, we can see that."},{"Start":"09:23.010 ","End":"09:28.190","Text":"Then the next example, 1/2 of a,"},{"Start":"09:28.190 ","End":"09:32.890","Text":"is just take the components of a and multiply each 1 by a 1/2,"},{"Start":"09:32.890 ","End":"09:35.370","Text":"so it\u0027ll be 1,2."},{"Start":"09:35.710 ","End":"09:39.670","Text":"For the third example, minus 2."},{"Start":"09:39.670 ","End":"09:42.915","Text":"If I multiply that by a,"},{"Start":"09:42.915 ","End":"09:45.950","Text":"again, negative makes no difference."},{"Start":"09:45.950 ","End":"09:49.700","Text":"Minus 2 times 2 is minus 4."},{"Start":"09:49.700 ","End":"09:54.360","Text":"Minus 2 times 4 is minus 8."},{"Start":"09:54.510 ","End":"09:59.035","Text":"I\u0027ll show you what these look like on a graph,"},{"Start":"09:59.035 ","End":"10:03.285","Text":"and here\u0027s our diagram."},{"Start":"10:03.285 ","End":"10:10.490","Text":"The original a, which is the 2,4, is in black."},{"Start":"10:10.490 ","End":"10:13.145","Text":"We just see the head of it."},{"Start":"10:13.145 ","End":"10:19.955","Text":"The 3a is this vector here, the blue 1."},{"Start":"10:19.955 ","End":"10:24.140","Text":"It goes all the way from the origin,"},{"Start":"10:24.140 ","End":"10:26.460","Text":"but we just didn\u0027t color it,"},{"Start":"10:26.460 ","End":"10:29.550","Text":"because y 1an\u0027t put 1 color on top of the other,"},{"Start":"10:29.680 ","End":"10:36.245","Text":"and 1/2 a is the green 1 and minus 2a is the red 1."},{"Start":"10:36.245 ","End":"10:41.330","Text":"Notice that they\u0027re all parallel to the original vector."},{"Start":"10:41.330 ","End":"10:44.120","Text":"If the scalar is positive,"},{"Start":"10:44.120 ","End":"10:47.135","Text":"it goes in exactly the same direction."},{"Start":"10:47.135 ","End":"10:49.100","Text":"If the scale is negative,"},{"Start":"10:49.100 ","End":"10:50.584","Text":"as in the last example,"},{"Start":"10:50.584 ","End":"10:52.250","Text":"it\u0027s in the opposite direction,"},{"Start":"10:52.250 ","End":"10:53.720","Text":"but still on the same line,"},{"Start":"10:53.720 ","End":"10:55.760","Text":"so it\u0027s still parallel."},{"Start":"10:55.760 ","End":"10:59.030","Text":"There actually is another case which is worth mentioning."},{"Start":"10:59.030 ","End":"11:01.690","Text":"What if the scalar is 0?"},{"Start":"11:01.690 ","End":"11:04.350","Text":"Let me add a fourth example here."},{"Start":"11:04.350 ","End":"11:08.710","Text":"0 and 0a."},{"Start":"11:08.720 ","End":"11:10.845","Text":"In our case,"},{"Start":"11:10.845 ","End":"11:17.100","Text":"it\u0027s 0 times 2 is 0."},{"Start":"11:17.100 ","End":"11:21.030","Text":"0 times 4 is 0."},{"Start":"11:21.030 ","End":"11:25.085","Text":"In fact, what we get is the 0 vector."},{"Start":"11:25.085 ","End":"11:29.030","Text":"This is true in general for vectors of any dimension,"},{"Start":"11:29.030 ","End":"11:39.200","Text":"that the scalar 0 times any vector will always give me the 0 vector."},{"Start":"11:39.200 ","End":"11:43.465","Text":"Just make a note, this is always in any dimension."},{"Start":"11:43.465 ","End":"11:47.285","Text":"I just made the picture smaller to get some more space."},{"Start":"11:47.285 ","End":"11:49.760","Text":"This brings us to another concept,"},{"Start":"11:49.760 ","End":"11:53.700","Text":"the concept of parallel."},{"Start":"11:54.120 ","End":"11:56.170","Text":"Based on this example,"},{"Start":"11:56.170 ","End":"11:59.065","Text":"we can generalize and say that if we have 2 vectors,"},{"Start":"11:59.065 ","End":"12:02.605","Text":"a and b vectors that they are parallel,"},{"Start":"12:02.605 ","End":"12:06.130","Text":"just means that 1 of them is a scalar times the other."},{"Start":"12:06.130 ","End":"12:13.390","Text":"That a might be equal to a scalar c times the other vector and usually"},{"Start":"12:13.390 ","End":"12:23.630","Text":"we exclude the 0 vector because it has no direction."},{"Start":"12:24.420 ","End":"12:26.800","Text":"Both of them should not be 0,"},{"Start":"12:26.800 ","End":"12:28.285","Text":"then 1 is parallel to the other."},{"Start":"12:28.285 ","End":"12:31.915","Text":"If 1 is a constant scalar times the other,"},{"Start":"12:31.915 ","End":"12:35.140","Text":"even if they\u0027re in opposite directions they\u0027re still parallel."},{"Start":"12:35.140 ","End":"12:36.970","Text":"Even if I draw them elsewhere,"},{"Start":"12:36.970 ","End":"12:39.670","Text":"if I drew this vector over here,"},{"Start":"12:39.670 ","End":"12:41.560","Text":"it would still be parallel."},{"Start":"12:41.560 ","End":"12:44.950","Text":"Take another example with parallel."},{"Start":"12:44.950 ","End":"12:50.425","Text":"Let\u0027s take an example of 2 vectors and see if we can see if they\u0027re parallel or not."},{"Start":"12:50.425 ","End":"12:54.325","Text":"The first 1 will be 3,"},{"Start":"12:54.325 ","End":"12:59.755","Text":"minus 5, 2 and that will be,"},{"Start":"12:59.755 ","End":"13:05.120","Text":"I\u0027ll call that a and the other 1 will be,"},{"Start":"13:05.120 ","End":"13:09.660","Text":"let\u0027s say, minus 9,"},{"Start":"13:09.660 ","End":"13:12.990","Text":"15, minus 6."},{"Start":"13:12.990 ","End":"13:14.775","Text":"This 1 I\u0027ll call it b."},{"Start":"13:14.775 ","End":"13:18.155","Text":"My question is, are a and b parallel?"},{"Start":"13:18.155 ","End":"13:22.720","Text":"We have to find a constant that multiplies 1 to give the other."},{"Start":"13:22.720 ","End":"13:27.130","Text":"It means we have to find the same constant that multiplies each of these 3 to give these."},{"Start":"13:27.130 ","End":"13:28.690","Text":"If we look at the first 1,"},{"Start":"13:28.690 ","End":"13:33.235","Text":"what do we multiply 3 by to get minus 9."},{"Start":"13:33.235 ","End":"13:35.890","Text":"It\u0027s minus 3."},{"Start":"13:35.890 ","End":"13:38.890","Text":"But I have to make sure that this minus 3 is good for the"},{"Start":"13:38.890 ","End":"13:42.985","Text":"other coordinates components also."},{"Start":"13:42.985 ","End":"13:45.745","Text":"Minus 3 times minus 5 is 15."},{"Start":"13:45.745 ","End":"13:48.610","Text":"Good. Minus 3 times 2 is minus 6."},{"Start":"13:48.610 ","End":"13:49.990","Text":"Good. I can say,"},{"Start":"13:49.990 ","End":"13:53.140","Text":"yeah, if I take c equals minus 3,"},{"Start":"13:53.140 ","End":"13:58.750","Text":"then c times a is equal to b."},{"Start":"13:58.750 ","End":"14:00.580","Text":"Doesn\u0027t really matter on which side you put"},{"Start":"14:00.580 ","End":"14:04.990","Text":"c. Let\u0027s take another example in 2-dimensions."},{"Start":"14:04.990 ","End":"14:08.540","Text":"Let\u0027s take 2,"},{"Start":"14:08.670 ","End":"14:19.105","Text":"6 and the other 1 I\u0027ll take as 1, 5."},{"Start":"14:19.105 ","End":"14:21.310","Text":"In 2-dimensions this is a,"},{"Start":"14:21.310 ","End":"14:25.900","Text":"this is b. I want to know if these 2 are parallel without drawing."},{"Start":"14:25.900 ","End":"14:29.230","Text":"I say, 1 is going to be a constant times the other."},{"Start":"14:29.230 ","End":"14:34.090","Text":"Let\u0027s see. I could multiply 2 by 1/2 to get 1."},{"Start":"14:34.090 ","End":"14:39.595","Text":"But if c is going to be 1/2 and I have to multiply it by 6,"},{"Start":"14:39.595 ","End":"14:41.320","Text":"1/2 times 6 is 3,"},{"Start":"14:41.320 ","End":"14:42.640","Text":"which is not 5,"},{"Start":"14:42.640 ","End":"14:46.990","Text":"so I can\u0027t find 1 c that\u0027s good for all the components so these are not"},{"Start":"14:46.990 ","End":"14:55.944","Text":"parallel and there is no such c. That\u0027s just how we can tell by looking at the numbers."},{"Start":"14:55.944 ","End":"14:59.860","Text":"Next minor topic I want to discuss,"},{"Start":"14:59.860 ","End":"15:02.245","Text":"just give it a name."},{"Start":"15:02.245 ","End":"15:07.225","Text":"A common task that we need to do with vectors and we will see this later on,"},{"Start":"15:07.225 ","End":"15:09.100","Text":"is that when we\u0027re given a vector,"},{"Start":"15:09.100 ","End":"15:13.480","Text":"we want to find a unit vector parallel to it."},{"Start":"15:13.480 ","End":"15:15.070","Text":"Not just parallel, but actually in"},{"Start":"15:15.070 ","End":"15:20.245","Text":"the same direction because there are 2 unit vectors 1 could be in the opposite direction."},{"Start":"15:20.245 ","End":"15:28.300","Text":"I brought in an example that we previously did about finding the magnitude of a vector."},{"Start":"15:28.300 ","End":"15:30.205","Text":"This was an example we already did."},{"Start":"15:30.205 ","End":"15:32.080","Text":"We took the 3, 4, 12,"},{"Start":"15:32.080 ","End":"15:37.165","Text":"we took it\u0027s magnitude using the formula and we got 13."},{"Start":"15:37.165 ","End":"15:47.515","Text":"Now, it turns out that if I take 1/13 times our vector, in other words,"},{"Start":"15:47.515 ","End":"15:57.730","Text":"we get, I\u0027ll call it w. W is in fact, let\u0027s see,"},{"Start":"15:57.730 ","End":"16:05.530","Text":"3/13, 4/13,"},{"Start":"16:06.090 ","End":"16:10.165","Text":"12/13, and a little arrow on top."},{"Start":"16:10.165 ","End":"16:15.160","Text":"Then it turns out that w meets our criterion,"},{"Start":"16:15.160 ","End":"16:18.670","Text":"that w is parallel to v. Well,"},{"Start":"16:18.670 ","End":"16:22.494","Text":"the parallel is obvious is when you take a vector and multiply it by a constant,"},{"Start":"16:22.494 ","End":"16:24.325","Text":"that\u0027s the definition of parallel."},{"Start":"16:24.325 ","End":"16:26.725","Text":"But why is it a unit vector?"},{"Start":"16:26.725 ","End":"16:28.885","Text":"It\u0027s almost obvious."},{"Start":"16:28.885 ","End":"16:31.015","Text":"Let\u0027s just do the computation."},{"Start":"16:31.015 ","End":"16:39.610","Text":"The magnitude of w is equal to the square root"},{"Start":"16:39.610 ","End":"16:45.175","Text":"of 3/13 squared plus"},{"Start":"16:45.175 ","End":"16:51.800","Text":"4/13 squared plus 12/13 squared."},{"Start":"16:51.900 ","End":"16:58.900","Text":"This will just equal the square root."},{"Start":"16:58.900 ","End":"17:07.335","Text":"Now, 13 squared goes in the denominator so I can write over 13 squared."},{"Start":"17:07.335 ","End":"17:13.690","Text":"Here I have 3 squared plus 4 squared plus 12 squared."},{"Start":"17:15.120 ","End":"17:18.940","Text":"Basically, what we get is just like here"},{"Start":"17:18.940 ","End":"17:23.515","Text":"because the square root of 3 squared plus 4 squared plus 12 squared is 13,"},{"Start":"17:23.515 ","End":"17:30.650","Text":"we get 13/13 which is exactly equal to 1."},{"Start":"17:33.150 ","End":"17:40.930","Text":"This means that this is a unit vector and it\u0027s also in the same direction."},{"Start":"17:40.930 ","End":"17:43.540","Text":"Perhaps I should have mentioned that with parallel,"},{"Start":"17:43.540 ","End":"17:47.500","Text":"if the c is positive then they\u0027re parallel and in"},{"Start":"17:47.500 ","End":"17:52.765","Text":"the same direction and if the c is negative,"},{"Start":"17:52.765 ","End":"17:56.080","Text":"it\u0027s parallel in the opposite direction."},{"Start":"17:56.080 ","End":"18:00.820","Text":"We distinguish c bigger than 0 and c less than 0."},{"Start":"18:00.820 ","End":"18:07.405","Text":"In general, if I\u0027m given a vector v and then I define a new vector,"},{"Start":"18:07.405 ","End":"18:10.070","Text":"I\u0027ll call it u."},{"Start":"18:10.560 ","End":"18:17.740","Text":"In fact, I think I should have called this 1 u because u for unit there,"},{"Start":"18:17.740 ","End":"18:19.285","Text":"just give it a name change."},{"Start":"18:19.285 ","End":"18:27.145","Text":"If I let u in general be 1/the magnitude of v,"},{"Start":"18:27.145 ","End":"18:31.750","Text":"little arrow here, and multiply it,"},{"Start":"18:31.750 ","End":"18:36.850","Text":"this is a scalar times the original vector v, then u,"},{"Start":"18:36.850 ","End":"18:46.825","Text":"this 1 will always be a unit vector and in the direction of v. That\u0027s how we do it."},{"Start":"18:46.825 ","End":"18:51.265","Text":"Just like here. We take a vector, find it\u0027s magnitude,"},{"Start":"18:51.265 ","End":"18:53.155","Text":"take 1/the magnitude,"},{"Start":"18:53.155 ","End":"18:54.550","Text":"multiply it by the vector,"},{"Start":"18:54.550 ","End":"18:57.760","Text":"we get a unit vector in the same direction."},{"Start":"18:57.760 ","End":"19:01.375","Text":"Moving on to the next topic."},{"Start":"19:01.375 ","End":"19:02.740","Text":"In the previous clip,"},{"Start":"19:02.740 ","End":"19:07.210","Text":"I mentioned the concept standard basis vectors and just to remind you,"},{"Start":"19:07.210 ","End":"19:09.415","Text":"I\u0027m going to copy paste."},{"Start":"19:09.415 ","End":"19:13.480","Text":"These are the standard basis vectors in 3D."},{"Start":"19:13.480 ","End":"19:16.765","Text":"It\u0027s similar in 2D and in higher dimensions."},{"Start":"19:16.765 ","End":"19:22.090","Text":"There\u0027s a 1 and the rest are 0s and in 3D they\u0027re called i, j,"},{"Start":"19:22.090 ","End":"19:25.300","Text":"and k and in 2D they\u0027re i and j,"},{"Start":"19:25.300 ","End":"19:28.850","Text":"I\u0027m not sure what letters we use in higher dimensions."},{"Start":"19:29.430 ","End":"19:32.140","Text":"I\u0027m going to do something backwards here."},{"Start":"19:32.140 ","End":"19:36.385","Text":"I\u0027m going to show you what I\u0027m heading towards and then I\u0027ll give you the theory."},{"Start":"19:36.385 ","End":"19:39.550","Text":"There\u0027s going to be another way to write vectors."},{"Start":"19:39.550 ","End":"19:43.930","Text":"If I have a vector like 3, 4, 5,"},{"Start":"19:43.930 ","End":"19:52.370","Text":"is I could say that this is equal to 3i plus 4j plus 5k."},{"Start":"19:54.330 ","End":"19:57.280","Text":"I\u0027m going to show you why it\u0027s true in general,"},{"Start":"19:57.280 ","End":"19:59.275","Text":"not just for these 3 numbers."},{"Start":"19:59.275 ","End":"20:02.020","Text":"Let\u0027s take a_1,"},{"Start":"20:02.020 ","End":"20:07.510","Text":"a_2, a_3 to be a vector in 3D."},{"Start":"20:07.510 ","End":"20:10.180","Text":"Now because of addition of vectors,"},{"Start":"20:10.180 ","End":"20:13.720","Text":"I can write this as a_1, 0,"},{"Start":"20:13.720 ","End":"20:19.150","Text":"0 plus 0, a_2,"},{"Start":"20:19.150 ","End":"20:25.180","Text":"0 plus 0, 0, a_3."},{"Start":"20:25.180 ","End":"20:28.540","Text":"We discussed addition of vectors."},{"Start":"20:28.540 ","End":"20:31.045","Text":"Basically, you just add the first component,"},{"Start":"20:31.045 ","End":"20:35.230","Text":"a_1 plus 0 plus 0 is a_1, and so on."},{"Start":"20:35.230 ","End":"20:40.765","Text":"The next thing we can do is we just learned about multiplication by scalars."},{"Start":"20:40.765 ","End":"20:45.835","Text":"I can say that this is the scalar a_1 times vector 1,"},{"Start":"20:45.835 ","End":"20:48.790","Text":"0, 0 because when we multiply by a scalar,"},{"Start":"20:48.790 ","End":"20:50.365","Text":"we multiply by each 1."},{"Start":"20:50.365 ","End":"20:56.165","Text":"The second 1, I can say is a_2 times 0, 1, 0."},{"Start":"20:56.165 ","End":"20:57.660","Text":"Just multiply each 1,"},{"Start":"20:57.660 ","End":"20:59.610","Text":"you get 0, a_2,"},{"Start":"20:59.610 ","End":"21:05.055","Text":"0 and the third 1 of course is a_3 times 0, 0, 1."},{"Start":"21:05.055 ","End":"21:07.725","Text":"But look, definition of this is i."},{"Start":"21:07.725 ","End":"21:15.940","Text":"This is just a_1 i plus a_2 j this 1 is j,"},{"Start":"21:15.940 ","End":"21:22.660","Text":"and the last 1 is what we called here k. That proves that a_1,"},{"Start":"21:22.660 ","End":"21:26.800","Text":"a_2, a_3 in general is equal to this and if I choose 3,"},{"Start":"21:26.800 ","End":"21:30.395","Text":"4, 5, then this is what I\u0027ll get."},{"Start":"21:30.395 ","End":"21:33.660","Text":"That\u0027s just another way of writing vectors,"},{"Start":"21:33.660 ","End":"21:38.985","Text":"the same numbers in an angular brackets and with commas in between."},{"Start":"21:38.985 ","End":"21:43.410","Text":"Another way is to write them with the standard basis vectors i,"},{"Start":"21:43.410 ","End":"21:47.685","Text":"j, and k. I just wanted to mention this for reference."},{"Start":"21:47.685 ","End":"21:57.805","Text":"I want to just end with some final arithmetic formulas that we haven\u0027t covered up to now."},{"Start":"21:57.805 ","End":"22:01.015","Text":"I\u0027ll just squeeze them in here."},{"Start":"22:01.015 ","End":"22:06.009","Text":"I want to start with the 1 I just actually used without saying."},{"Start":"22:06.009 ","End":"22:10.105","Text":"When we add 3 different vectors,"},{"Start":"22:10.105 ","End":"22:14.140","Text":"it doesn\u0027t matter which 2 you take first,"},{"Start":"22:14.140 ","End":"22:15.490","Text":"we learnt addition of 2."},{"Start":"22:15.490 ","End":"22:18.040","Text":"It doesn\u0027t matter if you add these 2 and then add this to the"},{"Start":"22:18.040 ","End":"22:21.625","Text":"third or you add these 2 and you take this plus this."},{"Start":"22:21.625 ","End":"22:23.320","Text":"That\u0027s like in arithmetic,"},{"Start":"22:23.320 ","End":"22:25.555","Text":"what we called the associative law."},{"Start":"22:25.555 ","End":"22:30.265","Text":"Then there is another rule is that it doesn\u0027t matter in which order you take 2 of them."},{"Start":"22:30.265 ","End":"22:35.750","Text":"V plus w or w plus v doesn\u0027t matter the order."},{"Start":"22:36.210 ","End":"22:41.710","Text":"Another rule is that if you take a vector and add the 0 vector,"},{"Start":"22:41.710 ","End":"22:44.335","Text":"you just get the vector itself."},{"Start":"22:44.335 ","End":"22:52.225","Text":"Another rule is that if you take the scalar 1 and multiply it by a vector,"},{"Start":"22:52.225 ","End":"22:54.415","Text":"it doesn\u0027t change the vector."},{"Start":"22:54.415 ","End":"22:58.330","Text":"If you take a scalar and multiply it by the sum,"},{"Start":"22:58.330 ","End":"23:01.165","Text":"you get a a distributive law."},{"Start":"23:01.165 ","End":"23:04.660","Text":"It\u0027s the same as if you multiplied each 1 of them by the scalar,"},{"Start":"23:04.660 ","End":"23:06.550","Text":"that might be 3."},{"Start":"23:06.550 ","End":"23:08.710","Text":"Instead of adding and then multiplying by 3,"},{"Start":"23:08.710 ","End":"23:12.235","Text":"we could take 3 times each of them and then add."},{"Start":"23:12.235 ","End":"23:15.145","Text":"The other thing is another distributive."},{"Start":"23:15.145 ","End":"23:19.330","Text":"If I have 2 plus 3 times a vector, in other words,"},{"Start":"23:19.330 ","End":"23:20.695","Text":"5 times a vector,"},{"Start":"23:20.695 ","End":"23:25.720","Text":"it\u0027s the same as twice the vector plus 3 times the vector, for example."},{"Start":"23:25.720 ","End":"23:29.740","Text":"I\u0027m not going to prove these and you should just have them for reference."},{"Start":"23:29.740 ","End":"23:32.360","Text":"There\u0027s nothing very deep about them."},{"Start":"23:33.990 ","End":"23:37.310","Text":"We\u0027re done with this clip."}],"ID":10642},{"Watched":false,"Name":"Exercise 1","Duration":"3m 54s","ChapterTopicVideoID":10301,"CourseChapterTopicPlaylistID":12289,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.680","Text":"In this exercise,"},{"Start":"00:01.680 ","End":"00:03.555","Text":"there\u0027s 3 exercises,"},{"Start":"00:03.555 ","End":"00:09.780","Text":"and we\u0027re practicing addition and subtraction of vectors,"},{"Start":"00:09.780 ","End":"00:15.690","Text":"multiplication by a scalar, and also magnitude."},{"Start":"00:15.690 ","End":"00:18.900","Text":"Begin with Part A."},{"Start":"00:18.900 ","End":"00:29.580","Text":"Part A, this is multiplication of a scalar 5 times a vector, 7, 4."},{"Start":"00:29.580 ","End":"00:35.325","Text":"What we do is just multiply the scalar by each component separately;"},{"Start":"00:35.325 ","End":"00:38.295","Text":"5 times 7, 35,"},{"Start":"00:38.295 ","End":"00:41.820","Text":"5 times 4, 20, that\u0027s it."},{"Start":"00:41.820 ","End":"00:48.905","Text":"In B, we have to do a bit of addition and subtraction as well as scalar multiplication."},{"Start":"00:48.905 ","End":"00:50.720","Text":"We\u0027ll do it in bits."},{"Start":"00:50.720 ","End":"00:58.890","Text":"6b is 6 times minus 2,"},{"Start":"00:58.890 ","End":"01:08.905","Text":"5 and then minus 3a is 3 times 7, 4."},{"Start":"01:08.905 ","End":"01:12.260","Text":"What we can do is, first of all,"},{"Start":"01:12.260 ","End":"01:16.340","Text":"let\u0027s do the scalar by vector multiplication."},{"Start":"01:16.340 ","End":"01:20.360","Text":"So 6 times minus 2 is minus 12,"},{"Start":"01:20.360 ","End":"01:23.165","Text":"6 times 5 is 30,"},{"Start":"01:23.165 ","End":"01:24.665","Text":"that\u0027s the first 1."},{"Start":"01:24.665 ","End":"01:25.970","Text":"The second 1;"},{"Start":"01:25.970 ","End":"01:29.070","Text":"3 times 7 is 21,"},{"Start":"01:29.070 ","End":"01:31.365","Text":"3 times 4 is 12."},{"Start":"01:31.365 ","End":"01:33.210","Text":"Okay, 2 scalar multiplications."},{"Start":"01:33.210 ","End":"01:34.645","Text":"Now subtraction."},{"Start":"01:34.645 ","End":"01:37.790","Text":"Subtraction, just subtract component-wise."},{"Start":"01:37.790 ","End":"01:41.425","Text":"Minus 12 minus 21,"},{"Start":"01:41.425 ","End":"01:47.930","Text":"I make that minus 33,"},{"Start":"01:47.930 ","End":"01:52.735","Text":"30 minus 12 is 18."},{"Start":"01:52.735 ","End":"01:59.860","Text":"Part C. Let\u0027s first"},{"Start":"01:59.860 ","End":"02:05.590","Text":"of all do the inside and then we\u0027ll take the norm,"},{"Start":"02:05.590 ","End":"02:07.960","Text":"or magnitude, or size,"},{"Start":"02:07.960 ","End":"02:11.240","Text":"different names for these bars."},{"Start":"02:12.680 ","End":"02:21.560","Text":"Let\u0027s see. I might as well keep them in already,"},{"Start":"02:21.560 ","End":"02:22.760","Text":"it\u0027s not that hard to write."},{"Start":"02:22.760 ","End":"02:27.325","Text":"What we have inside is 9 times 7,"},{"Start":"02:27.325 ","End":"02:36.910","Text":"4 plus 4 times minus 2, 5."},{"Start":"02:38.720 ","End":"02:43.625","Text":"We can actually do it all in our heads just component-wise."},{"Start":"02:43.625 ","End":"02:48.305","Text":"The first component we have 9 times 7 plus 4 times minus 2,"},{"Start":"02:48.305 ","End":"02:55.360","Text":"63 minus 8 is 55."},{"Start":"02:56.570 ","End":"02:58.860","Text":"Then the second component;"},{"Start":"02:58.860 ","End":"03:01.125","Text":"9 times 4 is 36,"},{"Start":"03:01.125 ","End":"03:07.170","Text":"4 times 5 is 20."},{"Start":"03:07.170 ","End":"03:10.560","Text":"So we get, what is it?"},{"Start":"03:10.560 ","End":"03:15.640","Text":"36 plus 20 is 56."},{"Start":"03:18.590 ","End":"03:29.640","Text":"This is equal to the square root of 55 squared plus 56 squared."},{"Start":"03:30.380 ","End":"03:35.795","Text":"This comes out to be the square root of 6,161."},{"Start":"03:35.795 ","End":"03:41.730","Text":"I would leave it like that but if you want a numerical result,"},{"Start":"03:41.730 ","End":"03:43.650","Text":"calculator will help,"},{"Start":"03:43.650 ","End":"03:53.800","Text":"78.492 something approximately. That\u0027s it."}],"ID":10643},{"Watched":false,"Name":"Exercise 2","Duration":"4m 57s","ChapterTopicVideoID":10297,"CourseChapterTopicPlaylistID":12289,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.060","Text":"In this exercise, we have to do some basic operations on a couple of 3D vectors."},{"Start":"00:06.060 ","End":"00:10.350","Text":"We need to do multiplication of scalar by a vector,"},{"Start":"00:10.350 ","End":"00:14.380","Text":"addition and subtraction of vectors and magnitude."},{"Start":"00:15.110 ","End":"00:18.765","Text":"Let\u0027s start with part a."},{"Start":"00:18.765 ","End":"00:23.025","Text":"What we have is that v is this and we want minus 4v,"},{"Start":"00:23.025 ","End":"00:31.785","Text":"so we want minus 4 times 6j minus 2k."},{"Start":"00:31.785 ","End":"00:33.900","Text":"Scalar by a vector,"},{"Start":"00:33.900 ","End":"00:37.650","Text":"we just multiply as you would expect,"},{"Start":"00:37.650 ","End":"00:43.305","Text":"we multiply the scalar by the j-component and by the k-component,"},{"Start":"00:43.305 ","End":"00:45.090","Text":"there is no i-component."},{"Start":"00:45.090 ","End":"00:48.040","Text":"We get minus 24j,"},{"Start":"00:49.360 ","End":"00:53.600","Text":"and minus 4 times minus 2 is plus 8k."},{"Start":"00:53.600 ","End":"00:56.090","Text":"That\u0027s simple as that."},{"Start":"00:56.090 ","End":"01:01.519","Text":"In b, we have a scalar multiplication and an addition."},{"Start":"01:01.519 ","End":"01:06.740","Text":"What we have here is 10 times the vector u which is 7i."},{"Start":"01:06.740 ","End":"01:12.800","Text":"I write minus"},{"Start":"01:12.800 ","End":"01:20.390","Text":"2j plus 4k and plus v,"},{"Start":"01:20.390 ","End":"01:30.405","Text":"which is 6j minus 2k and this is equal to."},{"Start":"01:30.405 ","End":"01:35.780","Text":"First of all, I\u0027ll do the multiplication of the scalar by the vector."},{"Start":"01:35.780 ","End":"01:43.380","Text":"This gives me 70i minus 20j plus 40k,"},{"Start":"01:43.810 ","End":"01:47.795","Text":"that\u0027s 1. Then the other one,"},{"Start":"01:47.795 ","End":"01:51.455","Text":"just as is 6j minus 2k."},{"Start":"01:51.455 ","End":"01:53.870","Text":"There is no i-component,"},{"Start":"01:53.870 ","End":"01:58.830","Text":"some people like to write 0i just to have it complete, possible."},{"Start":"01:58.830 ","End":"02:02.505","Text":"This equals component-wise 70i,"},{"Start":"02:02.505 ","End":"02:04.155","Text":"there is no i here,"},{"Start":"02:04.155 ","End":"02:06.045","Text":"so it\u0027s just 70i."},{"Start":"02:06.045 ","End":"02:11.520","Text":"Then minus 20 plus 6 with the j,"},{"Start":"02:11.520 ","End":"02:15.990","Text":"so that\u0027s minus 14j,"},{"Start":"02:15.990 ","End":"02:24.090","Text":"and 40 minus 2 is 38k."},{"Start":"02:24.090 ","End":"02:28.605","Text":"Now, I\u0027ll go fix those arrows. There we are."},{"Start":"02:28.605 ","End":"02:31.230","Text":"Finally in part c,"},{"Start":"02:31.230 ","End":"02:34.860","Text":"we also have a magnitude."},{"Start":"02:34.860 ","End":"02:37.545","Text":"What we have is minus 8,"},{"Start":"02:37.545 ","End":"02:45.520","Text":"and then u is 7i minus 2j plus 4k."},{"Start":"02:45.860 ","End":"02:53.860","Text":"Minus 3 vector v is 6j minus 2k."},{"Start":"02:56.240 ","End":"02:59.115","Text":"This is equal to?"},{"Start":"02:59.115 ","End":"03:03.965","Text":"First of all, I\u0027ll do the scalar product, scalar with vector."},{"Start":"03:03.965 ","End":"03:13.680","Text":"Minus 8 with this just component-wise, minus 56i,"},{"Start":"03:14.050 ","End":"03:21.770","Text":"and then plus 16j minus 32k,"},{"Start":"03:21.770 ","End":"03:28.070","Text":"and then I\u0027ll take it as a minus and multiply the"},{"Start":"03:28.070 ","End":"03:34.890","Text":"3 so I have 18j minus 6k."},{"Start":"03:35.500 ","End":"03:40.920","Text":"Now, I have to do the subtraction."},{"Start":"03:41.650 ","End":"03:45.565","Text":"What I have is minus 56i,"},{"Start":"03:45.565 ","End":"03:47.490","Text":"there is nothing here with i,"},{"Start":"03:47.490 ","End":"03:50.140","Text":"so it stays minus 56i."},{"Start":"03:51.290 ","End":"03:56.925","Text":"16j minus 18j is minus 2j,"},{"Start":"03:56.925 ","End":"04:05.170","Text":"and minus 32 plus 6 is minus 26k."},{"Start":"04:05.330 ","End":"04:11.910","Text":"That\u0027s just up to the magnitude."},{"Start":"04:11.910 ","End":"04:13.880","Text":"Now, we have to do the magnitude."},{"Start":"04:13.880 ","End":"04:19.130","Text":"This comes out to be usual formula;"},{"Start":"04:19.130 ","End":"04:22.190","Text":"the square root of each one squared."},{"Start":"04:22.190 ","End":"04:25.719","Text":"Obviously, the minuses don\u0027t make any difference."},{"Start":"04:25.719 ","End":"04:36.474","Text":"56 squared plus 2 squared plus 26 squared and this is equal to,"},{"Start":"04:36.474 ","End":"04:41.820","Text":"I make this 3816,"},{"Start":"04:41.820 ","End":"04:44.664","Text":"and if you have to have a numerical answer,"},{"Start":"04:44.664 ","End":"04:50.570","Text":"then 61.77 something, roughly."},{"Start":"04:50.570 ","End":"04:54.650","Text":"Not so important, the idea is to know how to do it."},{"Start":"04:54.650 ","End":"04:57.750","Text":"That\u0027s it for this exercise."}],"ID":10639},{"Watched":false,"Name":"Exercise 3","Duration":"3m 27s","ChapterTopicVideoID":10298,"CourseChapterTopicPlaylistID":12289,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"This exercise has 2 parts to it,"},{"Start":"00:02.805 ","End":"00:04.835","Text":"lets just start with part 1."},{"Start":"00:04.835 ","End":"00:10.360","Text":"We\u0027re given this vector and we want a unit vector that points in the same direction."},{"Start":"00:10.360 ","End":"00:12.490","Text":"Let me first check, we might be lucky,"},{"Start":"00:12.490 ","End":"00:14.995","Text":"this might be a unit vector already."},{"Start":"00:14.995 ","End":"00:20.380","Text":"In any event, we\u0027ll need the magnitude of v and this is equal to,"},{"Start":"00:20.380 ","End":"00:22.060","Text":"just using the formula,"},{"Start":"00:22.060 ","End":"00:29.784","Text":"the square root of 1 squared and minus 4 squared and a squared,"},{"Start":"00:29.784 ","End":"00:38.515","Text":"which is just 1 plus 16 plus 64."},{"Start":"00:38.515 ","End":"00:43.400","Text":"This comes out as 81."},{"Start":"00:43.400 ","End":"00:48.660","Text":"It\u0027s the square root of 81, which is 9."},{"Start":"00:48.660 ","End":"00:50.770","Text":"This is not equal to 1,"},{"Start":"00:50.770 ","End":"00:52.480","Text":"so it\u0027s not a unit vector,"},{"Start":"00:52.480 ","End":"00:55.930","Text":"so let\u0027s divide by it and that\u0027s how you get a unit vector."},{"Start":"00:55.930 ","End":"00:59.275","Text":"If I take 1/9 of v,"},{"Start":"00:59.275 ","End":"01:01.225","Text":"that should do the job."},{"Start":"01:01.225 ","End":"01:07.140","Text":"This is equal to 1/9 minus,"},{"Start":"01:07.140 ","End":"01:09.405","Text":"sorry, the I forgot,"},{"Start":"01:09.405 ","End":"01:14.970","Text":"I minus 4/9 j"},{"Start":"01:14.970 ","End":"01:21.595","Text":"plus 8/9 k. That\u0027s the answer."},{"Start":"01:21.595 ","End":"01:27.700","Text":"Optionally, you could check that this really is a unit vector using this formula again,"},{"Start":"01:27.700 ","End":"01:31.010","Text":"but I\u0027m going to skip that."},{"Start":"01:31.260 ","End":"01:34.975","Text":"Pretty confident that is a unit vector."},{"Start":"01:34.975 ","End":"01:38.180","Text":"In part 2,"},{"Start":"01:40.340 ","End":"01:42.760","Text":"similar to part 1,"},{"Start":"01:42.760 ","End":"01:44.889","Text":"we want a vector in the same direction,"},{"Start":"01:44.889 ","End":"01:46.690","Text":"but this time not a unit vector,"},{"Start":"01:46.690 ","End":"01:49.270","Text":"a vector with magnitude 10."},{"Start":"01:49.270 ","End":"01:53.195","Text":"The other difference is that this is in 3D and this is in 2D,"},{"Start":"01:53.195 ","End":"01:56.470","Text":"and here we\u0027re using angular brackets and here we\u0027re using i, j,"},{"Start":"01:56.470 ","End":"02:02.755","Text":"k. The same idea lets see what the magnitude of W is."},{"Start":"02:02.755 ","End":"02:10.189","Text":"This is equal to the square root of minus 2 squared plus 5 squared."},{"Start":"02:10.189 ","End":"02:13.700","Text":"That\u0027s the square root of 29,"},{"Start":"02:13.700 ","End":"02:16.880","Text":"not a whole number but root 29."},{"Start":"02:16.880 ","End":"02:21.890","Text":"Now, if I wanted a unit vector,"},{"Start":"02:21.890 ","End":"02:26.390","Text":"then I would take 1 over square root of"},{"Start":"02:26.390 ","End":"02:32.360","Text":"29 of w. That\u0027s if I wanted a unit vector,"},{"Start":"02:32.360 ","End":"02:35.195","Text":"but I don\u0027t, I want a vector with magnitude 10,"},{"Start":"02:35.195 ","End":"02:40.340","Text":"so I\u0027m going to multiply this by 10 and multiply it by 10."},{"Start":"02:40.340 ","End":"02:48.120","Text":"I\u0027ll get 10 over root 29 times w. This will come out to"},{"Start":"02:48.120 ","End":"02:53.400","Text":"be just the k angular brackets"},{"Start":"02:53.400 ","End":"02:59.690","Text":"minus 2 times 10 over 29 is minus 20 over root 29."},{"Start":"02:59.690 ","End":"03:01.355","Text":"Did I just say 29?"},{"Start":"03:01.355 ","End":"03:09.650","Text":"Comma and then 5 times the 10 over root 29,"},{"Start":"03:09.650 ","End":"03:14.360","Text":"which is 50 over root 29."},{"Start":"03:14.360 ","End":"03:20.390","Text":"That\u0027ll do it. The idea again is divide it by its magnitude,"},{"Start":"03:20.390 ","End":"03:27.570","Text":"but then multiply by 10. That\u0027s it."}],"ID":10640},{"Watched":false,"Name":"Exercise 4","Duration":"3m 54s","ChapterTopicVideoID":10299,"CourseChapterTopicPlaylistID":12289,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.900","Text":"In this exercise, we\u0027re given in each of the parts a and b,"},{"Start":"00:06.900 ","End":"00:08.340","Text":"a pair of vectors."},{"Start":"00:08.340 ","End":"00:10.530","Text":"We have to decide if they\u0027re parallel."},{"Start":"00:10.530 ","End":"00:13.065","Text":"Let\u0027s start with part a."},{"Start":"00:13.065 ","End":"00:15.300","Text":"In order for these 2 be parallel,"},{"Start":"00:15.300 ","End":"00:18.255","Text":"one has to be some scalar times the other."},{"Start":"00:18.255 ","End":"00:23.520","Text":"Let\u0027s say that w is going to equal sum k"},{"Start":"00:23.520 ","End":"00:30.745","Text":"times v. Usually we require k not equal to 0."},{"Start":"00:30.745 ","End":"00:32.310","Text":"If k was 0,"},{"Start":"00:32.310 ","End":"00:33.930","Text":"then w would be 0."},{"Start":"00:33.930 ","End":"00:38.360","Text":"It\u0027s debatable whether the 0 vector is parallel to another vector."},{"Start":"00:38.360 ","End":"00:39.995","Text":"Some people might say yes,"},{"Start":"00:39.995 ","End":"00:43.190","Text":"usually we take k to be non-zero."},{"Start":"00:43.190 ","End":"00:45.580","Text":"Suppose there is such a k,"},{"Start":"00:45.580 ","End":"00:48.785","Text":"in that case then component-wise,"},{"Start":"00:48.785 ","End":"00:51.920","Text":"we can get 3 equations."},{"Start":"00:51.920 ","End":"00:54.875","Text":"Let\u0027s take the first component, the i component."},{"Start":"00:54.875 ","End":"01:01.340","Text":"It would follow that 15 has to be k times 6."},{"Start":"01:01.340 ","End":"01:04.025","Text":"Just looking at the first component,"},{"Start":"01:04.025 ","End":"01:12.080","Text":"in which case we would get that k is equal to 15 over 6,"},{"Start":"01:12.080 ","End":"01:16.625","Text":"which is divide top and bottom by 3,"},{"Start":"01:16.625 ","End":"01:22.030","Text":"5 over 2 and 15 over 6."},{"Start":"01:22.250 ","End":"01:28.370","Text":"That means that w would have to be 5 over"},{"Start":"01:28.370 ","End":"01:34.250","Text":"2 times v. If this is the case, well, let\u0027s check."},{"Start":"01:34.250 ","End":"01:35.915","Text":"We don\u0027t know it\u0027s the case."},{"Start":"01:35.915 ","End":"01:38.480","Text":"We know it\u0027s true for the first component. Let\u0027s see."},{"Start":"01:38.480 ","End":"01:48.720","Text":"We just multiply out 5 over 2v is equal to 5 over 2 times 6 is 15."},{"Start":"01:48.720 ","End":"01:50.780","Text":"So far so good,"},{"Start":"01:50.780 ","End":"01:52.460","Text":"but that\u0027s what we expected."},{"Start":"01:52.460 ","End":"01:55.100","Text":"Let\u0027s see what happens with the other 2 components."},{"Start":"01:55.100 ","End":"02:00.750","Text":"5 over 2 times 4 is in fact 10."},{"Start":"02:00.750 ","End":"02:02.120","Text":"Yeah, and as a minus,"},{"Start":"02:02.120 ","End":"02:04.865","Text":"so we get minus 10j."},{"Start":"02:04.865 ","End":"02:09.650","Text":"If I take 5 over 2 times 16,"},{"Start":"02:09.650 ","End":"02:12.350","Text":"16 over 2 is 8 times 5 is 40."},{"Start":"02:12.350 ","End":"02:15.390","Text":"Yeah, so we\u0027ve got minus 40k,"},{"Start":"02:15.940 ","End":"02:19.490","Text":"and is this equal to w?"},{"Start":"02:19.490 ","End":"02:21.770","Text":"Well, the answer is yes."},{"Start":"02:21.770 ","End":"02:24.845","Text":"There\u0027s no question mark about it."},{"Start":"02:24.845 ","End":"02:31.005","Text":"These are equal and so yes, they are parallel."},{"Start":"02:31.005 ","End":"02:35.465","Text":"Now let\u0027s do the same thing in part b."},{"Start":"02:35.465 ","End":"02:40.180","Text":"If they\u0027re parallel, then let\u0027s say that the second is some"},{"Start":"02:40.180 ","End":"02:45.620","Text":"constant times the first constant, I mean scalar."},{"Start":"02:46.770 ","End":"02:55.345","Text":"Let\u0027s take it t b is some k times a."},{"Start":"02:55.345 ","End":"02:58.870","Text":"Now if I apply it to the first component,"},{"Start":"02:58.870 ","End":"03:04.645","Text":"then I\u0027ve got that 6 is equal to k times 3,"},{"Start":"03:04.645 ","End":"03:08.710","Text":"which gives me that k equals 2."},{"Start":"03:08.710 ","End":"03:12.280","Text":"Now let\u0027s see, we want it to work on all the components."},{"Start":"03:12.280 ","End":"03:20.310","Text":"My question is, does b really equal 2 times a?"},{"Start":"03:20.310 ","End":"03:22.215","Text":"Where I put the k equals 2 here."},{"Start":"03:22.215 ","End":"03:28.500","Text":"Well, let\u0027s say 2a is equal to its angular bracket form."},{"Start":"03:28.500 ","End":"03:32.310","Text":"We\u0027ve got 6 minus 4."},{"Start":"03:32.310 ","End":"03:34.290","Text":"So far so good."},{"Start":"03:34.290 ","End":"03:37.430","Text":"Then the third component is 10."},{"Start":"03:37.430 ","End":"03:39.515","Text":"Not so good."},{"Start":"03:39.515 ","End":"03:46.380","Text":"This is not equal to b."},{"Start":"03:46.380 ","End":"03:48.315","Text":"The answer is no,"},{"Start":"03:48.315 ","End":"03:53.860","Text":"not parallel. That\u0027s it."}],"ID":10641}],"Thumbnail":null,"ID":12289},{"Name":"Vectors Dot Product","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors - Dot Product","Duration":"20m 22s","ChapterTopicVideoID":10303,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Continuing with vectors, I\u0027m going to talk about something called"},{"Start":"00:03.300 ","End":"00:08.190","Text":"the dot product and let me get straight to the definition."},{"Start":"00:08.190 ","End":"00:12.225","Text":"But I\u0027ll define it in 3-dimensions and it will be similar in other dimensions."},{"Start":"00:12.225 ","End":"00:17.970","Text":"If I have 1 vector a and let\u0027s say a has components a1,"},{"Start":"00:17.970 ","End":"00:20.715","Text":"a2, and a3,"},{"Start":"00:20.715 ","End":"00:22.965","Text":"and I have another vector b,"},{"Start":"00:22.965 ","End":"00:28.260","Text":"which is naturally b1, b2, b3,"},{"Start":"00:28.260 ","End":"00:32.655","Text":"then I\u0027m going to define the dot product,"},{"Start":"00:32.655 ","End":"00:39.820","Text":"a dot with b to be actually a number a scalar,"},{"Start":"00:39.820 ","End":"00:47.375","Text":"a1b1 plus a2b2 plus a3b3."},{"Start":"00:47.375 ","End":"00:48.710","Text":"The first with the first,"},{"Start":"00:48.710 ","End":"00:49.790","Text":"the second with the second,"},{"Start":"00:49.790 ","End":"00:53.315","Text":"the third with the third multiply and then add."},{"Start":"00:53.315 ","End":"00:57.300","Text":"I\u0027ll get some examples in a moment,"},{"Start":"00:57.300 ","End":"01:05.720","Text":"I just want you to notice that we take a vector and we dot product it with a vector,"},{"Start":"01:05.720 ","End":"01:08.885","Text":"but the answer is a number, a scalar."},{"Start":"01:08.885 ","End":"01:12.110","Text":"Later there\u0027ll be another product called the cross-product,"},{"Start":"01:12.110 ","End":"01:14.940","Text":"where a vector times a vector will be a vector, but not yet."},{"Start":"01:14.940 ","End":"01:20.930","Text":"Examples. Let\u0027s say we have 0,4"},{"Start":"01:20.930 ","End":"01:28.099","Text":"minus 2 times 2 minus 1,"},{"Start":"01:28.099 ","End":"01:36.555","Text":"7 then we get 0 times 2 is 0,"},{"Start":"01:36.555 ","End":"01:39.885","Text":"4 times 1 is minus 4,"},{"Start":"01:39.885 ","End":"01:41.040","Text":"adding minus 4,"},{"Start":"01:41.040 ","End":"01:42.910","Text":"so I just put minus 4."},{"Start":"01:42.910 ","End":"01:45.460","Text":"Minus 2 times 7"},{"Start":"01:45.460 ","End":"01:55.855","Text":"is minus 14 and all together the answer is minus 18."},{"Start":"01:55.855 ","End":"02:01.550","Text":"Let\u0027s take an example in 2-dimensions;"},{"Start":"02:01.680 ","End":"02:09.620","Text":"5 minus 8 dot 1, 2."},{"Start":"02:09.620 ","End":"02:11.610","Text":"There\u0027s just 2 things to add,"},{"Start":"02:11.610 ","End":"02:13.625","Text":"5 times 1, is 5,"},{"Start":"02:13.625 ","End":"02:17.920","Text":"minus 8 times 2 is minus 16 minus 11."},{"Start":"02:17.920 ","End":"02:21.055","Text":"Now let\u0027s take an example with the other notation."},{"Start":"02:21.055 ","End":"02:31.705","Text":"Let\u0027s take 3j minus 7k dot-product with"},{"Start":"02:31.705 ","End":"02:39.590","Text":"2i plus 3j plus"},{"Start":"02:39.590 ","End":"02:45.925","Text":"k. Supposed to put arrows on these but okay, fine."},{"Start":"02:45.925 ","End":"02:51.480","Text":"Now in this case, notice that there is no i term,"},{"Start":"02:51.480 ","End":"02:57.075","Text":"so I can think of a 0i here,"},{"Start":"02:57.075 ","End":"03:01.340","Text":"even though it\u0027s not there and also notice that if it\u0027s just k on its own,"},{"Start":"03:01.340 ","End":"03:04.125","Text":"it\u0027s like it\u0027s a 1 here."},{"Start":"03:04.125 ","End":"03:11.400","Text":"What I get is 0 times 2 is 0 the i with the i and the j with the j,"},{"Start":"03:11.400 ","End":"03:14.610","Text":"3 times 3 is 9,"},{"Start":"03:14.610 ","End":"03:22.110","Text":"and the k with the k minus 7 times 1 is minus 7 so the answer is 2."},{"Start":"03:22.110 ","End":"03:25.460","Text":"Also be careful if for some reason they\u0027re not given in the right order,"},{"Start":"03:25.460 ","End":"03:26.900","Text":"make sure you order them i, j,"},{"Start":"03:26.900 ","End":"03:29.615","Text":"k. Here there\u0027s a 0,"},{"Start":"03:29.615 ","End":"03:33.040","Text":"here there\u0027s a 1 that\u0027s things to look out for."},{"Start":"03:34.670 ","End":"03:42.410","Text":"I\u0027ll give 1 more example just to show you that it works in 4 dimensions."},{"Start":"03:42.410 ","End":"03:52.664","Text":"We\u0027ll take 9,5 minus 4,2 dot-product with minus 3,"},{"Start":"03:52.664 ","End":"03:57.135","Text":"minus 2, 7 minus 1."},{"Start":"03:57.135 ","End":"03:58.970","Text":"Same thing in 4 dimensions."},{"Start":"03:58.970 ","End":"04:01.805","Text":"This with this minus 27,"},{"Start":"04:01.805 ","End":"04:04.820","Text":"5 times minus 2 minus 10."},{"Start":"04:04.820 ","End":"04:06.485","Text":"Minus 4 times 7,"},{"Start":"04:06.485 ","End":"04:09.695","Text":"minus 28, 2 times minus 1,"},{"Start":"04:09.695 ","End":"04:16.900","Text":"minus 2, I make it minus 67."},{"Start":"04:17.860 ","End":"04:20.810","Text":"Here are some formulas."},{"Start":"04:20.810 ","End":"04:22.560","Text":"Write the word formulas,"},{"Start":"04:22.560 ","End":"04:26.775","Text":"actually the correct word is formulae Latin and there are"},{"Start":"04:26.775 ","End":"04:31.250","Text":"6 of them and I\u0027ll just quickly go over them."},{"Start":"04:31.250 ","End":"04:32.900","Text":"I just want you to have them."},{"Start":"04:32.900 ","End":"04:38.345","Text":"There\u0027s a distributive law that looks a little bit like,"},{"Start":"04:38.345 ","End":"04:41.960","Text":"if it\u0027s multiplication and addition and"},{"Start":"04:41.960 ","End":"04:45.215","Text":"with the regular algebra then it looks just like this."},{"Start":"04:45.215 ","End":"04:49.550","Text":"The product of u with the sum of v plus w is"},{"Start":"04:49.550 ","End":"04:54.785","Text":"u dot v plus w is u dot v plus u dot w. It looks intuitive."},{"Start":"04:54.785 ","End":"04:57.049","Text":"If I take a constant, a scalar,"},{"Start":"04:57.049 ","End":"05:00.395","Text":"I multiply it by the first 1,"},{"Start":"05:00.395 ","End":"05:04.600","Text":"or by the second 1 or by the product, it\u0027s all the same."},{"Start":"05:04.600 ","End":"05:08.405","Text":"It doesn\u0027t matter where you put the constant, the scalar."},{"Start":"05:08.405 ","End":"05:12.800","Text":"The order doesn\u0027t matter with a dot-product."},{"Start":"05:12.800 ","End":"05:16.070","Text":"Obviously, if I interchange the order,"},{"Start":"05:16.070 ","End":"05:17.810","Text":"I\u0027ll just get b1a1,"},{"Start":"05:17.810 ","End":"05:21.095","Text":"which is the same as a1b1 and so on."},{"Start":"05:21.095 ","End":"05:27.350","Text":"Dot-product with the 0 vector means that if 1 of them is 0 like 000,"},{"Start":"05:27.350 ","End":"05:31.350","Text":"the product is going to be 0, the dot-product."},{"Start":"05:31.780 ","End":"05:33.995","Text":"Now what does this say?"},{"Start":"05:33.995 ","End":"05:39.860","Text":"The dot product with a vector with itself is the magnitude of the vector squared."},{"Start":"05:39.860 ","End":"05:43.085","Text":"Let\u0027s take a look at that 1 take it in 3D."},{"Start":"05:43.085 ","End":"05:49.670","Text":"Suppose my vector v is a1, a2, a3."},{"Start":"05:49.670 ","End":"05:51.755","Text":"These are the 3 components."},{"Start":"05:51.755 ","End":"05:55.460","Text":"We mentioned something called the magnitude of a vector where we put it"},{"Start":"05:55.460 ","End":"05:59.285","Text":"in bars and we defined it to be"},{"Start":"05:59.285 ","End":"06:07.265","Text":"the square root of a1 squared plus a2 squared plus a3 squared."},{"Start":"06:07.265 ","End":"06:11.010","Text":"Let\u0027s see that we really get an equality."},{"Start":"06:11.020 ","End":"06:22.500","Text":"V dot V vectors is going to equal a1, a2,"},{"Start":"06:22.500 ","End":"06:31.995","Text":"a3 dot again with a1, a2,"},{"Start":"06:31.995 ","End":"06:38.630","Text":"a3 and this with this is a1 squared,"},{"Start":"06:38.630 ","End":"06:41.900","Text":"a2 with a2, a2 squared,"},{"Start":"06:41.900 ","End":"06:45.885","Text":"a3 with a3, is a3 squared."},{"Start":"06:45.885 ","End":"06:48.015","Text":"That\u0027s the left-hand side."},{"Start":"06:48.015 ","End":"06:49.560","Text":"On the other hand,"},{"Start":"06:49.560 ","End":"06:52.050","Text":"the right-hand side,"},{"Start":"06:52.050 ","End":"07:01.465","Text":"magnitude of v squared is this thing squared."},{"Start":"07:01.465 ","End":"07:04.535","Text":"If I do that, the square root of something squared,"},{"Start":"07:04.535 ","End":"07:08.480","Text":"this is just equal to the thing without the square root so it\u0027s a1"},{"Start":"07:08.480 ","End":"07:12.930","Text":"squared plus a2 squared plus a3 squared."},{"Start":"07:12.930 ","End":"07:15.015","Text":"We get the same result,"},{"Start":"07:15.015 ","End":"07:17.190","Text":"so these really are equal."},{"Start":"07:17.190 ","End":"07:18.660","Text":"I mean, this equals this,"},{"Start":"07:18.660 ","End":"07:25.050","Text":"so left-hand side equals right-hand side, so we\u0027ve verified."},{"Start":"07:25.050 ","End":"07:30.095","Text":"Now look, this says that if we get 0,"},{"Start":"07:30.095 ","End":"07:32.870","Text":"the vector must be 0, why is that?"},{"Start":"07:32.870 ","End":"07:35.675","Text":"Because if v dot v is 0,"},{"Start":"07:35.675 ","End":"07:41.735","Text":"v dot v is just a1 squared plus a2 squared plus a3 squared."},{"Start":"07:41.735 ","End":"07:48.935","Text":"Suppose a1 squared plus a2 squared plus a3 squared equals 0."},{"Start":"07:48.935 ","End":"07:50.840","Text":"Now each of these is non-negative,"},{"Start":"07:50.840 ","End":"07:52.685","Text":"bigger or equal to 0."},{"Start":"07:52.685 ","End":"07:56.420","Text":"The only way non-negatives can be added"},{"Start":"07:56.420 ","End":"08:00.620","Text":"together to be 0 is that all of them have to be 0."},{"Start":"08:00.620 ","End":"08:04.995","Text":"This means that a1 is 0, a2 is 0,"},{"Start":"08:04.995 ","End":"08:07.685","Text":"and that a3 is 0 in short,"},{"Start":"08:07.685 ","End":"08:10.610","Text":"that the vector v is 0,"},{"Start":"08:10.610 ","End":"08:12.440","Text":"because all its components are 0."},{"Start":"08:12.440 ","End":"08:14.315","Text":"That explains that 1."},{"Start":"08:14.315 ","End":"08:19.310","Text":"It turns out that there is actually a geometric or"},{"Start":"08:19.310 ","End":"08:25.940","Text":"a trigonometric interpretation of the dot-product,"},{"Start":"08:25.940 ","End":"08:30.515","Text":"and I need to bring in a diagram for that."},{"Start":"08:30.515 ","End":"08:32.600","Text":"Here\u0027s the picture."},{"Start":"08:32.600 ","End":"08:34.725","Text":"This is the x-axis,"},{"Start":"08:34.725 ","End":"08:40.125","Text":"the y-axis, I take 2 vectors; a and b."},{"Start":"08:40.125 ","End":"08:41.160","Text":"We\u0027ll do it in 2D,"},{"Start":"08:41.160 ","End":"08:44.850","Text":"it\u0027s easier to sketch and let\u0027s say they have an angle,"},{"Start":"08:44.850 ","End":"08:47.600","Text":"call it theta between them."},{"Start":"08:47.600 ","End":"08:52.055","Text":"The magnitude is just a between 2 bars."},{"Start":"08:52.055 ","End":"08:54.035","Text":"Magnitude is the length of the vector."},{"Start":"08:54.035 ","End":"08:56.720","Text":"If I didn\u0027t mention it before then I\u0027m mentioning it now."},{"Start":"08:56.720 ","End":"08:59.920","Text":"The magnitude of b is the length of b,"},{"Start":"08:59.920 ","End":"09:03.650","Text":"so this is just magnitude of b."},{"Start":"09:03.650 ","End":"09:08.680","Text":"Now it turns out that there\u0027s an important formula that"},{"Start":"09:08.680 ","End":"09:16.790","Text":"a dot b is also equal to this length,"},{"Start":"09:16.790 ","End":"09:19.055","Text":"which is this,"},{"Start":"09:19.055 ","End":"09:21.700","Text":"times just a regular, not a dot-product,"},{"Start":"09:21.700 ","End":"09:24.955","Text":"just a times multiplication of numbers"},{"Start":"09:24.955 ","End":"09:34.250","Text":"b times the cosine of the angle in between them."},{"Start":"09:34.850 ","End":"09:38.935","Text":"It\u0027s not just the product of the lengths,"},{"Start":"09:38.935 ","End":"09:42.810","Text":"the product of the lengths times the cosine of the angle in between them."},{"Start":"09:42.810 ","End":"09:49.190","Text":"I\u0027m going to assume that the angle is between 0 and 180 degrees or in"},{"Start":"09:49.190 ","End":"09:55.910","Text":"radians between 0 and Pi because if it\u0027s bigger than a 180 degrees,"},{"Start":"09:55.910 ","End":"09:58.565","Text":"I can just look at it from the other side."},{"Start":"09:58.565 ","End":"10:04.115","Text":"That\u0027s the result and I\u0027m not going to prove it;"},{"Start":"10:04.115 ","End":"10:07.460","Text":"1 of the uses of this formula or in"},{"Start":"10:07.460 ","End":"10:13.025","Text":"geometric interpretation is to find the angle between 2 vectors,"},{"Start":"10:13.025 ","End":"10:17.195","Text":"because I can write it as cosine of"},{"Start":"10:17.195 ","End":"10:22.235","Text":"theta is equal to and if I take this over to the other side,"},{"Start":"10:22.235 ","End":"10:29.645","Text":"I get a dot b over magnitude of a,"},{"Start":"10:29.645 ","End":"10:30.960","Text":"magnitude of b,"},{"Start":"10:30.960 ","End":"10:35.780","Text":"and I\u0027ll just put the vector sign over each of these."},{"Start":"10:35.780 ","End":"10:40.380","Text":"I\u0027ll show you an example of how we use it to get to the angle."},{"Start":"10:40.380 ","End":"10:47.080","Text":"We get to the cosine of the angle and then on the calculator we do the arc cosine."},{"Start":"10:47.080 ","End":"10:51.700","Text":"In the example, let\u0027s take a and b both 3-dimensional."},{"Start":"10:51.700 ","End":"10:58.680","Text":"Let\u0027s take a to be 3 minus 4 minus 1,"},{"Start":"10:58.680 ","End":"11:05.030","Text":"and we\u0027ll take b as 0,5."},{"Start":"11:05.030 ","End":"11:09.670","Text":"If we use this formula we get that the cosine of Theta,"},{"Start":"11:09.670 ","End":"11:11.740","Text":"which is the angle between these 2."},{"Start":"11:11.740 ","End":"11:15.130","Text":"Cosine of Theta is, first of all,"},{"Start":"11:15.130 ","End":"11:21.475","Text":"a.b, which is 3 times 0 is 0,"},{"Start":"11:21.475 ","End":"11:25.945","Text":"minus 4 times 5 is minus 20,"},{"Start":"11:25.945 ","End":"11:33.700","Text":"minus 1 times 2 is minus 2 all this over."},{"Start":"11:33.700 ","End":"11:40.930","Text":"The magnitude of a is the square root of 3 squared."},{"Start":"11:40.930 ","End":"11:48.460","Text":"We can ignore the minus plus 4 squared plus 1 squared times the square root,"},{"Start":"11:48.460 ","End":"11:54.475","Text":"0 squared plus 5 squared plus 2 squared."},{"Start":"11:54.475 ","End":"11:56.770","Text":"Let\u0027s see what this is equal to."},{"Start":"11:56.770 ","End":"12:05.350","Text":"This is minus 22 divided by 9 plus 16 plus 1."},{"Start":"12:05.350 ","End":"12:13.900","Text":"We\u0027ve got the square root of 26 times the square root of 25 plus 4 is 29."},{"Start":"12:13.900 ","End":"12:16.555","Text":"If we do this on the calculator,"},{"Start":"12:16.555 ","End":"12:21.770","Text":"this comes out approximately 0.8011927."},{"Start":"12:25.080 ","End":"12:27.910","Text":"Sorry, I forgot the minus."},{"Start":"12:27.910 ","End":"12:31.450","Text":"Then leave this on the calculator to have many places."},{"Start":"12:31.450 ","End":"12:34.190","Text":"Let\u0027s do the arc cosine."},{"Start":"12:34.200 ","End":"12:41.230","Text":"We need the arc cosine of the above or rather Theta."},{"Start":"12:41.230 ","End":"12:45.475","Text":"Theta is equal to the arc cosine of the,"},{"Start":"12:45.475 ","End":"12:48.265","Text":"I don\u0027t want to copy it, just whatever this is."},{"Start":"12:48.265 ","End":"12:51.130","Text":"Depending on what your calculator is set on,"},{"Start":"12:51.130 ","End":"12:58.405","Text":"if it\u0027s on degrees the answer you\u0027ll get is 143.24 so that\u0027s degrees."},{"Start":"12:58.405 ","End":"13:00.340","Text":"If it\u0027s set to radians,"},{"Start":"13:00.340 ","End":"13:04.300","Text":"you\u0027ll get 2.5 radians."},{"Start":"13:04.300 ","End":"13:06.400","Text":"We write with a little c here,"},{"Start":"13:06.400 ","End":"13:09.685","Text":"depending on how you want your result."},{"Start":"13:09.685 ","End":"13:13.420","Text":"Another thing we can deduce from this formula is we"},{"Start":"13:13.420 ","End":"13:17.245","Text":"can tell when 2 vectors are perpendicular."},{"Start":"13:17.245 ","End":"13:21.880","Text":"Actually, instead of the word perpendicular in this context,"},{"Start":"13:21.880 ","End":"13:23.470","Text":"we are using another word."},{"Start":"13:23.470 ","End":"13:25.135","Text":"We use the word orthogonal,"},{"Start":"13:25.135 ","End":"13:27.100","Text":"so I\u0027m going to, from now on,"},{"Start":"13:27.100 ","End":"13:32.080","Text":"say that 2 vectors are orthogonal and you\u0027ll know that I mean perpendicular,"},{"Start":"13:32.080 ","End":"13:35.170","Text":"which means that 90 degrees to each other."},{"Start":"13:35.170 ","End":"13:38.350","Text":"Let\u0027s assume that these vectors are not 0."},{"Start":"13:38.350 ","End":"13:40.270","Text":"If 1 of them is 0 vector,"},{"Start":"13:40.270 ","End":"13:42.430","Text":"I don\u0027t know what it means to be perpendicular."},{"Start":"13:42.430 ","End":"13:45.220","Text":"Assume we\u0027re talking about non-zero vectors,"},{"Start":"13:45.220 ","End":"13:48.985","Text":"then if the lines are orthogonal,"},{"Start":"13:48.985 ","End":"13:55.300","Text":"it means that Theta is 90 degrees and then cosine of 90 degrees is 0."},{"Start":"13:55.300 ","End":"13:59.470","Text":"I\u0027ll just remind you that cosine of 90 degrees is 0,"},{"Start":"13:59.470 ","End":"14:02.620","Text":"which means that we get that this thing is 0."},{"Start":"14:02.620 ","End":"14:04.660","Text":"If the vectors are not 0,"},{"Start":"14:04.660 ","End":"14:06.145","Text":"the magnitudes are not 0,"},{"Start":"14:06.145 ","End":"14:13.090","Text":"it must mean that a dot-product with b is 0, and vice versa."},{"Start":"14:13.090 ","End":"14:14.980","Text":"If the dot-product is 0,"},{"Start":"14:14.980 ","End":"14:16.405","Text":"then the cosine is 0,"},{"Start":"14:16.405 ","End":"14:23.965","Text":"so the angle is 90 degrees or Pi over 2 if we\u0027re talking about radians."},{"Start":"14:23.965 ","End":"14:26.905","Text":"The condition for orthogonal,"},{"Start":"14:26.905 ","End":"14:35.170","Text":"and we\u0027re talking about vectors a and b is that a.b equals 0."},{"Start":"14:35.170 ","End":"14:38.770","Text":"That\u0027s the condition for perpendicular or orthogonal."},{"Start":"14:38.770 ","End":"14:41.230","Text":"While we\u0027re at it, we might as well talk about"},{"Start":"14:41.230 ","End":"14:44.695","Text":"the condition for 2 vectors to be parallel."},{"Start":"14:44.695 ","End":"14:48.400","Text":"There\u0027s 2 ways for vectors to be parallel."},{"Start":"14:48.400 ","End":"14:54.940","Text":"Either they point in the same direction or they\u0027re in exactly opposite directions."},{"Start":"14:54.940 ","End":"14:59.274","Text":"In other words, for 2 vectors to be parallel,"},{"Start":"14:59.274 ","End":"15:03.370","Text":"Theta\u0027s got to be equal to either 0 degrees,"},{"Start":"15:03.370 ","End":"15:08.125","Text":"I\u0027ll talk in degrees or 180 degrees."},{"Start":"15:08.125 ","End":"15:13.600","Text":"Each of these has an interpretation in terms of the dot-product,"},{"Start":"15:13.600 ","End":"15:15.505","Text":"just move this aside."},{"Start":"15:15.505 ","End":"15:19.765","Text":"What it means is that if Theta\u0027s 0 degrees,"},{"Start":"15:19.765 ","End":"15:22.315","Text":"and we put it in this formula here,"},{"Start":"15:22.315 ","End":"15:27.775","Text":"cosine of 0 is 1."},{"Start":"15:27.775 ","End":"15:37.120","Text":"We get the formula that the dot-product is equal to just the magnitude of the first,"},{"Start":"15:37.120 ","End":"15:43.014","Text":"times the magnitude of the second, and vector signs."},{"Start":"15:43.014 ","End":"15:46.285","Text":"That\u0027s for 1 way of being parallel."},{"Start":"15:46.285 ","End":"15:49.915","Text":"The other way of being parallel is for the 180 degrees."},{"Start":"15:49.915 ","End":"15:52.765","Text":"Cosine of 180 degrees is minus 1,"},{"Start":"15:52.765 ","End":"15:59.170","Text":"so we would get that a.b vector"},{"Start":"15:59.170 ","End":"16:05.980","Text":"is minus the magnitude of a times the magnitude of b and we can test this."},{"Start":"16:05.980 ","End":"16:13.620","Text":"Let me give an example with the numbers. Let\u0027s check."},{"Start":"16:13.620 ","End":"16:15.765","Text":"In my first example,"},{"Start":"16:15.765 ","End":"16:24.940","Text":"I\u0027ll take a to equal 6 minus 2 minus 1."},{"Start":"16:25.430 ","End":"16:32.380","Text":"Let\u0027s take b to equal 2, 5, 2."},{"Start":"16:32.380 ","End":"16:34.990","Text":"I\u0027m taking examples in 3 dimensions."},{"Start":"16:34.990 ","End":"16:38.260","Text":"Let\u0027s see what is a.b."},{"Start":"16:38.260 ","End":"16:43.795","Text":"Let\u0027s see if they are orthogonal or parallel or maybe neither."},{"Start":"16:43.795 ","End":"16:49.180","Text":"So a.b is 6 times 2 is 12,"},{"Start":"16:49.180 ","End":"16:52.405","Text":"minus 2 times 5 is minus 10,"},{"Start":"16:52.405 ","End":"16:56.320","Text":"minus 1 times 2 is minus 2."},{"Start":"16:56.320 ","End":"16:58.510","Text":"This is equal to 0."},{"Start":"16:58.510 ","End":"17:04.010","Text":"I know that these 2 vectors are orthogonal."},{"Start":"17:05.760 ","End":"17:09.535","Text":"Now let\u0027s take another example."},{"Start":"17:09.535 ","End":"17:13.585","Text":"The other example, I\u0027ll use the other notation,"},{"Start":"17:13.585 ","End":"17:17.965","Text":"u is 2i minus j. Yeah,"},{"Start":"17:17.965 ","End":"17:19.630","Text":"this is going to be a 2D example."},{"Start":"17:19.630 ","End":"17:21.729","Text":"This was like a 3D example."},{"Start":"17:21.729 ","End":"17:27.640","Text":"This is a 2D example and the other vector we\u0027ll call it V and it will"},{"Start":"17:27.640 ","End":"17:37.585","Text":"be minus a 1/2i plus a 1/4j."},{"Start":"17:37.585 ","End":"17:39.670","Text":"It\u0027s easier to work with this notation,"},{"Start":"17:39.670 ","End":"17:45.580","Text":"so let me just convert it right away 2 minus 1 is u,"},{"Start":"17:45.580 ","End":"17:54.580","Text":"and here I have minus 1/2, 1/4."},{"Start":"17:54.580 ","End":"17:57.860","Text":"If I want to know what is u.v,"},{"Start":"17:59.040 ","End":"18:04.510","Text":"I\u0027ll take this representation it\u0027s easier and do it in our heads,"},{"Start":"18:04.510 ","End":"18:07.180","Text":"2 times minus 1/2."},{"Start":"18:07.180 ","End":"18:08.590","Text":"Hold on, I\u0027ll write it,"},{"Start":"18:08.590 ","End":"18:11.305","Text":"2 times minus a 1/2 is minus 1,"},{"Start":"18:11.305 ","End":"18:16.555","Text":"and minus 1 times a 1/4"},{"Start":"18:16.555 ","End":"18:23.245","Text":"is minus a 1/4 and I\u0027ll write it as an improper fraction,"},{"Start":"18:23.245 ","End":"18:25.525","Text":"minus 5 over 4."},{"Start":"18:25.525 ","End":"18:29.155","Text":"Anyway, it\u0027s not 0, so these 2 are not orthogonal."},{"Start":"18:29.155 ","End":"18:35.485","Text":"The next thing we might want to check is, are they parallel?"},{"Start":"18:35.485 ","End":"18:37.720","Text":"Notice that for the parallel,"},{"Start":"18:37.720 ","End":"18:41.140","Text":"we compare the dot-product with the product of the magnitudes and"},{"Start":"18:41.140 ","End":"18:44.710","Text":"it either comes out the same or the minus and that\u0027s going to be good."},{"Start":"18:44.710 ","End":"18:49.150","Text":"Let\u0027s try now to see what is magnitude of"},{"Start":"18:49.150 ","End":"18:57.310","Text":"u times magnitude of v. This is a dot-product,"},{"Start":"18:57.310 ","End":"19:01.405","Text":"this is just a regular dot for multiplication of scalars."},{"Start":"19:01.405 ","End":"19:11.150","Text":"What do we get? We get magnitude of u is the square root of 2 squared plus 1 squared."},{"Start":"19:11.970 ","End":"19:18.640","Text":"Yes, 2D, there\u0027s only 2 terms times the square root of"},{"Start":"19:18.640 ","End":"19:25.630","Text":"a 1/2 squared plus a 1/4 squared."},{"Start":"19:25.630 ","End":"19:29.905","Text":"Let\u0027s see, this is the square root of 5,"},{"Start":"19:29.905 ","End":"19:35.950","Text":"and this is the square root of 1/4 plus a 1/16."},{"Start":"19:35.950 ","End":"19:37.330","Text":"I put it all over 16."},{"Start":"19:37.330 ","End":"19:38.485","Text":"It\u0027s 4 plus 1,"},{"Start":"19:38.485 ","End":"19:43.430","Text":"it\u0027s 5/16, so we have 5/16."},{"Start":"19:43.430 ","End":"19:45.250","Text":"What this equals is,"},{"Start":"19:45.250 ","End":"19:54.890","Text":"the square root of 5 times the square root of 5 over the square root of 16."},{"Start":"19:55.850 ","End":"20:03.535","Text":"This equals 5/4 because this times this is 5 and this is 4."},{"Start":"20:03.535 ","End":"20:07.300","Text":"Now, this is not the same as this,"},{"Start":"20:07.300 ","End":"20:10.000","Text":"it\u0027s the minus of this."},{"Start":"20:10.000 ","End":"20:14.635","Text":"It turns out that they are parallel and in fact,"},{"Start":"20:14.635 ","End":"20:17.590","Text":"they are in opposite directions."},{"Start":"20:17.590 ","End":"20:23.210","Text":"I think we\u0027re done with that topic and let\u0027s take a break now."}],"ID":10646},{"Watched":false,"Name":"Vectors - Dot Product (continued)","Duration":"15m 36s","ChapterTopicVideoID":10302,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"I\u0027m going to go onto the next topic,"},{"Start":"00:02.430 ","End":"00:07.000","Text":"which is called projections."},{"Start":"00:07.610 ","End":"00:16.245","Text":"Let\u0027s say I have 2 vectors, a and b."},{"Start":"00:16.245 ","End":"00:21.645","Text":"Then I\u0027m going to define a concept called the projection."},{"Start":"00:21.645 ","End":"00:25.410","Text":"Let\u0027s see, projection of b onto a."},{"Start":"00:25.410 ","End":"00:36.450","Text":"We\u0027re going to denote it as proj of b onto a."},{"Start":"00:36.450 ","End":"00:40.100","Text":"It gets a bit tedious to draw these arrows all the time."},{"Start":"00:40.100 ","End":"00:43.160","Text":"I suppose you could leave them out sometimes."},{"Start":"00:43.230 ","End":"00:47.850","Text":"I\u0027ll give you a diagram of what this means."},{"Start":"00:48.320 ","End":"00:51.770","Text":"If this is a and this is b,"},{"Start":"00:51.770 ","End":"00:59.150","Text":"the projection is you take the vector and you drop a perpendicular onto a."},{"Start":"00:59.150 ","End":"01:00.710","Text":"It\u0027s as if, I don\u0027t know,"},{"Start":"01:00.710 ","End":"01:06.560","Text":"you had a projector like the sun or a light source from above."},{"Start":"01:06.560 ","End":"01:10.220","Text":"This would be like the shadow that b would form onto a."},{"Start":"01:10.220 ","End":"01:13.040","Text":"If a is the ground and this is up in the air,"},{"Start":"01:13.040 ","End":"01:15.125","Text":"however you want to imagine it."},{"Start":"01:15.125 ","End":"01:21.660","Text":"But you take the tip of b and you drop a perpendicular line to the vector a."},{"Start":"01:22.060 ","End":"01:24.080","Text":"What could happen though,"},{"Start":"01:24.080 ","End":"01:26.270","Text":"is for an obtuse angle,"},{"Start":"01:26.270 ","End":"01:30.440","Text":"you get that the projection is actually not onto a itself,"},{"Start":"01:30.440 ","End":"01:32.060","Text":"but the continuation of a."},{"Start":"01:32.060 ","End":"01:35.675","Text":"In fact, the projection in this case,"},{"Start":"01:35.675 ","End":"01:39.420","Text":"which is this vector in blue as it is here,"},{"Start":"01:39.420 ","End":"01:42.200","Text":"in this case it\u0027s in the opposite direction from a,"},{"Start":"01:42.200 ","End":"01:45.730","Text":"and in this case it\u0027s in the same direction of a."},{"Start":"01:45.730 ","End":"01:48.220","Text":"These are the 2 situations."},{"Start":"01:48.220 ","End":"01:49.760","Text":"If it\u0027s 90 degrees,"},{"Start":"01:49.760 ","End":"01:53.300","Text":"then the projection\u0027s just going to be 0."},{"Start":"01:53.300 ","End":"01:56.240","Text":"Now, the formal definition,"},{"Start":"01:56.240 ","End":"01:57.950","Text":"not the picture definition,"},{"Start":"01:57.950 ","End":"02:05.495","Text":"is that the projection of b onto a is equal to,"},{"Start":"02:05.495 ","End":"02:12.390","Text":"it\u0027s going to be some scalar times a. I know that there\u0027s going to be an a in here."},{"Start":"02:12.470 ","End":"02:15.830","Text":"The scalar is going to be as follows."},{"Start":"02:15.830 ","End":"02:20.120","Text":"It\u0027s going to be the dot product of a with"},{"Start":"02:20.120 ","End":"02:28.340","Text":"b over the magnitude of a squared."},{"Start":"02:28.340 ","End":"02:35.105","Text":"Notice that the projection of b onto a is not the same as the projection of a onto b."},{"Start":"02:35.105 ","End":"02:37.520","Text":"If I did the projection of a onto b,"},{"Start":"02:37.520 ","End":"02:40.280","Text":"I would be drawing a perpendicular onto b."},{"Start":"02:40.280 ","End":"02:46.230","Text":"It would be parallel to b it\u0027d be in the same direction as b."},{"Start":"02:46.230 ","End":"02:47.855","Text":"It\u0027d be something different."},{"Start":"02:47.855 ","End":"02:50.690","Text":"If you just change a with b and b with a everywhere,"},{"Start":"02:50.690 ","End":"02:52.490","Text":"you\u0027ll get the opposite formula."},{"Start":"02:52.490 ","End":"02:54.275","Text":"You don\u0027t need an extra formula."},{"Start":"02:54.275 ","End":"02:56.875","Text":"Let\u0027s take a numerical example."},{"Start":"02:56.875 ","End":"03:00.720","Text":"I wrote down the 2 vectors;"},{"Start":"03:00.720 ","End":"03:03.240","Text":"a is this, b is this."},{"Start":"03:03.240 ","End":"03:11.660","Text":"What I want is the projection of b onto a."},{"Start":"03:11.660 ","End":"03:13.975","Text":"What does that equal?"},{"Start":"03:13.975 ","End":"03:17.060","Text":"You know what? Let\u0027s also do it the other way around."},{"Start":"03:17.060 ","End":"03:20.030","Text":"I\u0027ll just show you that it\u0027s different, completely different."},{"Start":"03:20.030 ","End":"03:25.970","Text":"We\u0027ll do also the projection onto b of a, of a onto b."},{"Start":"03:25.970 ","End":"03:27.980","Text":"What does that equal?"},{"Start":"03:27.980 ","End":"03:32.850","Text":"In the first 1, we just use the formula straight as is."},{"Start":"03:33.820 ","End":"03:37.425","Text":"Let\u0027s copy them out here, these things."},{"Start":"03:37.425 ","End":"03:42.260","Text":"Now, we\u0027ll start projection of b onto a using this formula."},{"Start":"03:42.260 ","End":"03:45.645","Text":"First of all, a dot b."},{"Start":"03:45.645 ","End":"03:48.635","Text":"We have a fraction here and on the top,"},{"Start":"03:48.635 ","End":"03:53.690","Text":"1 times 2 is 2 plus 0 times 1 is 0,"},{"Start":"03:53.690 ","End":"03:57.330","Text":"minus 2 times minus 1 is plus 2."},{"Start":"03:57.330 ","End":"04:02.315","Text":"Here, the magnitude of a squared is the magnitude of this."},{"Start":"04:02.315 ","End":"04:06.305","Text":"It\u0027s just 1 squared plus 0 squared plus 2 squared."},{"Start":"04:06.305 ","End":"04:08.600","Text":"For the magnitude I would take the square root,"},{"Start":"04:08.600 ","End":"04:10.030","Text":"but then I square it again."},{"Start":"04:10.030 ","End":"04:12.555","Text":"Don\u0027t bother even doing the square root."},{"Start":"04:12.555 ","End":"04:16.910","Text":"All this times the vector a,"},{"Start":"04:16.910 ","End":"04:22.200","Text":"which is 1 comma 0 minus 2. Now, what do we have here?"},{"Start":"04:22.200 ","End":"04:23.955","Text":"2 plus 2 is 4,"},{"Start":"04:23.955 ","End":"04:26.610","Text":"1 plus 4 is 5."},{"Start":"04:26.610 ","End":"04:28.245","Text":"This is 4,"},{"Start":"04:28.245 ","End":"04:37.200","Text":"this part here is 4/5 of this 1,0 minus 2,"},{"Start":"04:37.200 ","End":"04:43.620","Text":"and this equals 4/5, 4/5 times 0 is 0."},{"Start":"04:43.620 ","End":"04:46.605","Text":"4/5 times minus 2 is minus 8/5."},{"Start":"04:46.605 ","End":"04:48.420","Text":"That\u0027s the first 1. Now,"},{"Start":"04:48.420 ","End":"04:50.000","Text":"how about the second 1?"},{"Start":"04:50.000 ","End":"04:52.510","Text":"We\u0027ll just reverse a and b everywhere."},{"Start":"04:52.510 ","End":"04:56.110","Text":"We still need the dot-product and it doesn\u0027t matter in what order."},{"Start":"04:56.110 ","End":"05:03.250","Text":"On this numerator, we still get 4 the 2 plus 0 plus 2 is going to be the same."},{"Start":"05:03.250 ","End":"05:05.380","Text":"But here already there\u0027s going to be a difference."},{"Start":"05:05.380 ","End":"05:07.735","Text":"We need the magnitude of b squared."},{"Start":"05:07.735 ","End":"05:13.380","Text":"This time it\u0027s 2 squared plus 1 squared plus 1 squared."},{"Start":"05:13.380 ","End":"05:15.240","Text":"Also here, a different vector."},{"Start":"05:15.240 ","End":"05:16.880","Text":"It\u0027s going to be vector b,"},{"Start":"05:16.880 ","End":"05:20.900","Text":"which is 2, 1 minus 1."},{"Start":"05:20.900 ","End":"05:24.620","Text":"This time it\u0027s equal to 2 plus 1,"},{"Start":"05:24.620 ","End":"05:27.655","Text":"sorry, 4 plus 1 plus 1 is 6,"},{"Start":"05:27.655 ","End":"05:35.520","Text":"4 over 6 is 2/3 of vector 2, 1 minus 1."},{"Start":"05:35.520 ","End":"05:40.305","Text":"This time we get 2 times 2/3 is 4/3,"},{"Start":"05:40.305 ","End":"05:42.945","Text":"1 times 2/3 is 2/3,"},{"Start":"05:42.945 ","End":"05:46.660","Text":"and minus 1 is minus 2/3."},{"Start":"05:46.670 ","End":"05:49.065","Text":"That\u0027s it for projections."},{"Start":"05:49.065 ","End":"05:51.100","Text":"Now, let\u0027s move on."},{"Start":"05:51.100 ","End":"05:55.160","Text":"The last topic under dot-product,"},{"Start":"05:55.160 ","End":"06:00.305","Text":"is something called direction cosines and direction angles."},{"Start":"06:00.305 ","End":"06:03.560","Text":"This is peculiar to 3D."},{"Start":"06:03.560 ","End":"06:06.650","Text":"Normally everything we\u0027ve said so far is okay for 2D,"},{"Start":"06:06.650 ","End":"06:08.585","Text":"3D, 4D, whatever."},{"Start":"06:08.585 ","End":"06:11.064","Text":"This 1 is, just for 3D."},{"Start":"06:11.064 ","End":"06:13.749","Text":"I\u0027ll show you a picture."},{"Start":"06:13.780 ","End":"06:18.380","Text":"In this picture we have a coordinate system, x, y,"},{"Start":"06:18.380 ","End":"06:23.670","Text":"and z, and we have a vector, call it a."},{"Start":"06:23.860 ","End":"06:33.575","Text":"What we want to know are the angles that this vector forms with the axis that say,"},{"Start":"06:33.575 ","End":"06:39.140","Text":"forms an angle of Alpha with the x-axis,"},{"Start":"06:39.140 ","End":"06:41.060","Text":"Beta with the y-axis,"},{"Start":"06:41.060 ","End":"06:43.000","Text":"and Gamma with the z-axis."},{"Start":"06:43.000 ","End":"06:46.040","Text":"We\u0027re usually more concerned with the cosine of the angles."},{"Start":"06:46.040 ","End":"06:48.450","Text":"Well, both of them."},{"Start":"06:48.940 ","End":"06:52.615","Text":"Turns out that there is a formula."},{"Start":"06:52.615 ","End":"06:58.280","Text":"Here\u0027s the formula. Well, 3 formulas for the cosine of each of these angles,"},{"Start":"06:58.280 ","End":"07:00.570","Text":"Alpha, Beta, and Gamma."},{"Start":"07:00.730 ","End":"07:05.180","Text":"Of course, once we have the cosine of the angle we\u0027ll be able to find the angles too."},{"Start":"07:05.180 ","End":"07:09.500","Text":"But let\u0027s look meanwhile what the cosine of Alpha is."},{"Start":"07:09.500 ","End":"07:15.255","Text":"It\u0027s a the vector dot with i,"},{"Start":"07:15.255 ","End":"07:20.670","Text":"the standard basis vector i and over the magnitude of a."},{"Start":"07:20.670 ","End":"07:23.855","Text":"Now then everything else is just the same except instead of i,"},{"Start":"07:23.855 ","End":"07:29.390","Text":"we have j and k. I\u0027m not going to repeat the meaning of i,"},{"Start":"07:29.390 ","End":"07:32.400","Text":"j and k. You\u0027re supposed to remember."},{"Start":"07:32.400 ","End":"07:36.025","Text":"This is like 100, 010, 001."},{"Start":"07:36.025 ","End":"07:39.810","Text":"Go back and look if you\u0027ve forgotten."},{"Start":"07:40.300 ","End":"07:46.080","Text":"There is an alternative formula for each of these."},{"Start":"07:46.090 ","End":"07:51.455","Text":"For example, well, let\u0027s say that a is a1,"},{"Start":"07:51.455 ","End":"07:56.575","Text":"a2, a3, just to give it some components coordinates."},{"Start":"07:56.575 ","End":"08:04.410","Text":"In that case, the first 1 would be a1,"},{"Start":"08:04.410 ","End":"08:12.790","Text":"a2, a3 dot i is 1, 0, 0."},{"Start":"08:13.020 ","End":"08:15.340","Text":"Well, leave the denominator alone."},{"Start":"08:15.340 ","End":"08:17.500","Text":"I just want to see what the numerator is."},{"Start":"08:17.500 ","End":"08:23.680","Text":"This thing is equal to a_1 times 1 plus a_2 times 0 plus a_3 times 0."},{"Start":"08:23.680 ","End":"08:24.910","Text":"Because we have 2 0s,"},{"Start":"08:24.910 ","End":"08:26.710","Text":"all we\u0027re left with is the a_1 times 1,"},{"Start":"08:26.710 ","End":"08:28.885","Text":"so we\u0027ve just got a_1."},{"Start":"08:28.885 ","End":"08:36.880","Text":"This thing turns out to equal a_1 over the magnitude of a."},{"Start":"08:36.880 ","End":"08:41.620","Text":"Similarly, here we\u0027d get a_2,"},{"Start":"08:41.620 ","End":"08:43.525","Text":"here we\u0027d get a_3,"},{"Start":"08:43.525 ","End":"08:45.939","Text":"and everything is over"},{"Start":"08:45.939 ","End":"08:54.805","Text":"the magnitude of vector a."},{"Start":"08:54.805 ","End":"08:56.710","Text":"It\u0027s a bit crowded here,"},{"Start":"08:56.710 ","End":"08:58.840","Text":"but I think you can follow,"},{"Start":"08:58.840 ","End":"09:03.980","Text":"put some separators here, 3 formulas."},{"Start":"09:04.050 ","End":"09:06.625","Text":"I\u0027m going to erase this."},{"Start":"09:06.625 ","End":"09:13.645","Text":"Now these 3 cosines are called direction cosines of the vector a,"},{"Start":"09:13.645 ","End":"09:16.375","Text":"and the Alpha, Beta,"},{"Start":"09:16.375 ","End":"09:19.885","Text":"Gamma themselves are called direction angles."},{"Start":"09:19.885 ","End":"09:24.535","Text":"Like I said, if we have the cosine on the calculator,"},{"Start":"09:24.535 ","End":"09:29.110","Text":"we can always do the arc cosine of a number to get from its cosine back to the angle."},{"Start":"09:29.110 ","End":"09:30.895","Text":"We\u0027ll see that in the example."},{"Start":"09:30.895 ","End":"09:36.490","Text":"Let\u0027s say the vector a was this 2, 1, minus 4."},{"Start":"09:36.490 ","End":"09:43.344","Text":"I want to find the direction cosines and the direction angles."},{"Start":"09:43.344 ","End":"09:47.515","Text":"Notice that in all these formulas we need the magnitude of a."},{"Start":"09:47.515 ","End":"09:49.285","Text":"Let\u0027s just do that first."},{"Start":"09:49.285 ","End":"09:54.415","Text":"Magnitude of a is the square root of what?"},{"Start":"09:54.415 ","End":"09:59.950","Text":"2 squared plus 1 squared and we ignore the minus 4 squared."},{"Start":"09:59.950 ","End":"10:10.220","Text":"That comes out to be 4 plus 1 plus 16, that would be 21."},{"Start":"10:10.800 ","End":"10:16.180","Text":"Cosine of Alpha, according to this formula a_1,"},{"Start":"10:16.180 ","End":"10:18.655","Text":"which is the first component,"},{"Start":"10:18.655 ","End":"10:24.160","Text":"2 over the magnitude of a square root of 21."},{"Start":"10:24.160 ","End":"10:29.320","Text":"Similarly, cosine Beta is 1 over square root of"},{"Start":"10:29.320 ","End":"10:37.400","Text":"21 and cosine Gamma minus 4 over root of 21."},{"Start":"10:38.040 ","End":"10:44.155","Text":"All we need now is to do this on the calculator."},{"Start":"10:44.155 ","End":"10:49.795","Text":"Then if we do the arc cosine we can get Alpha,"},{"Start":"10:49.795 ","End":"10:51.355","Text":"Beta, and Gamma."},{"Start":"10:51.355 ","End":"10:54.160","Text":"It depends on what your calculator is set to,"},{"Start":"10:54.160 ","End":"10:56.510","Text":"if it\u0027s degrees or radians."},{"Start":"10:56.510 ","End":"10:58.680","Text":"If it\u0027s set to degrees,"},{"Start":"10:58.680 ","End":"11:03.905","Text":"then this comes out to 64.123 degrees."},{"Start":"11:03.905 ","End":"11:08.290","Text":"Here, 77.396 degrees."},{"Start":"11:08.290 ","End":"11:11.095","Text":"I\u0027m rounding to 3 places of course."},{"Start":"11:11.095 ","End":"11:18.415","Text":"The last 1 is 150.794 degrees."},{"Start":"11:18.415 ","End":"11:22.420","Text":"Notice that the negative ones come out bigger than 90 degrees."},{"Start":"11:22.420 ","End":"11:24.860","Text":"That\u0027s how it works."},{"Start":"11:25.200 ","End":"11:27.970","Text":"If you had it set to radians,"},{"Start":"11:27.970 ","End":"11:30.790","Text":"I\u0027ll just show you what you would get."},{"Start":"11:30.790 ","End":"11:35.590","Text":"Well, here they are, I just wrote them out for you in case you were doing it in radians."},{"Start":"11:35.590 ","End":"11:37.885","Text":"Both are okay here."},{"Start":"11:37.885 ","End":"11:39.760","Text":"The cosines, of course, are the same."},{"Start":"11:39.760 ","End":"11:43.045","Text":"It\u0027s just that when you take the arc cosine or inverse cosine,"},{"Start":"11:43.045 ","End":"11:46.480","Text":"then it depends on what your calculator is set to."},{"Start":"11:46.480 ","End":"11:50.290","Text":"We\u0027re almost done. I just want to show you some formulas."},{"Start":"11:50.290 ","End":"11:53.500","Text":"There\u0027s 3 of them actually I wanted to just leave you with."},{"Start":"11:53.500 ","End":"11:58.450","Text":"One of them is that the vector a is equal to the magnitude of"},{"Start":"11:58.450 ","End":"12:06.925","Text":"a times the vector made up of cosine Alpha,"},{"Start":"12:06.925 ","End":"12:11.485","Text":"cosine Beta, cosine Gamma."},{"Start":"12:11.485 ","End":"12:14.240","Text":"I\u0027m going to highlight it."},{"Start":"12:16.530 ","End":"12:19.810","Text":"I\u0027m even going to show you why this is so, let\u0027s see."},{"Start":"12:19.810 ","End":"12:28.030","Text":"The left hand side would be a_1, a_2, a_3."},{"Start":"12:28.030 ","End":"12:35.620","Text":"The question is is this equal to, let\u0027s see."},{"Start":"12:35.620 ","End":"12:39.085","Text":"The right-hand side is magnitude of a."},{"Start":"12:39.085 ","End":"12:47.545","Text":"Now, cosine Alpha is a_1 over magnitude of a."},{"Start":"12:47.545 ","End":"12:55.795","Text":"Then we have a_2 and a_3 and each of them is over magnitude of"},{"Start":"12:55.795 ","End":"13:04.075","Text":"a. I say that"},{"Start":"13:04.075 ","End":"13:07.870","Text":"this is equal because we said that if you multiply a scalar by a vector,"},{"Start":"13:07.870 ","End":"13:09.340","Text":"you multiply each component."},{"Start":"13:09.340 ","End":"13:11.440","Text":"This cancels with this, with this, with this."},{"Start":"13:11.440 ","End":"13:14.780","Text":"So this 1 is true."},{"Start":"13:15.210 ","End":"13:19.375","Text":"The second of the 3 is that this vector here,"},{"Start":"13:19.375 ","End":"13:20.680","Text":"let\u0027s call it u,"},{"Start":"13:20.680 ","End":"13:22.555","Text":"I have a reason for calling it u,"},{"Start":"13:22.555 ","End":"13:26.980","Text":"which is the cosine of Alpha,"},{"Start":"13:26.980 ","End":"13:30.835","Text":"cosine Beta, cosine Gamma."},{"Start":"13:30.835 ","End":"13:33.040","Text":"This is a unit vector."},{"Start":"13:33.040 ","End":"13:35.720","Text":"That\u0027s why I called it u."},{"Start":"13:36.180 ","End":"13:39.775","Text":"Once again, I\u0027m going to explain to you why."},{"Start":"13:39.775 ","End":"13:42.550","Text":"I changed my mind about the highlighting."},{"Start":"13:42.550 ","End":"13:49.730","Text":"The reason is that u is just a over the magnitude of a,"},{"Start":"13:51.660 ","End":"13:54.310","Text":"and this thing here is a scalar."},{"Start":"13:54.310 ","End":"14:01.810","Text":"If I take the magnitude of a vector over a positive scalar,"},{"Start":"14:01.810 ","End":"14:04.540","Text":"then this thing which is u,"},{"Start":"14:04.540 ","End":"14:07.615","Text":"if I take the magnitude of this,"},{"Start":"14:07.615 ","End":"14:11.530","Text":"whenever you have a positive constant times a vector,"},{"Start":"14:11.530 ","End":"14:13.300","Text":"you can take it outside."},{"Start":"14:13.300 ","End":"14:21.205","Text":"So it\u0027s 1 over the magnitude of a times the magnitude of a."},{"Start":"14:21.205 ","End":"14:26.785","Text":"Well, and this is obviously just equal to 1 so that\u0027s why it\u0027s a unit vector."},{"Start":"14:26.785 ","End":"14:29.395","Text":"If I want to rephrase this,"},{"Start":"14:29.395 ","End":"14:31.105","Text":"to say that this is a unit vector,"},{"Start":"14:31.105 ","End":"14:35.920","Text":"it means the square root of this squared plus this squared plus this squared is 1."},{"Start":"14:35.920 ","End":"14:41.140","Text":"What that says is that cosine squared Alpha plus cosine"},{"Start":"14:41.140 ","End":"14:47.920","Text":"squared Beta plus cosine squared Gamma is equal to 1."},{"Start":"14:47.920 ","End":"14:49.960","Text":"I should have put the square root here,"},{"Start":"14:49.960 ","End":"14:52.120","Text":"but if the square root of it is 1,"},{"Start":"14:52.120 ","End":"14:53.890","Text":"then it is 1 also."},{"Start":"14:53.890 ","End":"14:59.809","Text":"Actually we\u0027ve even verified all these 3 formulas."},{"Start":"15:00.720 ","End":"15:10.300","Text":"This has been verified and I\u0027ve explained this also and let\u0027s just label them."},{"Start":"15:10.300 ","End":"15:15.220","Text":"Maybe this would be 1,2"},{"Start":"15:15.220 ","End":"15:21.039","Text":"and 3 and just useful formulas."},{"Start":"15:21.039 ","End":"15:24.020","Text":"I\u0027ll just erase the proofs."},{"Start":"15:24.510 ","End":"15:29.260","Text":"Here they are for future reference and we\u0027re done with"},{"Start":"15:29.260 ","End":"15:35.660","Text":"the subject of dot products and the sub topic of direction cosines."}],"ID":10647},{"Watched":false,"Name":"Exercise 1","Duration":"2m 46s","ChapterTopicVideoID":10304,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"This exercise is in 3 parts and each of them is an a and a b vectors,"},{"Start":"00:05.490 ","End":"00:07.679","Text":"and we have to find the dot product."},{"Start":"00:07.679 ","End":"00:09.645","Text":"In number 1,"},{"Start":"00:09.645 ","End":"00:15.810","Text":"both of them are in 2 dimensions and we have angular bracket notation,"},{"Start":"00:15.810 ","End":"00:20.790","Text":"and we just use the formula a dot b,"},{"Start":"00:20.790 ","End":"00:23.880","Text":"we multiply component-wise and add."},{"Start":"00:23.880 ","End":"00:34.230","Text":"We\u0027ve got 5 times 4 plus negative 4 times 3,"},{"Start":"00:34.230 ","End":"00:38.370","Text":"and let\u0027s see, this is 20 minus 12,"},{"Start":"00:38.370 ","End":"00:39.945","Text":"which is 8,"},{"Start":"00:39.945 ","End":"00:42.210","Text":"and that\u0027s all there is."},{"Start":"00:42.210 ","End":"00:47.715","Text":"In 2, they\u0027re both 3-dimensional vectors,"},{"Start":"00:47.715 ","End":"00:49.639","Text":"we have the ijk notation,"},{"Start":"00:49.639 ","End":"00:51.430","Text":"but it\u0027s the same idea,"},{"Start":"00:51.430 ","End":"00:55.950","Text":"we just multiply component-wise and add."},{"Start":"00:55.950 ","End":"00:58.675","Text":"We\u0027ve got 8 times 6,"},{"Start":"00:58.675 ","End":"01:01.820","Text":"and then we have plus 6 times minus 4,"},{"Start":"01:01.820 ","End":"01:05.045","Text":"so write it as minus 6 times 4,"},{"Start":"01:05.045 ","End":"01:07.970","Text":"and then also a minus and a plus,"},{"Start":"01:07.970 ","End":"01:11.525","Text":"so minus 3 times 7."},{"Start":"01:11.525 ","End":"01:19.210","Text":"That will give us 48 minus 24 minus 21,"},{"Start":"01:19.210 ","End":"01:24.700","Text":"I make that 48 minus 45 is 3."},{"Start":"01:24.700 ","End":"01:26.920","Text":"In the third question,"},{"Start":"01:26.920 ","End":"01:31.780","Text":"we\u0027re not given directly what the vectors a and b are,"},{"Start":"01:31.780 ","End":"01:33.625","Text":"we\u0027re just given some hints about them."},{"Start":"01:33.625 ","End":"01:38.575","Text":"That this has magnitude 4 magnitude 3 and we know the angle between them."},{"Start":"01:38.575 ","End":"01:41.935","Text":"But remember there\u0027s another way for dot-product."},{"Start":"01:41.935 ","End":"01:46.020","Text":"If you have a vector or 2 vectors,"},{"Start":"01:46.020 ","End":"01:50.595","Text":"one of them a and 1 of them b."},{"Start":"01:50.595 ","End":"01:53.940","Text":"If we know the angle between them, Theta,"},{"Start":"01:53.940 ","End":"02:00.925","Text":"then the dot product a dot b is equal to the magnitude of"},{"Start":"02:00.925 ","End":"02:09.880","Text":"a times the magnitude of b times the cosine of the angle between them,"},{"Start":"02:09.880 ","End":"02:12.290","Text":"and that\u0027s what we\u0027re going to use in this."},{"Start":"02:12.290 ","End":"02:19.200","Text":"We get that a dot b is magnitude of a is 4,"},{"Start":"02:19.200 ","End":"02:22.095","Text":"magnitude of b is 3,"},{"Start":"02:22.095 ","End":"02:23.885","Text":"and we\u0027re not given the cosine,"},{"Start":"02:23.885 ","End":"02:25.010","Text":"we\u0027re given the angle,"},{"Start":"02:25.010 ","End":"02:29.265","Text":"so I need cosine of Pi over 3."},{"Start":"02:29.265 ","End":"02:37.955","Text":"Now, Pi over 3 is 60 degrees and the cosine of 60 degrees is 1/2,"},{"Start":"02:37.955 ","End":"02:41.570","Text":"so we get 4 times 3 times a 1/2,"},{"Start":"02:41.570 ","End":"02:46.680","Text":"and that is equal to 6. That\u0027s it."}],"ID":10648},{"Watched":false,"Name":"Exercise 2","Duration":"6m 2s","ChapterTopicVideoID":10305,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.600","Text":"Here we have 2 exercises in 1 and each of"},{"Start":"00:03.600 ","End":"00:06.765","Text":"them we have to find the angle between the 2 vectors."},{"Start":"00:06.765 ","End":"00:11.910","Text":"Let\u0027s give the angle a name we\u0027d use, Greek letter Theta."},{"Start":"00:11.910 ","End":"00:15.960","Text":"We just use a formula in a,"},{"Start":"00:15.960 ","End":"00:19.410","Text":"we would get the cosine of Theta,"},{"Start":"00:19.410 ","End":"00:22.740","Text":"the formula gives us the cosine and later we have to take"},{"Start":"00:22.740 ","End":"00:27.330","Text":"the arc cosine using the calculator or otherwise."},{"Start":"00:27.330 ","End":"00:31.350","Text":"Anyway, the cosine is the dot product of the 2 vectors."},{"Start":"00:31.350 ","End":"00:35.220","Text":"In this case, it would be a.b,"},{"Start":"00:35.220 ","End":"00:43.345","Text":"and we have to divide by the magnitude of a times the magnitude of b."},{"Start":"00:43.345 ","End":"00:48.980","Text":"There\u0027s 3 calculations, the dot product and magnitude for each."},{"Start":"00:48.980 ","End":"00:53.635","Text":"Let\u0027s see what we get, a.b."},{"Start":"00:53.635 ","End":"00:56.285","Text":"These are 2-dimensional vectors."},{"Start":"00:56.285 ","End":"01:00.170","Text":"Just multiply component-wise and add."},{"Start":"01:00.170 ","End":"01:07.550","Text":"This is going to be 3 times 7 plus 5 times 6."},{"Start":"01:07.550 ","End":"01:09.440","Text":"That\u0027s the numerator."},{"Start":"01:09.440 ","End":"01:17.925","Text":"Now, magnitude of a is going to be the square root of 3 squared plus 5 squared."},{"Start":"01:17.925 ","End":"01:26.655","Text":"The magnitude of b will be the square root of 7 squared plus 6 squared."},{"Start":"01:26.655 ","End":"01:29.455","Text":"Let\u0027s see what we get,"},{"Start":"01:29.455 ","End":"01:31.580","Text":"3 times 7 is 21,"},{"Start":"01:31.580 ","End":"01:33.875","Text":"5 times 6 is 30."},{"Start":"01:33.875 ","End":"01:37.935","Text":"That makes this 51,"},{"Start":"01:37.935 ","End":"01:40.860","Text":"3 squared is 9, 5 squared,"},{"Start":"01:40.860 ","End":"01:46.305","Text":"is 25 so we get the square root of 34,"},{"Start":"01:46.305 ","End":"01:50.660","Text":"7 squared is 49, 6 squared is 36."},{"Start":"01:50.660 ","End":"01:52.770","Text":"That makes it what?"},{"Start":"01:52.770 ","End":"01:58.105","Text":"85 square root of 85."},{"Start":"01:58.105 ","End":"02:01.310","Text":"We could just compute this on the calculator,"},{"Start":"02:01.310 ","End":"02:06.230","Text":"but I just happen to notice that all these numbers are divisible by 17."},{"Start":"02:06.230 ","End":"02:09.005","Text":"This is 17 times 3,"},{"Start":"02:09.005 ","End":"02:13.635","Text":"and this is 17 times 2 and 17 times 5."},{"Start":"02:13.635 ","End":"02:16.490","Text":"I can take out root 17 and root 17."},{"Start":"02:16.490 ","End":"02:18.605","Text":"It\u0027ll cancel out with the 17."},{"Start":"02:18.605 ","End":"02:23.870","Text":"I\u0027ll get 3 over square root of 2,"},{"Start":"02:23.870 ","End":"02:29.090","Text":"square root of 5."},{"Start":"02:29.090 ","End":"02:31.880","Text":"That would be 3 over square root of 10."},{"Start":"02:31.880 ","End":"02:33.830","Text":"If I didn\u0027t have a calculator,"},{"Start":"02:33.830 ","End":"02:36.950","Text":"that would not be too hard to compute anyway."},{"Start":"02:36.950 ","End":"02:41.210","Text":"Let\u0027s just say it\u0027s 3 over root 10."},{"Start":"02:41.210 ","End":"02:45.785","Text":"Then I get that cosine of Theta is,"},{"Start":"02:45.785 ","End":"02:48.325","Text":"I do this on the calculator."},{"Start":"02:48.325 ","End":"02:51.960","Text":"It comes out 0.948 something."},{"Start":"02:51.960 ","End":"02:58.985","Text":"This is not really important because while it\u0027s on the calculator in its exact form,"},{"Start":"02:58.985 ","End":"03:02.150","Text":"we can just take the inverse cosine,"},{"Start":"03:02.150 ","End":"03:04.220","Text":"depending on your calculator,"},{"Start":"03:04.220 ","End":"03:09.240","Text":"it would be shift or inverse with the cosine."},{"Start":"03:09.740 ","End":"03:14.270","Text":"Then depending on what your calculator is set to degrees or radians."},{"Start":"03:14.270 ","End":"03:15.620","Text":"If it\u0027s set to degrees,"},{"Start":"03:15.620 ","End":"03:19.730","Text":"it comes out to approximately 18 degrees."},{"Start":"03:19.730 ","End":"03:25.590","Text":"Well, 18.435 degrees."},{"Start":"03:25.590 ","End":"03:27.965","Text":"But if you did it in radians,"},{"Start":"03:27.965 ","End":"03:29.780","Text":"then I would get"},{"Start":"03:29.780 ","End":"03:38.970","Text":"0.321 to 3 decimal places radians."},{"Start":"03:38.970 ","End":"03:42.555","Text":"Just write a little c for circular measure."},{"Start":"03:42.555 ","End":"03:45.640","Text":"That\u0027s part a."},{"Start":"03:45.950 ","End":"03:51.410","Text":"Similarly in part b we need the cosine of the angle."},{"Start":"03:51.410 ","End":"03:54.440","Text":"Using the same idea,"},{"Start":"03:54.440 ","End":"03:57.380","Text":"the same formula, well it won\u0027t be a.b,"},{"Start":"03:57.380 ","End":"03:59.480","Text":"it will be v.w."},{"Start":"03:59.480 ","End":"04:02.995","Text":"But the same idea, we need the dot products from the numerator."},{"Start":"04:02.995 ","End":"04:11.130","Text":"We need 1 times 5 from this and this and then minus 2 times 6,"},{"Start":"04:11.130 ","End":"04:16.015","Text":"minus 3 times 7 over,"},{"Start":"04:16.015 ","End":"04:18.200","Text":"here we get the square root,"},{"Start":"04:18.200 ","End":"04:22.350","Text":"we\u0027re in 3D so we get 3 terms here,"},{"Start":"04:22.350 ","End":"04:26.990","Text":"1 squared plus 2 squared plus 3 squared."},{"Start":"04:26.990 ","End":"04:29.735","Text":"I ignore the minus because of squaring."},{"Start":"04:29.735 ","End":"04:40.140","Text":"Then the square root of 5 squared plus 6 squared plus 7 squared."},{"Start":"04:40.330 ","End":"04:45.770","Text":"Numerator 5 minus 12 minus 21,"},{"Start":"04:45.770 ","End":"04:51.880","Text":"which is 5 minus 33 minus 28."},{"Start":"04:51.880 ","End":"04:56.700","Text":"On the denominator, 1 plus 4 plus 9"},{"Start":"04:56.700 ","End":"05:04.995","Text":"would be let\u0027s see 14 under the square root sign."},{"Start":"05:04.995 ","End":"05:09.625","Text":"Here we\u0027ve got the square root of"},{"Start":"05:09.625 ","End":"05:16.575","Text":"25 and 36 and 49 and I make that 110."},{"Start":"05:16.575 ","End":"05:19.300","Text":"Then if you compute this on the calculator,"},{"Start":"05:19.300 ","End":"05:26.680","Text":"we get approximately minus 0.7135 approximately."},{"Start":"05:26.680 ","End":"05:29.050","Text":"But then we straight away,"},{"Start":"05:29.050 ","End":"05:31.470","Text":"take the inverse cosine."},{"Start":"05:31.470 ","End":"05:34.495","Text":"We\u0027ve got that Theta equals,"},{"Start":"05:34.495 ","End":"05:39.410","Text":"and I\u0027ll do it in both degrees and radians,"},{"Start":"05:39.410 ","End":"05:44.900","Text":"135.521 degrees"},{"Start":"05:44.900 ","End":"05:53.085","Text":"and 2.36528 radians approximately."},{"Start":"05:53.085 ","End":"05:57.050","Text":"Either way, if you have to choose,"},{"Start":"05:57.050 ","End":"06:02.610","Text":"go for radians. That\u0027s it."}],"ID":10649},{"Watched":false,"Name":"Exercise 3","Duration":"7m 34s","ChapterTopicVideoID":10306,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"This exercise is 3 in 1."},{"Start":"00:02.640 ","End":"00:04.680","Text":"In each case, we have a pair of vectors,"},{"Start":"00:04.680 ","End":"00:08.895","Text":"and we have to decide if they\u0027re parallel or orthogonal."},{"Start":"00:08.895 ","End":"00:13.514","Text":"Orthogonal meaning perpendicular or neither."},{"Start":"00:13.514 ","End":"00:17.145","Text":"There\u0027s more than one way to do this."},{"Start":"00:17.145 ","End":"00:20.700","Text":"I just want to mention one way which I\u0027m not going to use."},{"Start":"00:20.700 ","End":"00:22.695","Text":"But if you like, you can use."},{"Start":"00:22.695 ","End":"00:24.645","Text":"In each case you could say,"},{"Start":"00:24.645 ","End":"00:30.345","Text":"let Theta be the angle between the 2 vectors."},{"Start":"00:30.345 ","End":"00:34.455","Text":"There is a standard formula for cosine of Theta."},{"Start":"00:34.455 ","End":"00:39.405","Text":"It\u0027s the dot product divided by the magnitude of 1 times the magnitude of the other."},{"Start":"00:39.405 ","End":"00:45.500","Text":"Then you could compute the cosine and then divide it into cases."},{"Start":"00:45.500 ","End":"00:52.730","Text":"Say, if the cosine comes out to be plus or minus 1,"},{"Start":"00:52.730 ","End":"01:01.140","Text":"then the angle is going to be either 0 or 180 degrees."},{"Start":"01:03.740 ","End":"01:06.090","Text":"If you like it in radians,"},{"Start":"01:06.090 ","End":"01:08.400","Text":"that\u0027s a 0 or Pi."},{"Start":"01:08.400 ","End":"01:12.370","Text":"Then you\u0027ll know that they are parallel."},{"Start":"01:13.010 ","End":"01:18.005","Text":"If the cosine is plus or minus 1, they\u0027re parallel."},{"Start":"01:18.005 ","End":"01:21.320","Text":"If the cosine comes out to be 0,"},{"Start":"01:21.320 ","End":"01:26.825","Text":"the angle whose cosine is 0 is 90 degrees or Pi over 2 in radians,"},{"Start":"01:26.825 ","End":"01:31.080","Text":"and then you know that they\u0027re orthogonal."},{"Start":"01:32.630 ","End":"01:35.909","Text":"If it\u0027s neither of these,"},{"Start":"01:35.909 ","End":"01:39.920","Text":"neither plus or minus 1 or 0,"},{"Start":"01:39.920 ","End":"01:44.075","Text":"then it\u0027s neither parallel nor orthogonal."},{"Start":"01:44.075 ","End":"01:45.740","Text":"That\u0027s just 1 way."},{"Start":"01:45.740 ","End":"01:47.270","Text":"I\u0027m not going to use this."},{"Start":"01:47.270 ","End":"01:52.805","Text":"It\u0027s just an alternative that you could try in addition later."},{"Start":"01:52.805 ","End":"01:57.510","Text":"In part a, let\u0027s see what we\u0027re going to try first."},{"Start":"01:57.510 ","End":"01:59.534","Text":"Let\u0027s try for parallel first."},{"Start":"01:59.534 ","End":"02:00.810","Text":"If they\u0027re parallel,"},{"Start":"02:00.810 ","End":"02:04.055","Text":"that means that one of them, say q,"},{"Start":"02:04.055 ","End":"02:10.075","Text":"is some non 0 number scalar k times the other."},{"Start":"02:10.075 ","End":"02:14.380","Text":"Let\u0027s see if we can find such a k."},{"Start":"02:15.380 ","End":"02:20.300","Text":"Now, we could try using the components and saying,"},{"Start":"02:20.300 ","End":"02:23.600","Text":"well, if there is such a k,"},{"Start":"02:23.600 ","End":"02:25.820","Text":"k has to be 5,"},{"Start":"02:25.820 ","End":"02:30.300","Text":"and because 5 times 1 will give me 5,"},{"Start":"02:30.300 ","End":"02:33.950","Text":"I can see that 5 times minus 2 is not minus 8."},{"Start":"02:33.950 ","End":"02:36.875","Text":"But sometimes you can see it more clearly."},{"Start":"02:36.875 ","End":"02:41.600","Text":"Because if I have here a plus, minus, plus,"},{"Start":"02:41.600 ","End":"02:46.880","Text":"if I take plus, minus, plus, and I multiply it by a positive,"},{"Start":"02:46.880 ","End":"02:51.050","Text":"then I\u0027ll get also plus minus plus."},{"Start":"02:51.050 ","End":"02:53.060","Text":"If I multiply by a negative,"},{"Start":"02:53.060 ","End":"02:55.385","Text":"I\u0027ll get minus, plus, minus."},{"Start":"02:55.385 ","End":"02:57.295","Text":"But this is neither."},{"Start":"02:57.295 ","End":"03:00.315","Text":"You can, just by looking at the signs,"},{"Start":"03:00.315 ","End":"03:03.300","Text":"if the middle 1 is the odd 1 out,"},{"Start":"03:03.300 ","End":"03:06.020","Text":"then when I multiply by plus or minus the middle one"},{"Start":"03:06.020 ","End":"03:08.795","Text":"is still going to be the odd one out as far as sign."},{"Start":"03:08.795 ","End":"03:11.795","Text":"That\u0027s easier than actually doing computations."},{"Start":"03:11.795 ","End":"03:14.010","Text":"Anyway, we\u0027ve concluded,"},{"Start":"03:14.010 ","End":"03:18.015","Text":"meanwhile, that they\u0027re not parallel."},{"Start":"03:18.015 ","End":"03:23.210","Text":"Let\u0027s see if it\u0027s possible that they are orthogonal."},{"Start":"03:23.210 ","End":"03:24.800","Text":"Now for orthogonal,"},{"Start":"03:24.800 ","End":"03:30.545","Text":"the technique is that if the dot product is 0, then they\u0027re orthogonal."},{"Start":"03:30.545 ","End":"03:35.725","Text":"Let\u0027s see what is p dot product with q."},{"Start":"03:35.725 ","End":"03:38.840","Text":"Remember, we multiply component-wise and add."},{"Start":"03:38.840 ","End":"03:44.540","Text":"This is equal to 1 times 5 minus 2 minus 8,"},{"Start":"03:44.540 ","End":"03:47.630","Text":"so it\u0027s going to be plus 2 times 8,"},{"Start":"03:47.630 ","End":"03:51.530","Text":"and a plus and a minus is a minus 3 times 7."},{"Start":"03:51.530 ","End":"03:56.410","Text":"Let\u0027s see what this is. This is 5 plus 16 minus 21,"},{"Start":"03:56.410 ","End":"03:58.650","Text":"it is equal to 0."},{"Start":"03:58.650 ","End":"04:03.560","Text":"If it\u0027s equal to 0, then it means that they are orthogonal."},{"Start":"04:03.560 ","End":"04:08.675","Text":"That\u0027s a. Now part b."},{"Start":"04:08.675 ","End":"04:12.380","Text":"Let\u0027s try for parallel first, the parallel,"},{"Start":"04:12.380 ","End":"04:16.790","Text":"then b is going to be some scalar times a."},{"Start":"04:16.790 ","End":"04:25.990","Text":"Yeah, I meant to say times a."},{"Start":"04:26.030 ","End":"04:28.100","Text":"If this is the case,"},{"Start":"04:28.100 ","End":"04:30.815","Text":"then they\u0027re equal component-wise,"},{"Start":"04:30.815 ","End":"04:45.560","Text":"like 7 is equal to k times 3 and 6 is equal to k times 5."},{"Start":"04:45.560 ","End":"04:48.905","Text":"There is no such k. For example,"},{"Start":"04:48.905 ","End":"04:55.565","Text":"we can just extract it from the 1st one and say that k must equal 7/3."},{"Start":"04:55.565 ","End":"04:57.890","Text":"But then if I substituted it in the 2nd,"},{"Start":"04:57.890 ","End":"05:06.825","Text":"I\u0027ll get that 6 equals 7/3 times 5,"},{"Start":"05:06.825 ","End":"05:11.950","Text":"and that\u0027s certainly not true, false."},{"Start":"05:12.080 ","End":"05:14.835","Text":"They\u0027re not parallel."},{"Start":"05:14.835 ","End":"05:17.245","Text":"Let\u0027s write that down."},{"Start":"05:17.245 ","End":"05:21.215","Text":"Next, I\u0027ll try for the orthogonal."},{"Start":"05:21.215 ","End":"05:26.300","Text":"Let\u0027s see if a dot product with b is 0."},{"Start":"05:26.300 ","End":"05:27.785","Text":"We\u0027ll see what it is."},{"Start":"05:27.785 ","End":"05:30.145","Text":"It\u0027s component-wise."},{"Start":"05:30.145 ","End":"05:32.210","Text":"I don\u0027t even have to multiply because I\u0027ve got"},{"Start":"05:32.210 ","End":"05:34.810","Text":"plus, plus and everything is a plus,"},{"Start":"05:34.810 ","End":"05:37.925","Text":"3 times 7 plus 5 times 6,"},{"Start":"05:37.925 ","End":"05:41.600","Text":"[inaudible] it\u0027s not equal to 0, no way."},{"Start":"05:41.600 ","End":"05:46.110","Text":"These 2 vectors are not orthogonal,"},{"Start":"05:46.110 ","End":"05:53.670","Text":"and so we write the answer is neither."},{"Start":"05:53.670 ","End":"05:58.635","Text":"Now, in part c, let\u0027s see."},{"Start":"05:58.635 ","End":"06:00.585","Text":"Let\u0027s try for parallel."},{"Start":"06:00.585 ","End":"06:06.860","Text":"If parallel, then w is going to equal a scalar times v."},{"Start":"06:06.860 ","End":"06:11.325","Text":"From the first component,"},{"Start":"06:11.325 ","End":"06:17.590","Text":"I have the minus 5 has got to equal k times 1."},{"Start":"06:17.780 ","End":"06:23.880","Text":"That means that k has to equal minus 5."},{"Start":"06:23.880 ","End":"06:28.420","Text":"Let\u0027s compute minus 5 times v,"},{"Start":"06:30.230 ","End":"06:34.640","Text":"and let\u0027s see if we get w minus 5 times v is"},{"Start":"06:34.640 ","End":"06:45.960","Text":"minus 5 times i minus 2j plus 3k."},{"Start":"06:45.960 ","End":"06:49.785","Text":"This is going to equal minus 5,"},{"Start":"06:49.785 ","End":"06:55.770","Text":"i minus 5 times minus 2 is plus 10j,"},{"Start":"06:55.770 ","End":"06:57.245","Text":"and the third component,"},{"Start":"06:57.245 ","End":"07:03.050","Text":"minus 5 times 3 is minus 15k,"},{"Start":"07:03.050 ","End":"07:07.940","Text":"and this is equal to w,"},{"Start":"07:07.940 ","End":"07:13.125","Text":"and so we are good for parallel."},{"Start":"07:13.125 ","End":"07:17.800","Text":"Parallel is the answer."},{"Start":"07:18.140 ","End":"07:25.325","Text":"In part a, just summarizing, we got orthogonal,"},{"Start":"07:25.325 ","End":"07:28.460","Text":"in part b, we got neither,"},{"Start":"07:28.460 ","End":"07:32.130","Text":"and in part c, we got parallel,"},{"Start":"07:32.130 ","End":"07:34.360","Text":"and we\u0027re done."}],"ID":10650},{"Watched":false,"Name":"Exercise 4","Duration":"5m 58s","ChapterTopicVideoID":10307,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.660","Text":"In this exercise, we have to compute the projection of 1 vector onto another."},{"Start":"00:06.660 ","End":"00:08.925","Text":"In case you\u0027ve forgotten,"},{"Start":"00:08.925 ","End":"00:14.145","Text":"I\u0027m going to copy the formula and a diagram from the tutorial."},{"Start":"00:14.145 ","End":"00:17.325","Text":"Here we are, and yeah,"},{"Start":"00:17.325 ","End":"00:20.730","Text":"we did it there with a and b. I\u0027m sure we\u0027ll manage to adapt that to"},{"Start":"00:20.730 ","End":"00:24.360","Text":"u and v. The first example is 2D,"},{"Start":"00:24.360 ","End":"00:25.890","Text":"the second 1 is 3D."},{"Start":"00:25.890 ","End":"00:29.220","Text":"Let\u0027s start with the 2D example."},{"Start":"00:29.220 ","End":"00:35.790","Text":"You will be like a and v will be like b in the formula."},{"Start":"00:35.790 ","End":"00:39.595","Text":"What we want, we want the projection."},{"Start":"00:39.595 ","End":"00:43.520","Text":"It\u0027s actually the projection of b onto a, in this case,"},{"Start":"00:43.520 ","End":"00:49.205","Text":"the projection of v onto u."},{"Start":"00:49.205 ","End":"00:52.340","Text":"That\u0027s how we say it. This is projected onto this."},{"Start":"00:52.340 ","End":"00:57.545","Text":"Anyway, It\u0027s equal by the formula [inaudible] ."},{"Start":"00:57.545 ","End":"01:04.759","Text":"In this exercise, we\u0027re going to practice projecting 1 vector onto another vector."},{"Start":"01:04.759 ","End":"01:11.460","Text":"Here we\u0027re going to have u and v. We have a 2-dimensional case and a 3-dimensional case."},{"Start":"01:11.460 ","End":"01:14.650","Text":"In case you\u0027ve forgotten what this is all about,"},{"Start":"01:14.650 ","End":"01:21.660","Text":"I\u0027ll bring in the formula and the diagram from the tutorial. Here we are."},{"Start":"01:21.660 ","End":"01:23.780","Text":"We did it there with a and b,"},{"Start":"01:23.780 ","End":"01:28.580","Text":"and here we have u and v. Same principles apply."},{"Start":"01:28.580 ","End":"01:33.470","Text":"Let\u0027s start with the first 1,"},{"Start":"01:33.470 ","End":"01:38.690","Text":"where u is a and v is b."},{"Start":"01:38.690 ","End":"01:48.624","Text":"What we want is the projection of v onto u."},{"Start":"01:48.624 ","End":"01:51.860","Text":"This will equal by the formula,"},{"Start":"01:51.860 ","End":"01:56.375","Text":"the dot-product, sorry, not a."},{"Start":"01:56.375 ","End":"01:59.465","Text":"Well, a is just u and I\u0027ll write it already in here."},{"Start":"01:59.465 ","End":"02:02.330","Text":"We need 4 minus 1,"},{"Start":"02:02.330 ","End":"02:05.975","Text":"which is u dot with the other vector,"},{"Start":"02:05.975 ","End":"02:15.465","Text":"1 comma 7 divided by the magnitude of u squared."},{"Start":"02:15.465 ","End":"02:22.350","Text":"We want the magnitude of 4 comma minus 1 squared."},{"Start":"02:22.350 ","End":"02:24.230","Text":"All this, this is a scalar,"},{"Start":"02:24.230 ","End":"02:28.400","Text":"all this times vector u in this case,"},{"Start":"02:28.400 ","End":"02:31.505","Text":"which is 4 comma minus 1."},{"Start":"02:31.505 ","End":"02:36.170","Text":"Let\u0027s see. The dot-product component-wise,"},{"Start":"02:36.170 ","End":"02:43.490","Text":"4 times 1 is 4 minus 7 altogether minus 3."},{"Start":"02:43.490 ","End":"02:48.815","Text":"The magnitude squared, we just take this squared plus this squared."},{"Start":"02:48.815 ","End":"02:53.550","Text":"4 squared plus 1 squared, in our heads,"},{"Start":"02:53.550 ","End":"03:01.620","Text":"that will be 17 times vector 4 comma minus 1."},{"Start":"03:01.620 ","End":"03:07.765","Text":"The answer will just be minus 3 over 17 times each of the components,"},{"Start":"03:07.765 ","End":"03:10.720","Text":"minus 12 over 17,"},{"Start":"03:10.720 ","End":"03:14.350","Text":"and then plus 3 over 17,"},{"Start":"03:14.350 ","End":"03:17.820","Text":"and that\u0027s the answer for a."},{"Start":"03:17.820 ","End":"03:21.540","Text":"Part b, a 3D case."},{"Start":"03:21.540 ","End":"03:24.225","Text":"Here we have the ijk notation."},{"Start":"03:24.225 ","End":"03:26.025","Text":"The same thing applies."},{"Start":"03:26.025 ","End":"03:33.845","Text":"What we want is the projection of v onto u,"},{"Start":"03:33.845 ","End":"03:36.875","Text":"which you write the u down here and the v up here."},{"Start":"03:36.875 ","End":"03:39.720","Text":"Again, we want a dot-product."},{"Start":"03:40.550 ","End":"03:43.680","Text":"Be easier for me without the ijk."},{"Start":"03:43.680 ","End":"03:51.275","Text":"I\u0027ll write it as 7, minus 1, 1 dot minus 2,"},{"Start":"03:51.275 ","End":"03:54.595","Text":"5 minus 6,"},{"Start":"03:54.595 ","End":"03:59.670","Text":"over this 1 squared, the u squared,"},{"Start":"03:59.670 ","End":"04:03.345","Text":"which will be 7,"},{"Start":"04:03.345 ","End":"04:08.550","Text":"1 minus 1 magnitude squared,"},{"Start":"04:08.550 ","End":"04:13.130","Text":"and all this times the 1 we\u0027re projecting onto, which is this."},{"Start":"04:13.130 ","End":"04:16.730","Text":"I\u0027ll write it also, 7, minus 1, 1."},{"Start":"04:16.730 ","End":"04:19.415","Text":"What do we get?"},{"Start":"04:19.415 ","End":"04:24.420","Text":"The dot-product, let\u0027s see."},{"Start":"04:24.420 ","End":"04:31.335","Text":"I\u0027ll just write it, minus 14, minus 5."},{"Start":"04:31.335 ","End":"04:32.970","Text":"They\u0027re all coming out negative,"},{"Start":"04:32.970 ","End":"04:35.580","Text":"minus 6 over,"},{"Start":"04:35.580 ","End":"04:40.910","Text":"and let\u0027s see, 7 squared plus 1 squared plus 1 squared."},{"Start":"04:40.910 ","End":"04:42.170","Text":"I\u0027m ignoring the minuses."},{"Start":"04:42.170 ","End":"04:43.820","Text":"Of course, we are squaring."},{"Start":"04:43.820 ","End":"04:50.790","Text":"All this times 7, minus 1, 1. Let\u0027s see."},{"Start":"04:50.790 ","End":"04:54.030","Text":"In the numerator, it\u0027s all minus,"},{"Start":"04:54.030 ","End":"04:56.745","Text":"so I\u0027ll add them up 12 and 5 and 6,"},{"Start":"04:56.745 ","End":"04:58.590","Text":"let\u0027s see, 12, 17,"},{"Start":"04:58.590 ","End":"05:00.720","Text":"23 at the top,"},{"Start":"05:00.720 ","End":"05:03.030","Text":"minus 23 on the bottom,"},{"Start":"05:03.030 ","End":"05:04.080","Text":"49, and 1,"},{"Start":"05:04.080 ","End":"05:07.450","Text":"and 1, 51."},{"Start":"05:07.910 ","End":"05:12.945","Text":"Just minus 23 over 51 times each of these,"},{"Start":"05:12.945 ","End":"05:16.930","Text":"minus 23 times 7."},{"Start":"05:17.360 ","End":"05:20.730","Text":"Lets see. [inaudible],140 plus 21,"},{"Start":"05:20.730 ","End":"05:27.930","Text":"a 161 minus a 161 over 51."},{"Start":"05:27.930 ","End":"05:30.273","Text":"Then here\u0027s just a minus,"},{"Start":"05:30.273 ","End":"05:34.145","Text":"so that makes it plus 23 over 51,"},{"Start":"05:34.145 ","End":"05:39.180","Text":"and here minus 23 over 51."},{"Start":"05:39.620 ","End":"05:43.650","Text":"That\u0027s it. Just following the formula."},{"Start":"05:43.650 ","End":"05:47.510","Text":"The diagram is if you want to have an idea what the meaning is,"},{"Start":"05:47.510 ","End":"05:51.814","Text":"basically we\u0027re taking the shadow of b onto a line"},{"Start":"05:51.814 ","End":"05:58.890","Text":"that\u0027s in the direction of vector a. That\u0027s it."}],"ID":10651},{"Watched":false,"Name":"Exercise 5","Duration":"5m 1s","ChapterTopicVideoID":10308,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"In this exercise, we\u0027re given a 3-dimensional vector."},{"Start":"00:03.825 ","End":"00:09.780","Text":"We have to find the direction cosines and the direction angles for this."},{"Start":"00:09.780 ","End":"00:14.445","Text":"This concept of direction cosines doesn\u0027t apply in 2D,"},{"Start":"00:14.445 ","End":"00:16.410","Text":"just have it in 3D."},{"Start":"00:16.410 ","End":"00:18.690","Text":"Just to remind you what it is,"},{"Start":"00:18.690 ","End":"00:21.810","Text":"I\u0027ll bring the diagram from the tutorial."},{"Start":"00:21.810 ","End":"00:24.180","Text":"Here\u0027s our diagram."},{"Start":"00:24.180 ","End":"00:30.360","Text":"This will be the vector V, the 1 in red."},{"Start":"00:30.360 ","End":"00:36.285","Text":"Let\u0027s assume it has components V1,"},{"Start":"00:36.285 ","End":"00:39.960","Text":"V2, V3,"},{"Start":"00:39.960 ","End":"00:45.495","Text":"or V1i plus V2j plus V3k."},{"Start":"00:45.495 ","End":"00:55.665","Text":"1 variation of the formula for the direction cosines is that the cosine of alpha,"},{"Start":"00:55.665 ","End":"00:57.600","Text":"they\u0027re in order alpha, beta, gamma."},{"Start":"00:57.600 ","End":"00:59.070","Text":"Alpha goes with 1B,"},{"Start":"00:59.070 ","End":"01:01.230","Text":"beta goes with 2, gamma goes with 3."},{"Start":"01:01.230 ","End":"01:11.160","Text":"Cosine alpha is going to be V1 over the magnitude of V. Cosine of beta will"},{"Start":"01:11.160 ","End":"01:18.255","Text":"be V2 over the magnitude of V and cosine of gamma will be"},{"Start":"01:18.255 ","End":"01:28.395","Text":"V3 over the magnitude of V. Here we have the V1 is 1,"},{"Start":"01:28.395 ","End":"01:31.830","Text":"V2 will be negative 2,"},{"Start":"01:31.830 ","End":"01:35.780","Text":"and V3 will be 3 and doesn\u0027t matter if you use the I,"},{"Start":"01:35.780 ","End":"01:38.795","Text":"j, k notation or the bracket notation."},{"Start":"01:38.795 ","End":"01:42.200","Text":"Notice that it\u0027s best to first compute"},{"Start":"01:42.200 ","End":"01:46.180","Text":"magnitude of V because I need this in all 3 computations."},{"Start":"01:46.180 ","End":"01:48.000","Text":"Let\u0027s see."},{"Start":"01:48.000 ","End":"01:53.415","Text":"The magnitude of V is just equal to the square root."},{"Start":"01:53.415 ","End":"01:56.385","Text":"I\u0027ll just add each component up and square."},{"Start":"01:56.385 ","End":"02:00.030","Text":"1 squared, never mind the minus."},{"Start":"02:00.030 ","End":"02:11.595","Text":"It will be plus 2 squared plus 3 squared which is 1 plus 4 plus 9 square root of 14."},{"Start":"02:11.595 ","End":"02:20.180","Text":"Cosine of alpha will be 1 over square root of 14."},{"Start":"02:20.180 ","End":"02:25.570","Text":"Here, we\u0027ll have minus 2 over square root"},{"Start":"02:25.570 ","End":"02:31.925","Text":"of 14 and here we\u0027ll have 3 over square root of 14."},{"Start":"02:31.925 ","End":"02:34.865","Text":"Now those are the direction cosines."},{"Start":"02:34.865 ","End":"02:37.205","Text":"Now I want the angles."},{"Start":"02:37.205 ","End":"02:38.750","Text":"For the angles,"},{"Start":"02:38.750 ","End":"02:42.625","Text":"we will need to use the calculator."},{"Start":"02:42.625 ","End":"02:49.730","Text":"Let me just write alpha equals beta equals and gamma equals."},{"Start":"02:49.730 ","End":"02:52.730","Text":"It didn\u0027t say degrees or radians."},{"Start":"02:52.730 ","End":"02:55.030","Text":"I might do it in both."},{"Start":"02:55.030 ","End":"02:58.520","Text":"I know though that the negative ones are going to be"},{"Start":"02:58.520 ","End":"03:03.365","Text":"bigger than 90 degrees and the positive ones will be less than 90 degrees."},{"Start":"03:03.365 ","End":"03:09.695","Text":"The direction cosine is always between 0 and 180 so let\u0027s start with this."},{"Start":"03:09.695 ","End":"03:12.845","Text":"What you do is you compute this on the calculator,"},{"Start":"03:12.845 ","End":"03:18.440","Text":"and then you do inverse cosine or shift"},{"Start":"03:18.440 ","End":"03:24.305","Text":"cosine or however your calculator works and if you have the calculator set for degrees,"},{"Start":"03:24.305 ","End":"03:32.990","Text":"you get something like 74.49 something degrees and if it\u0027s set to radians,"},{"Start":"03:32.990 ","End":"03:38.565","Text":"I make it 1.300 in radians."},{"Start":"03:38.565 ","End":"03:41.300","Text":"Now the other 1 as I said,"},{"Start":"03:41.300 ","End":"03:43.400","Text":"we\u0027re expecting an obtuse angle,"},{"Start":"03:43.400 ","End":"03:46.385","Text":"meaning bigger than 90 degrees."},{"Start":"03:46.385 ","End":"03:55.775","Text":"This comes out to be 122.31 something in degrees and if you want it in radians,"},{"Start":"03:55.775 ","End":"04:03.525","Text":"2.134 something, something, something radians."},{"Start":"04:03.525 ","End":"04:05.535","Text":"The last 1, gamma,"},{"Start":"04:05.535 ","End":"04:11.260","Text":"the angle between the V and the z-axis,"},{"Start":"04:13.970 ","End":"04:22.710","Text":"it\u0027ll make this 36.699 something"},{"Start":"04:22.710 ","End":"04:33.075","Text":"degrees and in radian 0.640 something radians."},{"Start":"04:33.075 ","End":"04:38.269","Text":"The exact answer is not important it\u0027s the method that\u0027s important."},{"Start":"04:38.269 ","End":"04:46.970","Text":"Really what we do is just take each of the components and divide by the magnitude of"},{"Start":"04:46.970 ","End":"04:48.990","Text":"the vector and then we get"},{"Start":"04:48.990 ","End":"04:57.545","Text":"the 3 cosines and then the inverse cosine or arc cosine will give us the actual angles,"},{"Start":"04:57.545 ","End":"05:00.425","Text":"whatever we want, degrees or radians."},{"Start":"05:00.425 ","End":"05:02.790","Text":"Okay, that\u0027s it.3"}],"ID":10652},{"Watched":false,"Name":"Exercise 6","Duration":"8m 32s","ChapterTopicVideoID":27752,"CourseChapterTopicPlaylistID":12290,"HasSubtitles":false,"VideoComments":[],"Subtitles":[],"ID":29044}],"Thumbnail":null,"ID":12290},{"Name":"Vectors Cross Product","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors - Cross Product","Duration":"19m 5s","ChapterTopicVideoID":10309,"CourseChapterTopicPlaylistID":12291,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"Continuing with vectors, we just finished with the dot product,"},{"Start":"00:03.840 ","End":"00:06.450","Text":"and now we\u0027re going to learn a cross product."},{"Start":"00:06.450 ","End":"00:08.985","Text":"There\u0027s some major differences."},{"Start":"00:08.985 ","End":"00:14.950","Text":"cross product is something that only works in 3D."},{"Start":"00:14.960 ","End":"00:19.845","Text":"Another major difference is that when we took"},{"Start":"00:19.845 ","End":"00:24.900","Text":"a vector and we did a dot product with another vector,"},{"Start":"00:24.900 ","End":"00:28.320","Text":"what we got was a scalar, a number"},{"Start":"00:28.320 ","End":"00:31.120","Text":"but when we take a vector,"},{"Start":"00:31.520 ","End":"00:36.125","Text":"this is how the cross product looks just like a multiplication,"},{"Start":"00:36.125 ","End":"00:38.610","Text":"and we cross product with another vector,"},{"Start":"00:38.610 ","End":"00:40.500","Text":"what we get is a vector,"},{"Start":"00:40.500 ","End":"00:43.035","Text":"also a 3D vector."},{"Start":"00:43.035 ","End":"00:46.680","Text":"Those are the major differences."},{"Start":"00:46.750 ","End":"00:49.250","Text":"Just like with the dot product,"},{"Start":"00:49.250 ","End":"00:50.900","Text":"we gave 2 definitions."},{"Start":"00:50.900 ","End":"00:56.270","Text":"One was like a formula for how to compute this given these,"},{"Start":"00:56.270 ","End":"01:00.560","Text":"and the other we had a diagram with the geometric meaning."},{"Start":"01:00.560 ","End":"01:04.220","Text":"I\u0027m going to give both here also,"},{"Start":"01:04.220 ","End":"01:10.775","Text":"there is a geometric trigonometric meaning and another drier definition."},{"Start":"01:10.775 ","End":"01:14.840","Text":"I\u0027m going to start with the geometric interpretation."},{"Start":"01:14.840 ","End":"01:17.105","Text":"When you have 2 vectors,"},{"Start":"01:17.105 ","End":"01:20.780","Text":"a and b, and I want to know what is a cross b,"},{"Start":"01:20.780 ","End":"01:29.370","Text":"what we do is we take a vector which is perpendicular to both."},{"Start":"01:30.650 ","End":"01:36.410","Text":"The size of this vector is determined by the formula that"},{"Start":"01:36.410 ","End":"01:43.750","Text":"the magnitude of a cross b is equal to the magnitude of a,"},{"Start":"01:43.750 ","End":"01:46.525","Text":"times the magnitude of b,"},{"Start":"01:46.525 ","End":"01:50.765","Text":"times the sine of the angle between them."},{"Start":"01:50.765 ","End":"01:58.110","Text":"Theta would be between 0 and 180 degrees or let\u0027s say Pi and radians."},{"Start":"01:59.840 ","End":"02:07.260","Text":"Notice that there are actually 2 vectors which are perpendicular and have this size;"},{"Start":"02:07.260 ","End":"02:10.370","Text":"it could face up or face down in this diagram."},{"Start":"02:10.370 ","End":"02:14.960","Text":"There\u0027s another rule which says whether you take it upwards or downwards,"},{"Start":"02:14.960 ","End":"02:17.630","Text":"and it\u0027s called the right-hand rule."},{"Start":"02:17.630 ","End":"02:21.635","Text":"This is a picture I found on the Internet of the right-hand rule."},{"Start":"02:21.635 ","End":"02:24.650","Text":"It basically says if your 4 fingers of your right hand is"},{"Start":"02:24.650 ","End":"02:28.220","Text":"pointing towards a and the middle finger is pointing towards b,"},{"Start":"02:28.220 ","End":"02:31.960","Text":"the direction of the thumb is what you take as the direction of this."},{"Start":"02:31.960 ","End":"02:37.220","Text":"Just by the way, notice that this really is a positive or at least non-negative number,"},{"Start":"02:37.220 ","End":"02:38.900","Text":"because between 0 and Pi,"},{"Start":"02:38.900 ","End":"02:43.045","Text":"the sine is positive or possibly 0."},{"Start":"02:43.045 ","End":"02:46.740","Text":"That\u0027s the geometric side of it,"},{"Start":"02:46.740 ","End":"02:52.350","Text":"now let\u0027s get to the more formula side of how to compute it."},{"Start":"02:52.790 ","End":"02:57.830","Text":"Let\u0027s say that the vector a is given by,"},{"Start":"02:57.830 ","End":"02:59.495","Text":"as usual, a_1,"},{"Start":"02:59.495 ","End":"03:00.940","Text":"a_2, a_3;"},{"Start":"03:00.940 ","End":"03:02.930","Text":"these are the components."},{"Start":"03:02.930 ","End":"03:11.545","Text":"Let\u0027s say that the vector b is b_1, b_2, b_3."},{"Start":"03:11.545 ","End":"03:19.560","Text":"Then the definition of a cross b is equal to,"},{"Start":"03:19.560 ","End":"03:22.390","Text":"and this is complicated,"},{"Start":"03:23.150 ","End":"03:32.780","Text":"the first component is a_2 times b_3 minus a_3b_2."},{"Start":"03:32.780 ","End":"03:34.850","Text":"I\u0027m going to try and show you there\u0027s a pattern here."},{"Start":"03:34.850 ","End":"03:38.809","Text":"In the first place, the first component,"},{"Start":"03:38.809 ","End":"03:41.890","Text":"I take the number next to 1,"},{"Start":"03:41.890 ","End":"03:43.710","Text":"after 1 is 2,"},{"Start":"03:43.710 ","End":"03:45.840","Text":"and then the number after that 3,"},{"Start":"03:45.840 ","End":"03:48.870","Text":"and is always an a and b and an a and a b with a minus."},{"Start":"03:48.870 ","End":"03:51.235","Text":"The next one, if I take a_2,"},{"Start":"03:51.235 ","End":"03:55.025","Text":"is I take what comes after 2 is 3."},{"Start":"03:55.025 ","End":"03:56.995","Text":"What comes after 3?"},{"Start":"03:56.995 ","End":"04:00.810","Text":"Well, we wrap around back to 1 again and then we do the reverse,"},{"Start":"04:00.810 ","End":"04:03.075","Text":"the a_1 with the b_3."},{"Start":"04:03.075 ","End":"04:05.450","Text":"Then when we get to the third place,"},{"Start":"04:05.450 ","End":"04:08.515","Text":"after 3 comes 1 because we\u0027re going cyclically."},{"Start":"04:08.515 ","End":"04:13.890","Text":"It\u0027s a_1, and after that is b_2 minus a_2b_1."},{"Start":"04:13.890 ","End":"04:17.255","Text":"It\u0027s tricky to remember."},{"Start":"04:17.255 ","End":"04:20.615","Text":"At the end, I\u0027m going to show you an alternative way;"},{"Start":"04:20.615 ","End":"04:22.400","Text":"there are actually several ways"},{"Start":"04:22.400 ","End":"04:25.700","Text":"but I don\u0027t want to assume you know what a determinant is,"},{"Start":"04:25.700 ","End":"04:30.900","Text":"so at the end I\u0027ll show you a way to compute it using determinants."},{"Start":"04:31.820 ","End":"04:37.440","Text":"One thing that\u0027s clear from this rule,"},{"Start":"04:37.440 ","End":"04:43.145","Text":"is that a cross b is not the same as b cross a."},{"Start":"04:43.145 ","End":"04:45.485","Text":"If I do b cross a,"},{"Start":"04:45.485 ","End":"04:47.450","Text":"the thumb will point down,"},{"Start":"04:47.450 ","End":"04:52.160","Text":"I\u0027ll get exactly the same magnitude but in the downward direction."},{"Start":"04:52.160 ","End":"05:01.640","Text":"In fact, b cross a is going to be the negative vector of a cross b,"},{"Start":"05:01.640 ","End":"05:03.875","Text":"same size but opposite direction."},{"Start":"05:03.875 ","End":"05:06.950","Text":"The order is important with the cross products,"},{"Start":"05:06.950 ","End":"05:11.390","Text":"as opposed to dot products where the order wasn\u0027t important."},{"Start":"05:11.390 ","End":"05:14.480","Text":"I\u0027d like to do a computational example."},{"Start":"05:14.480 ","End":"05:22.170","Text":"Let\u0027s take a to be the vector, 2, 1,"},{"Start":"05:22.170 ","End":"05:32.025","Text":"minus 1, and let\u0027s take the vector b to equal minus 3, 4, 1."},{"Start":"05:32.025 ","End":"05:39.390","Text":"Let\u0027s see if we can compute what is the cross product of these 2 vectors."},{"Start":"05:39.470 ","End":"05:48.120","Text":"I\u0027ll highlight at least this part of the formula just to make it clearer."},{"Start":"05:49.400 ","End":"05:53.480","Text":"Looking at the highlighted portion,"},{"Start":"05:53.480 ","End":"06:00.265","Text":"the first component, a_2 times b_3 is 1 times 1."},{"Start":"06:00.265 ","End":"06:04.350","Text":"Well, maybe I\u0027ll write it 1 times 1."},{"Start":"06:04.350 ","End":"06:12.065","Text":"Then a_3b_2, this times this minus,"},{"Start":"06:12.065 ","End":"06:16.580","Text":"minus 1 times 4 comma,"},{"Start":"06:16.580 ","End":"06:18.575","Text":"and we\u0027ll compute them all at the end."},{"Start":"06:18.575 ","End":"06:20.585","Text":"Next one, a_3,"},{"Start":"06:20.585 ","End":"06:23.655","Text":"b_1, this with this."},{"Start":"06:23.655 ","End":"06:33.795","Text":"It\u0027s this with this minus 1 times minus 3 minus,"},{"Start":"06:33.795 ","End":"06:36.465","Text":"we can always look at it here,"},{"Start":"06:36.465 ","End":"06:39.475","Text":"a_1b_3, 2 times 1."},{"Start":"06:39.475 ","End":"06:44.425","Text":"Well, just to put it in brackets instead of the dot."},{"Start":"06:44.425 ","End":"06:50.985","Text":"Then the last component would be just a_1b_2 minus a_2b_1."},{"Start":"06:50.985 ","End":"06:54.825","Text":"This times this 2 times 4,"},{"Start":"06:54.825 ","End":"07:02.790","Text":"minus 1 times minus 3."},{"Start":"07:02.790 ","End":"07:06.440","Text":"What does this come out to be? Let\u0027s see."},{"Start":"07:06.440 ","End":"07:13.630","Text":"1 plus 4 is 5,"},{"Start":"07:14.930 ","End":"07:19.185","Text":"well, the angular bracket."},{"Start":"07:19.185 ","End":"07:27.945","Text":"Next one, plus 3 minus 2 is 1."},{"Start":"07:27.945 ","End":"07:35.130","Text":"The left one is 8 plus 3 is 11. That\u0027s the answer."},{"Start":"07:35.130 ","End":"07:37.790","Text":"If someone asked you what is b cross a,"},{"Start":"07:37.790 ","End":"07:39.380","Text":"you could say it\u0027s minus 5,"},{"Start":"07:39.380 ","End":"07:42.980","Text":"minus 1, minus 11, for example."},{"Start":"07:42.980 ","End":"07:47.765","Text":"I\u0027d like to show you another property of the cross product."},{"Start":"07:47.765 ","End":"07:50.240","Text":"If we took 2 vectors,"},{"Start":"07:50.240 ","End":"07:52.435","Text":"let\u0027s say they\u0027re not 0,"},{"Start":"07:52.435 ","End":"07:55.770","Text":"and if they\u0027re parallel,"},{"Start":"07:55.770 ","End":"07:58.670","Text":"parallel means could be in the same direction or in"},{"Start":"07:58.670 ","End":"08:01.970","Text":"opposite directions within the same line, if they\u0027re parallel,"},{"Start":"08:01.970 ","End":"08:03.860","Text":"then Theta would either be"},{"Start":"08:03.860 ","End":"08:08.840","Text":"0 or 180 degrees depending on whether they\u0027re same direction or precisely the opposite."},{"Start":"08:08.840 ","End":"08:11.540","Text":"If Theta\u0027s 180 degrees or Pi,"},{"Start":"08:11.540 ","End":"08:13.700","Text":"then sine Theta is 0,"},{"Start":"08:13.700 ","End":"08:18.285","Text":"so that a cross b would be 0."},{"Start":"08:18.285 ","End":"08:19.730","Text":"The reverse is true."},{"Start":"08:19.730 ","End":"08:22.069","Text":"If this is 0 and these 2 are non-zero,"},{"Start":"08:22.069 ","End":"08:24.020","Text":"then sine Theta is 0."},{"Start":"08:24.020 ","End":"08:29.530","Text":"Theta has to be 0 or Pi or plus multiples of 2Pi,"},{"Start":"08:29.530 ","End":"08:30.740","Text":"but in this range,"},{"Start":"08:30.740 ","End":"08:32.375","Text":"only 0 and Pi."},{"Start":"08:32.375 ","End":"08:34.760","Text":"I\u0027ll write that down."},{"Start":"08:34.760 ","End":"08:40.235","Text":"That a is parallel to b if and only if"},{"Start":"08:40.235 ","End":"08:46.250","Text":"a cross b is equal to,"},{"Start":"08:46.250 ","End":"08:47.990","Text":"well, it\u0027s not the 0 scalar,"},{"Start":"08:47.990 ","End":"08:50.300","Text":"it\u0027s the 0 vector unlike the dot product."},{"Start":"08:50.300 ","End":"09:00.370","Text":"I\u0027m assuming that a and b are both nonzero vectors."},{"Start":"09:02.080 ","End":"09:07.340","Text":"Now, I want to show you a typical problem that is often asked."},{"Start":"09:07.340 ","End":"09:09.590","Text":"It\u0027s quite useful in physics,"},{"Start":"09:09.590 ","End":"09:11.300","Text":"but also in mathematics."},{"Start":"09:11.300 ","End":"09:15.875","Text":"There\u0027s a concept called a normal vector to a plane."},{"Start":"09:15.875 ","End":"09:19.290","Text":"When you have a plane,"},{"Start":"09:19.420 ","End":"09:21.500","Text":"the word normal vector,"},{"Start":"09:21.500 ","End":"09:22.685","Text":"you don\u0027t have to remember this,"},{"Start":"09:22.685 ","End":"09:25.120","Text":"but every plane,"},{"Start":"09:25.120 ","End":"09:30.130","Text":"there\u0027s a direction called a normal or it could be the opposite,"},{"Start":"09:30.130 ","End":"09:36.440","Text":"it means it\u0027s perpendicular to all the plane which means perpendicular to any vector."},{"Start":"09:36.440 ","End":"09:40.700","Text":"For example, if I had inside the plane a vector"},{"Start":"09:40.700 ","End":"09:47.185","Text":"a and I had another vector inside the plane b,"},{"Start":"09:47.185 ","End":"09:53.570","Text":"then 1 way of finding a normal vector would be to take"},{"Start":"09:53.570 ","End":"10:01.295","Text":"the cross product of a with b. I don\u0027t know why the term normal is used,"},{"Start":"10:01.295 ","End":"10:04.835","Text":"besides, you have the terms perpendicular and orthogonal."},{"Start":"10:04.835 ","End":"10:07.910","Text":"But often we have a question where we\u0027re given 2 vectors,"},{"Start":"10:07.910 ","End":"10:10.440","Text":"and we want to find something that\u0027s perpendicular to both,"},{"Start":"10:10.440 ","End":"10:14.880","Text":"so the cross product is the perfect way to do that."},{"Start":"10:14.880 ","End":"10:17.865","Text":"Let\u0027s take an example problem."},{"Start":"10:17.865 ","End":"10:21.480","Text":"Suppose I have 3 points in the plane,"},{"Start":"10:21.480 ","End":"10:23.190","Text":"now these are points that\u0027s round brackets,"},{"Start":"10:23.190 ","End":"10:27.600","Text":"P is 1, 0,"},{"Start":"10:27.600 ","End":"10:32.115","Text":"0, let\u0027s say Q is the point 1, 1,"},{"Start":"10:32.115 ","End":"10:38.865","Text":"1, and R is the point 2 minus 1, 3."},{"Start":"10:38.865 ","End":"10:44.820","Text":"We want to find a normal to the plane."},{"Start":"10:44.820 ","End":"10:49.140","Text":"I wrote that these are 3 points in a plane,"},{"Start":"10:49.140 ","End":"10:51.950","Text":"and I\u0027m going to find a vector orthogonal to the plane."},{"Start":"10:51.950 ","End":"10:55.280","Text":"I didn\u0027t use the word normal, orthogonal or perpendicular."},{"Start":"10:55.280 ","End":"10:57.570","Text":"What we do is,"},{"Start":"10:57.570 ","End":"11:07.690","Text":"let\u0027s suppose this is my P and this is my Q and this is R inside the plane, P,"},{"Start":"11:07.690 ","End":"11:18.440","Text":"Q, and R. What we do is we take the vector that goes from P to Q,"},{"Start":"11:18.440 ","End":"11:22.345","Text":"and we call that say a,"},{"Start":"11:22.345 ","End":"11:28.010","Text":"and the vector from P to R and call that let\u0027s say b."},{"Start":"11:28.010 ","End":"11:34.440","Text":"Then a perpendicular vector that we want,"},{"Start":"11:34.440 ","End":"11:37.140","Text":"if it\u0027s perpendicular to these 2 vectors,"},{"Start":"11:37.140 ","End":"11:39.095","Text":"it\u0027ll be perpendicular to the whole plane."},{"Start":"11:39.095 ","End":"11:45.060","Text":"What we can do is just do a cross b."},{"Start":"11:45.240 ","End":"11:48.740","Text":"How do we find a and b?"},{"Start":"11:49.140 ","End":"11:55.460","Text":"Well, a is just what we call PQ,"},{"Start":"11:56.910 ","End":"11:59.800","Text":"the vector from P to Q."},{"Start":"11:59.800 ","End":"12:02.680","Text":"If you remember, the way we do this is we take"},{"Start":"12:02.680 ","End":"12:06.250","Text":"the coordinates of Q and subtract the coordinates of"},{"Start":"12:06.250 ","End":"12:14.080","Text":"P so 1,1,1 less 1,0,0 I get 1 minus 1 is 0,"},{"Start":"12:14.080 ","End":"12:15.880","Text":"1 minus 0 is 1,"},{"Start":"12:15.880 ","End":"12:21.820","Text":"1 minus 0 is 1 and b similarly joins P to"},{"Start":"12:21.820 ","End":"12:30.235","Text":"R. Similarly we take away the P from R,"},{"Start":"12:30.235 ","End":"12:34.390","Text":"so 2 minus 1 is 1, minus 1,"},{"Start":"12:34.390 ","End":"12:39.040","Text":"less 0 is minus 1 and 3 less 0 is 3."},{"Start":"12:39.040 ","End":"12:46.880","Text":"Now all we have to do is to do the cross product of a with b."},{"Start":"12:49.440 ","End":"12:55.975","Text":"The formula disappeared so I\u0027ll just copy it from above."},{"Start":"12:55.975 ","End":"12:57.910","Text":"Now we\u0027ve got a_2,"},{"Start":"12:57.910 ","End":"13:01.340","Text":"b_3 minus a_3, b_2."},{"Start":"13:02.550 ","End":"13:07.059","Text":"This times this minus this times this."},{"Start":"13:07.059 ","End":"13:09.115","Text":"I\u0027ll do it all in one stage."},{"Start":"13:09.115 ","End":"13:13.225","Text":"This times this is 3 minus minus 1."},{"Start":"13:13.225 ","End":"13:16.000","Text":"3 minus minus 1 is 4."},{"Start":"13:16.000 ","End":"13:20.215","Text":"Next I see I need 3 and 1 so a_3,"},{"Start":"13:20.215 ","End":"13:29.140","Text":"b_1 minus the other diagonal so it\u0027s 1 times 1 is 1 minus 0, so that\u0027s 1."},{"Start":"13:29.140 ","End":"13:36.865","Text":"For the last one I need 1 and 2, a_1 and b_2."},{"Start":"13:36.865 ","End":"13:39.085","Text":"0 times minus 1 is 0,"},{"Start":"13:39.085 ","End":"13:43.225","Text":"less 1, so that\u0027s minus 1."},{"Start":"13:43.225 ","End":"13:48.009","Text":"Here we have a vector which is orthogonal to the plane,"},{"Start":"13:48.009 ","End":"13:52.000","Text":"or also called the normal vector to a plane."},{"Start":"13:52.000 ","End":"13:59.065","Text":"Next, I want to give you some formulas that you should have anyway and here they are."},{"Start":"13:59.065 ","End":"14:07.070","Text":"We assume that u and v and w here are any vectors."},{"Start":"14:07.070 ","End":"14:12.380","Text":"We already discussed this we use a and b but the cross-product,"},{"Start":"14:12.380 ","End":"14:14.310","Text":"if you do it switch the order,"},{"Start":"14:14.310 ","End":"14:16.230","Text":"it becomes minus the vector,"},{"Start":"14:16.230 ","End":"14:19.005","Text":"the vector in the opposite direction."},{"Start":"14:19.005 ","End":"14:23.855","Text":"There\u0027s a distributive rules similar to what we had with the.product,"},{"Start":"14:23.855 ","End":"14:26.950","Text":"just like with multiplication and addition in algebra,"},{"Start":"14:26.950 ","End":"14:28.600","Text":"that\u0027s the way to remember it."},{"Start":"14:28.600 ","End":"14:37.405","Text":"U cross v plus w so you do it cross with v separately and with w separately and you add."},{"Start":"14:37.405 ","End":"14:41.530","Text":"A constant can go in anywhere."},{"Start":"14:41.530 ","End":"14:48.580","Text":"You can multiply the constant by the first or the second or by the product all the same."},{"Start":"14:48.580 ","End":"14:51.940","Text":"The last one is a strange-looking one."},{"Start":"14:51.940 ","End":"14:56.545","Text":"It turns out that if I put a. here and across here,"},{"Start":"14:56.545 ","End":"14:59.275","Text":"or a cross here and a.here,"},{"Start":"14:59.275 ","End":"15:00.520","Text":"I get the same answer."},{"Start":"15:00.520 ","End":"15:02.245","Text":"The answer will be a scalar."},{"Start":"15:02.245 ","End":"15:07.060","Text":"This is a vector and a vector.a vector is a scalar and same here."},{"Start":"15:07.060 ","End":"15:10.180","Text":"There\u0027s actually a nice way of computing this,"},{"Start":"15:10.180 ","End":"15:14.230","Text":"but you have to know determinants so I\u0027m going to leave that also to the end."},{"Start":"15:14.230 ","End":"15:17.890","Text":"Actually, I will soon show you a use for this."},{"Start":"15:17.890 ","End":"15:24.895","Text":"I\u0027ll show you another meaning of the cross-product or how it\u0027s useful in geometry."},{"Start":"15:24.895 ","End":"15:28.090","Text":"I\u0027ll show you the diagram and then I\u0027ll explain."},{"Start":"15:28.090 ","End":"15:30.070","Text":"Here is the diagram I wanted to show you."},{"Start":"15:30.070 ","End":"15:31.270","Text":"We\u0027re back to a, b,"},{"Start":"15:31.270 ","End":"15:32.440","Text":"and c instead of u, v,"},{"Start":"15:32.440 ","End":"15:37.510","Text":"and w. This is what we call parallelepiped."},{"Start":"15:37.510 ","End":"15:41.815","Text":"When you take 3 vectors and you complete the parallelogram."},{"Start":"15:41.815 ","End":"15:45.265","Text":"Let\u0027s first take a look at just this frontier."},{"Start":"15:45.265 ","End":"15:46.630","Text":"That\u0027s a parallelogram."},{"Start":"15:46.630 ","End":"15:49.105","Text":"I\u0027ll just shade it so you know which one I\u0027m talking about."},{"Start":"15:49.105 ","End":"15:51.670","Text":"This side here, it\u0027s a parallelogram."},{"Start":"15:51.670 ","End":"15:58.150","Text":"Turns out that the formula for the area of the parallelogram is given"},{"Start":"15:58.150 ","End":"16:05.485","Text":"by the magnitude of vector a, cross vector b."},{"Start":"16:05.485 ","End":"16:07.645","Text":"That\u0027s one use of a cross product."},{"Start":"16:07.645 ","End":"16:11.050","Text":"If I take the cross product and then I take its magnitude,"},{"Start":"16:11.050 ","End":"16:16.600","Text":"I can find the area of parallelogram of 2 vectors."},{"Start":"16:16.600 ","End":"16:20.460","Text":"The other thing is if I now forget about the area,"},{"Start":"16:20.460 ","End":"16:22.890","Text":"but talk about the volume,"},{"Start":"16:22.890 ","End":"16:28.690","Text":"this whole parallelepiped it is a three-dimensional figure and it has a volume."},{"Start":"16:28.740 ","End":"16:33.370","Text":"The volume is given by this here."},{"Start":"16:33.370 ","End":"16:36.370","Text":"Well, we\u0027re using a, b, and c. It\u0027s equal to"},{"Start":"16:36.370 ","End":"16:47.830","Text":"a.product with b cross c. Not quite."},{"Start":"16:47.830 ","End":"16:49.660","Text":"This thing could be negative."},{"Start":"16:49.660 ","End":"16:53.605","Text":"We need the absolute value because the volume has to be positive."},{"Start":"16:53.605 ","End":"16:57.460","Text":"Now actually this formula with the volume gives us"},{"Start":"16:57.460 ","End":"17:02.410","Text":"a way to find out if 3 vectors are in the same plane."},{"Start":"17:02.410 ","End":"17:05.380","Text":"If the 3 vectors are in the same plane,"},{"Start":"17:05.380 ","End":"17:09.850","Text":"then the volume is going to be 0 because this is going to be flat."},{"Start":"17:09.850 ","End":"17:13.645","Text":"Let\u0027s do a problem with that."},{"Start":"17:13.645 ","End":"17:17.515","Text":"Let me take 3 vectors."},{"Start":"17:17.515 ","End":"17:26.935","Text":"Let\u0027s take vector a to equal 1, 4 minus 7,"},{"Start":"17:26.935 ","End":"17:34.195","Text":"and we\u0027ll take vector b to equal 2 minus 1,"},{"Start":"17:34.195 ","End":"17:44.215","Text":"4 and let\u0027s take vector c to equal 0 minus 9, 18."},{"Start":"17:44.215 ","End":"17:47.500","Text":"Are these 3 vectors in the same plane?"},{"Start":"17:47.500 ","End":"17:59.095","Text":"First, we need to compute b cross c and this is equal to,"},{"Start":"17:59.095 ","End":"18:01.000","Text":"I did it for you."},{"Start":"18:01.000 ","End":"18:03.415","Text":"This is what it comes out to be."},{"Start":"18:03.415 ","End":"18:07.450","Text":"For example, the first one we would take from the formula,"},{"Start":"18:07.450 ","End":"18:10.570","Text":"a_2, b_3 minus a_3,"},{"Start":"18:10.570 ","End":"18:13.855","Text":"b_2 minus 1 times 18,"},{"Start":"18:13.855 ","End":"18:20.275","Text":"and then less minus 36 from this times this comes up plus 18 and so on."},{"Start":"18:20.275 ","End":"18:27.100","Text":"Next thing we need to do is to do a.with"},{"Start":"18:27.100 ","End":"18:37.435","Text":"b cross c. This comes out to be this thing above 18 minus 36,"},{"Start":"18:37.435 ","End":"18:44.260","Text":"18.product with c, which is 0 minus 9,"},{"Start":"18:44.260 ","End":"18:48.730","Text":"18 and at least one of them is a 0."},{"Start":"18:48.730 ","End":"18:52.270","Text":"It\u0027s this with this, this with this and this with this."},{"Start":"18:52.270 ","End":"18:59.290","Text":"I\u0027II write it,18 times 0 plus 36 times 9."},{"Start":"18:59.290 ","End":"19:02.335","Text":"I\u0027m writing plus because it\u0027s minus and minus,"},{"Start":"19:02.335 ","End":"19:08.560","Text":"sorry, it\u0027s minus 18 and then minus 18 times 18."},{"Start":"19:08.560 ","End":"19:15.910","Text":"Anyway, turns out this is 324 and this is 324 and this is 0 so it ends up being 0,"},{"Start":"19:15.910 ","End":"19:22.255","Text":"which means that the answer is yes they are in the same plane."},{"Start":"19:22.255 ","End":"19:25.945","Text":"Now, I\u0027m basically done here."},{"Start":"19:25.945 ","End":"19:33.115","Text":"If you have studied determinants or you know what a determinant is,"},{"Start":"19:33.115 ","End":"19:36.670","Text":"then you\u0027re welcome to stay and I\u0027ll talk"},{"Start":"19:36.670 ","End":"19:40.480","Text":"about some shortcut formulas for finding the cross product as"},{"Start":"19:40.480 ","End":"19:49.640","Text":"well as an easy way of finding this product.Stay or not, you\u0027re welcome."}],"ID":10653},{"Watched":false,"Name":"Vectors - Cross Product (continued)","Duration":"6m 42s","ChapterTopicVideoID":10313,"CourseChapterTopicPlaylistID":12291,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.470 ","End":"00:06.940","Text":"If you\u0027re continuing and I\u0027m assuming you know something about determinants."},{"Start":"00:07.370 ","End":"00:12.795","Text":"This formula has a simpler expression using determinants,"},{"Start":"00:12.795 ","End":"00:14.850","Text":"and here\u0027s the formula,"},{"Start":"00:14.850 ","End":"00:18.105","Text":"and I\u0027m going to show you how we use this."},{"Start":"00:18.105 ","End":"00:21.255","Text":"I just copied the example we did earlier,"},{"Start":"00:21.255 ","End":"00:24.420","Text":"and I want to use the same example at where I can check"},{"Start":"00:24.420 ","End":"00:28.275","Text":"the answer using the method with determinants."},{"Start":"00:28.275 ","End":"00:30.840","Text":"In our case with this example,"},{"Start":"00:30.840 ","End":"00:34.360","Text":"what we would get would be the determinant."},{"Start":"00:34.360 ","End":"00:39.360","Text":"It\u0027s always i, j,"},{"Start":"00:39.360 ","End":"00:44.370","Text":"and k. Then on this row,"},{"Start":"00:44.370 ","End":"00:48.570","Text":"next I put the 2, 1,"},{"Start":"00:48.570 ","End":"00:56.550","Text":"minus 1 and then the minus 3, 4, 1."},{"Start":"00:56.550 ","End":"01:00.295","Text":"Even though determinants there\u0027s more than 1 way of doing it,"},{"Start":"01:00.295 ","End":"01:04.505","Text":"I\u0027m going to use the method of cofactors."},{"Start":"01:04.505 ","End":"01:08.840","Text":"This is going to equal, let\u0027s see,"},{"Start":"01:08.840 ","End":"01:13.250","Text":"we take the i rather, you know what?"},{"Start":"01:13.250 ","End":"01:15.185","Text":"I\u0027ll move this to the right."},{"Start":"01:15.185 ","End":"01:20.390","Text":"Here I put the cofactor of i,"},{"Start":"01:20.390 ","End":"01:22.820","Text":"which is like deleting the column and row with it."},{"Start":"01:22.820 ","End":"01:26.340","Text":"I\u0027ve got 1 minus 1, 4, 1."},{"Start":"01:26.690 ","End":"01:29.040","Text":"Then we alternate signs."},{"Start":"01:29.040 ","End":"01:33.770","Text":"It has to be minus something with j and"},{"Start":"01:33.770 ","End":"01:37.865","Text":"then plus the cofactor of k with"},{"Start":"01:37.865 ","End":"01:43.610","Text":"k. In here I need to put a c for j. I cross this out and this out."},{"Start":"01:43.610 ","End":"01:45.530","Text":"I\u0027ve got 2 minus 1, minus 3,"},{"Start":"01:45.530 ","End":"01:50.195","Text":"1, 2 minus 1, minus 3, 1."},{"Start":"01:50.195 ","End":"01:53.105","Text":"Then for k, I need this."},{"Start":"01:53.105 ","End":"01:58.735","Text":"It\u0027s 2, 1 minus 3, 4."},{"Start":"01:58.735 ","End":"02:05.704","Text":"That equals, now the determinant of a 2 by 2 is this diagonal less this diagonal."},{"Start":"02:05.704 ","End":"02:07.790","Text":"It\u0027s 1 minus,"},{"Start":"02:07.790 ","End":"02:12.490","Text":"minus 4, which is 5."},{"Start":"02:12.490 ","End":"02:15.105","Text":"This 1 is 5."},{"Start":"02:15.105 ","End":"02:21.250","Text":"Listed the determinants, 2 plus 3."},{"Start":"02:21.560 ","End":"02:25.730","Text":"No, sorry, it\u0027s 2 minus this diagonal."},{"Start":"02:25.730 ","End":"02:29.825","Text":"The diagonal is 3, 2 minus 3, minus 1, sorry."},{"Start":"02:29.825 ","End":"02:31.970","Text":"Then 8 minus,"},{"Start":"02:31.970 ","End":"02:34.850","Text":"minus 3 is 11."},{"Start":"02:34.850 ","End":"02:37.870","Text":"What we end up getting is"},{"Start":"02:37.870 ","End":"02:45.170","Text":"5i plus 1j"},{"Start":"02:45.170 ","End":"02:48.755","Text":"and plus vector"},{"Start":"02:48.755 ","End":"02:50.970","Text":"plus 11k."},{"Start":"02:52.180 ","End":"03:00.500","Text":"I would say that this and this are the same,"},{"Start":"03:00.500 ","End":"03:02.270","Text":"just a different form of it."},{"Start":"03:02.270 ","End":"03:03.950","Text":"You\u0027ve got the right answer."},{"Start":"03:03.950 ","End":"03:09.415","Text":"So this is an easier formula if you know determinants."},{"Start":"03:09.415 ","End":"03:11.810","Text":"I\u0027m going to continue on a fresh page."},{"Start":"03:11.810 ","End":"03:20.840","Text":"I remember I said something when I introduced the concept of a.b cross c,"},{"Start":"03:20.840 ","End":"03:23.665","Text":"it was in a formula."},{"Start":"03:23.665 ","End":"03:26.450","Text":"Also, we used it to compute the volume."},{"Start":"03:26.450 ","End":"03:28.880","Text":"I said there\u0027s an easier way to compute this."},{"Start":"03:28.880 ","End":"03:32.420","Text":"This has a formula using determinants,"},{"Start":"03:32.420 ","End":"03:37.875","Text":"that this is the determinant of a_1, a_2,"},{"Start":"03:37.875 ","End":"03:41.925","Text":"a_3, b_1, b_2,"},{"Start":"03:41.925 ","End":"03:48.760","Text":"b_3, and c_1, c_2, c_3."},{"Start":"03:48.760 ","End":"03:52.310","Text":"I didn\u0027t write it again there we assume that a is the vector a_1,"},{"Start":"03:52.310 ","End":"03:53.900","Text":"a_2, a_3, and so on,"},{"Start":"03:53.900 ","End":"04:02.089","Text":"b and c. I\u0027m going to test it out on the example we used earlier."},{"Start":"04:02.089 ","End":"04:06.470","Text":"Here\u0027s the example we had earlier with these 3 vectors."},{"Start":"04:06.470 ","End":"04:10.080","Text":"This time let\u0027s do it using a determinant."},{"Start":"04:10.720 ","End":"04:15.575","Text":"In our case, what we have is,"},{"Start":"04:15.575 ","End":"04:18.860","Text":"let\u0027s see,1, 4 minus 7,"},{"Start":"04:18.860 ","End":"04:21.380","Text":"and then 2 minus 1,"},{"Start":"04:21.380 ","End":"04:28.500","Text":"4, and then 0 minus 9, 18."},{"Start":"04:32.780 ","End":"04:37.775","Text":"Again, I\u0027m going to use cofactors,"},{"Start":"04:37.775 ","End":"04:43.430","Text":"and I\u0027m going to use the last row rather than the first row,"},{"Start":"04:43.430 ","End":"04:46.730","Text":"which I usually use because there\u0027s a 0 in here and it\u0027ll save us"},{"Start":"04:46.730 ","End":"04:51.000","Text":"one of the terms in the sum."},{"Start":"04:51.000 ","End":"04:54.650","Text":"What we get is that this is equal to,"},{"Start":"04:54.650 ","End":"04:58.415","Text":"I\u0027ll write it out though it\u0027s 0 times the cofactor,"},{"Start":"04:58.415 ","End":"05:02.360","Text":"which is, doesn\u0027t really matter because it\u0027s 0 but 4,"},{"Start":"05:02.360 ","End":"05:06.040","Text":"7 minus 1, 4."},{"Start":"05:06.040 ","End":"05:09.700","Text":"Then there\u0027s a minus in the middle."},{"Start":"05:11.120 ","End":"05:14.045","Text":"That would be minus,"},{"Start":"05:14.045 ","End":"05:17.509","Text":"minus 9 times its cofactor,"},{"Start":"05:17.509 ","End":"05:20.580","Text":"which is 1, 7, 2, 4."},{"Start":"05:22.460 ","End":"05:27.820","Text":"Then plus 18 times its cofactor,"},{"Start":"05:27.820 ","End":"05:31.640","Text":"which is 1, 4, 2 minus 1."},{"Start":"05:32.510 ","End":"05:34.980","Text":"This is equal to,"},{"Start":"05:34.980 ","End":"05:37.775","Text":"well, this is 0."},{"Start":"05:37.775 ","End":"05:40.465","Text":"Doesn\u0027t matter what this is."},{"Start":"05:40.465 ","End":"05:45.055","Text":"This cofactor is, I\u0027ll just put a question mark."},{"Start":"05:45.055 ","End":"05:47.365","Text":"I don\u0027t care because I\u0027m going to multiply it by 0."},{"Start":"05:47.365 ","End":"05:51.235","Text":"I hope this is a minus 7, 4 plus 14."},{"Start":"05:51.235 ","End":"05:53.335","Text":"This is 18."},{"Start":"05:53.335 ","End":"05:57.560","Text":"Then minus 1,"},{"Start":"05:57.560 ","End":"06:01.485","Text":"minus 8, this is minus 9."},{"Start":"06:01.485 ","End":"06:06.780","Text":"Basically, what we get is 0 plus 9 times"},{"Start":"06:06.780 ","End":"06:16.200","Text":"18 and minus 18 times 9."},{"Start":"06:16.200 ","End":"06:20.285","Text":"I don\u0027t even have to compute what 9 times 18 is."},{"Start":"06:20.285 ","End":"06:24.860","Text":"I see that this actually is equal to 0,"},{"Start":"06:24.860 ","End":"06:27.180","Text":"which is what we got,"},{"Start":"06:27.180 ","End":"06:30.180","Text":"also, the other way."},{"Start":"06:30.180 ","End":"06:35.085","Text":"That\u0027s good. That\u0027s really it."},{"Start":"06:35.085 ","End":"06:37.100","Text":"For those of you who studied determinants,"},{"Start":"06:37.100 ","End":"06:40.345","Text":"there are some shortcut formulas."},{"Start":"06:40.345 ","End":"06:43.240","Text":"We\u0027re done with the cross-product."}],"ID":10657},{"Watched":false,"Name":"Exercise 1","Duration":"6m 8s","ChapterTopicVideoID":10310,"CourseChapterTopicPlaylistID":12291,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.715","Text":"The purpose of this exercise is just to practice computing the cross-product,"},{"Start":"00:05.715 ","End":"00:09.765","Text":"and there\u0027s more than 1 way to do this."},{"Start":"00:09.765 ","End":"00:11.865","Text":"In number 1,"},{"Start":"00:11.865 ","End":"00:14.250","Text":"I\u0027m not going to use determinants,"},{"Start":"00:14.250 ","End":"00:16.740","Text":"and in number 2, I will."},{"Start":"00:16.740 ","End":"00:20.460","Text":"I copied the formula from the tutorial,"},{"Start":"00:20.460 ","End":"00:22.890","Text":"the 1 that doesn\u0027t use determinants."},{"Start":"00:22.890 ","End":"00:26.370","Text":"If we have the components for a and the components for b,"},{"Start":"00:26.370 ","End":"00:33.845","Text":"we just make 3 separate computations for each of the components of the cross product."},{"Start":"00:33.845 ","End":"00:36.665","Text":"Let\u0027s fill these in,"},{"Start":"00:36.665 ","End":"00:41.220","Text":"a_2 is minus 2,"},{"Start":"00:41.220 ","End":"00:45.505","Text":"and also here minus 2,"},{"Start":"00:45.505 ","End":"00:49.800","Text":"b_3 would be 7,"},{"Start":"00:50.240 ","End":"00:56.560","Text":"and b_3 also here would be 7."},{"Start":"00:56.750 ","End":"01:02.680","Text":"Then we have a_3 is 5,"},{"Start":"01:07.820 ","End":"01:12.075","Text":"I\u0027ll just fill this out for you."},{"Start":"01:12.075 ","End":"01:15.060","Text":"I just copied a_1, a_2, a_3,"},{"Start":"01:15.060 ","End":"01:16.320","Text":"b_1, b_2, b_3,"},{"Start":"01:16.320 ","End":"01:17.880","Text":"etc., from here,"},{"Start":"01:17.880 ","End":"01:23.715","Text":"and so a cross b is equal to, let\u0027s see."},{"Start":"01:23.715 ","End":"01:30.790","Text":"The first component would be minus 14,"},{"Start":"01:31.190 ","End":"01:35.235","Text":"then it would be plus 20."},{"Start":"01:35.235 ","End":"01:37.095","Text":"Then we have what?"},{"Start":"01:37.095 ","End":"01:43.190","Text":"30 minus 21, and"},{"Start":"01:43.190 ","End":"01:51.280","Text":"then minus 12 plus 12."},{"Start":"01:53.630 ","End":"01:56.280","Text":"That is equal to,"},{"Start":"01:56.280 ","End":"01:58.230","Text":"let\u0027s see, 6,"},{"Start":"01:58.230 ","End":"02:03.855","Text":"30 minus 21 is 9, and here 0."},{"Start":"02:03.855 ","End":"02:07.710","Text":"Mostly, it\u0027s just boring computation."},{"Start":"02:07.710 ","End":"02:13.940","Text":"The second part, you\u0027re not expected to do another computation,"},{"Start":"02:13.940 ","End":"02:19.835","Text":"you\u0027re expected to remember that if you change the order of the cross-product,"},{"Start":"02:19.835 ","End":"02:22.115","Text":"you have a minus sign added,"},{"Start":"02:22.115 ","End":"02:25.220","Text":"it\u0027s minus a cross b."},{"Start":"02:25.220 ","End":"02:28.370","Text":"In this case, we just put a minus in front of everything,"},{"Start":"02:28.370 ","End":"02:30.095","Text":"and it\u0027s minus 6,"},{"Start":"02:30.095 ","End":"02:34.290","Text":"minus 9, minus 0."},{"Start":"02:34.600 ","End":"02:37.280","Text":"In part 2,"},{"Start":"02:37.280 ","End":"02:39.785","Text":"it\u0027s given in i, j, k form."},{"Start":"02:39.785 ","End":"02:45.320","Text":"But this time I will use determinants in doing the cross product."},{"Start":"02:45.320 ","End":"02:49.730","Text":"I gave a formula that for a cross product,"},{"Start":"02:49.730 ","End":"02:54.290","Text":"what you do is write a 3 by 3 determinant,"},{"Start":"02:54.290 ","End":"02:58.090","Text":"and here we put i,"},{"Start":"02:58.090 ","End":"03:02.250","Text":"j, and k. On the next row,"},{"Start":"03:02.250 ","End":"03:04.380","Text":"we put the first 1,"},{"Start":"03:04.380 ","End":"03:06.540","Text":"which is the u,"},{"Start":"03:06.540 ","End":"03:10.815","Text":"so 3, just the coefficients,"},{"Start":"03:10.815 ","End":"03:14.910","Text":"minus 1 and 5."},{"Start":"03:14.910 ","End":"03:16.475","Text":"Then the other 1,"},{"Start":"03:16.475 ","End":"03:17.690","Text":"there is no i,"},{"Start":"03:17.690 ","End":"03:20.525","Text":"so that means it\u0027s a 0 here,"},{"Start":"03:20.525 ","End":"03:24.445","Text":"then 4, and then minus 2."},{"Start":"03:24.445 ","End":"03:28.390","Text":"Then we use the method of co-factors."},{"Start":"03:28.390 ","End":"03:30.770","Text":"Basically, what we do is this,"},{"Start":"03:30.770 ","End":"03:33.755","Text":"I\u0027ll just write down the format."},{"Start":"03:33.755 ","End":"03:35.990","Text":"It\u0027ll be a 2 by 2 determinant here."},{"Start":"03:35.990 ","End":"03:38.015","Text":"I\u0027ll fill it out in a moment."},{"Start":"03:38.015 ","End":"03:43.545","Text":"I, and then there will be a minus another 2 by 2 determinant,"},{"Start":"03:43.545 ","End":"03:51.200","Text":"j and there will be a plus a 2 by 2 determinant k. What we do to get"},{"Start":"03:51.200 ","End":"03:58.740","Text":"the bit in front of the i is just eliminate the row and column with the i,"},{"Start":"03:58.740 ","End":"04:01.280","Text":"so we\u0027re left with this part here,"},{"Start":"04:01.280 ","End":"04:03.335","Text":"which is minus 1,"},{"Start":"04:03.335 ","End":"04:07.460","Text":"5, 4, minus 2."},{"Start":"04:07.460 ","End":"04:11.690","Text":"Then for j, eliminate the row and column with the j,"},{"Start":"04:11.690 ","End":"04:15.690","Text":"we\u0027re left with 3, 5, 0, minus 2."},{"Start":"04:17.920 ","End":"04:22.215","Text":"For k, we just get this corner here."},{"Start":"04:22.215 ","End":"04:25.480","Text":"3, minus 1, 0, 4."},{"Start":"04:27.950 ","End":"04:36.825","Text":"Then a 2 by 2 determinant is just this diagonal\u0027s product minus this diagonal."},{"Start":"04:36.825 ","End":"04:43.505","Text":"Here we have 2 minus 20,"},{"Start":"04:43.505 ","End":"04:51.675","Text":"which gives us minus 18i minus, let\u0027s see."},{"Start":"04:51.675 ","End":"04:54.740","Text":"This times this is minus 6,"},{"Start":"04:54.740 ","End":"04:57.905","Text":"this times this is nothing."},{"Start":"04:57.905 ","End":"05:02.455","Text":"We have minus, minus 6,"},{"Start":"05:02.455 ","End":"05:06.140","Text":"so that\u0027s plus 6j."},{"Start":"05:07.340 ","End":"05:12.300","Text":"Then this 1 is 12 less nothing,"},{"Start":"05:12.300 ","End":"05:15.130","Text":"so it\u0027s just 12k."},{"Start":"05:16.490 ","End":"05:18.645","Text":"That\u0027s the answer,"},{"Start":"05:18.645 ","End":"05:20.250","Text":"and it\u0027s in i, j,"},{"Start":"05:20.250 ","End":"05:26.200","Text":"k form, which I would expect if the original question was given in i,"},{"Start":"05:26.200 ","End":"05:27.715","Text":"j, k form."},{"Start":"05:27.715 ","End":"05:34.720","Text":"That\u0027s u cross v. If I"},{"Start":"05:34.720 ","End":"05:42.520","Text":"want v cross u,"},{"Start":"05:42.520 ","End":"05:52.850","Text":"then v cross u is just the negative of u cross v,"},{"Start":"05:52.850 ","End":"06:03.215","Text":"so it\u0027s going to be 18i minus 6j minus 12k."},{"Start":"06:03.215 ","End":"06:08.250","Text":"Just practice in computation, and we\u0027re done."}],"ID":10654},{"Watched":false,"Name":"Exercise 2","Duration":"6m 51s","ChapterTopicVideoID":10311,"CourseChapterTopicPlaylistID":12291,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we want to find a vector that\u0027s orthogonal"},{"Start":"00:04.950 ","End":"00:09.555","Text":"or perpendicular to the plane containing these 3 points."},{"Start":"00:09.555 ","End":"00:14.440","Text":"Just made up the example using the numbers from 1 through 9."},{"Start":"00:14.450 ","End":"00:19.600","Text":"Now, how is the cross-product going to help us here?"},{"Start":"00:19.600 ","End":"00:22.940","Text":"Well, let\u0027s see, first of all, what it means."},{"Start":"00:22.940 ","End":"00:25.620","Text":"Have a plane,"},{"Start":"00:25.690 ","End":"00:27.920","Text":"and we have 3 points, P,"},{"Start":"00:27.920 ","End":"00:30.510","Text":"Q and R. P, Q,"},{"Start":"00:30.510 ","End":"00:32.355","Text":"and R label them P,"},{"Start":"00:32.355 ","End":"00:35.265","Text":"Q, R. Nothing is to scale here."},{"Start":"00:35.265 ","End":"00:41.750","Text":"In general, there is a single plane that goes through any 3 points."},{"Start":"00:41.750 ","End":"00:44.390","Text":"Yeah, there\u0027s an exception if all 3 points"},{"Start":"00:44.390 ","End":"00:47.630","Text":"happen to be on the same line, that wouldn\u0027t work."},{"Start":"00:47.630 ","End":"00:51.380","Text":"But let\u0027s assume that we\u0027re not in an exceptional case."},{"Start":"00:51.380 ","End":"00:57.080","Text":"2 vectors that would be parallel to the plane could take any combination."},{"Start":"00:57.080 ","End":"01:02.045","Text":"But let\u0027s say I use PQ, and PR."},{"Start":"01:02.045 ","End":"01:08.475","Text":"I can get these 2 vectors and then I\u0027ll take the cross-product."},{"Start":"01:08.475 ","End":"01:11.695","Text":"I\u0027ll get a vector perpendicular to these 2."},{"Start":"01:11.695 ","End":"01:17.935","Text":"If a vector is perpendicular to 2 vectors in the plane that are not parallel,"},{"Start":"01:17.935 ","End":"01:21.570","Text":"then it\u0027s going to be perpendicular to the whole plane."},{"Start":"01:21.570 ","End":"01:23.620","Text":"Here\u0027s what we do. First of all,"},{"Start":"01:23.620 ","End":"01:26.570","Text":"let\u0027s see what vector PQ is."},{"Start":"01:26.570 ","End":"01:28.755","Text":"Call that PQ,"},{"Start":"01:28.755 ","End":"01:37.785","Text":"and that would be the components of Q minus the components of P,"},{"Start":"01:37.785 ","End":"01:42.670","Text":"so what we\u0027ll get would be 4 minus 1."},{"Start":"01:46.230 ","End":"01:51.220","Text":"In this exercise, we need to find a vector that\u0027s orthogonal"},{"Start":"01:51.220 ","End":"01:56.095","Text":"or perpendicular to the plane containing these 3 points."},{"Start":"01:56.095 ","End":"02:00.310","Text":"We\u0027re going to use the cross product vectors to help us here."},{"Start":"02:00.310 ","End":"02:07.690","Text":"First of all is a quick sketch that we have a plane that goes through 3 points."},{"Start":"02:07.690 ","End":"02:11.245","Text":"This isn\u0027t always the case."},{"Start":"02:11.245 ","End":"02:15.930","Text":"It\u0027s true if the 3 points are not on a straight line,"},{"Start":"02:15.930 ","End":"02:17.520","Text":"which they\u0027re not here."},{"Start":"02:17.520 ","End":"02:19.100","Text":"If 3 points are not,"},{"Start":"02:19.100 ","End":"02:20.770","Text":"what is called co-linear,"},{"Start":"02:20.770 ","End":"02:24.475","Text":"then there will be a plane through these 3 points."},{"Start":"02:24.475 ","End":"02:28.860","Text":"The idea is to get 2 vectors,"},{"Start":"02:28.860 ","End":"02:30.990","Text":"just choose 2 pairs of these points,"},{"Start":"02:30.990 ","End":"02:35.160","Text":"let say from P to Q and from P to"},{"Start":"02:35.160 ","End":"02:41.235","Text":"R. Then if you take the cross product of these 2,"},{"Start":"02:41.235 ","End":"02:44.900","Text":"we\u0027ll get a third vector which will be"},{"Start":"02:44.900 ","End":"02:50.975","Text":"perpendicular to these 2 and will therefore be perpendicular to the whole plane."},{"Start":"02:50.975 ","End":"02:55.860","Text":"Here it goes. PQ, the vector,"},{"Start":"02:55.860 ","End":"02:59.260","Text":"we get this by subtracting"},{"Start":"03:01.250 ","End":"03:09.470","Text":"the components of Q minus the components of P. We\u0027ll get 6 minus 1,"},{"Start":"03:09.470 ","End":"03:16.840","Text":"5 minus 2, 4 minus 3."},{"Start":"03:17.060 ","End":"03:20.380","Text":"That should be actually vector."},{"Start":"03:20.380 ","End":"03:23.945","Text":"Yeah, we\u0027re using angular brackets for vectors."},{"Start":"03:23.945 ","End":"03:30.225","Text":"In other words, 5, 3, 1."},{"Start":"03:30.225 ","End":"03:36.575","Text":"The other 1, I\u0027ll take PR though you could have take an RQ or some other possibility."},{"Start":"03:36.575 ","End":"03:41.195","Text":"We\u0027re taking PR and that will equal."},{"Start":"03:41.195 ","End":"03:49.295","Text":"I\u0027ll just do the final and suggest taking R minus P coordinate ys,"},{"Start":"03:49.295 ","End":"03:52.295","Text":"7 minus 1 is 6,"},{"Start":"03:52.295 ","End":"03:56.765","Text":"8 minus 2 is 6,"},{"Start":"03:56.765 ","End":"04:00.890","Text":"and 9 minus 3 is also 6."},{"Start":"04:00.890 ","End":"04:06.470","Text":"What we need now is the cross product PQ,"},{"Start":"04:06.470 ","End":"04:11.390","Text":"cross product with PR to get a perpendicular to both"},{"Start":"04:11.390 ","End":"04:16.030","Text":"of these and if it\u0027s perpendicular to 2 non-parallel vectors in the plane,"},{"Start":"04:16.030 ","End":"04:18.095","Text":"it\u0027s perpendicular to the whole plane."},{"Start":"04:18.095 ","End":"04:20.360","Text":"What this will equal,"},{"Start":"04:20.360 ","End":"04:27.460","Text":"if we use the determinant notation with i, j,"},{"Start":"04:27.460 ","End":"04:31.245","Text":"k here, I put i, j,"},{"Start":"04:31.245 ","End":"04:35.840","Text":"k. Here I put the first 1,"},{"Start":"04:35.840 ","End":"04:39.410","Text":"which is 5, 3, 1,"},{"Start":"04:39.410 ","End":"04:44.070","Text":"and here 6, 6, 6."},{"Start":"04:44.110 ","End":"04:49.835","Text":"What I\u0027ll do is several techniques for doing this."},{"Start":"04:49.835 ","End":"04:52.490","Text":"I\u0027ll do the co-factor method."},{"Start":"04:52.490 ","End":"04:58.230","Text":"What goes with i is you erase the i column in row 1,"},{"Start":"04:58.230 ","End":"05:03.645","Text":"we\u0027re left with the determinant 3, 1, 6, 6."},{"Start":"05:03.645 ","End":"05:06.055","Text":"This is the coefficient of i."},{"Start":"05:06.055 ","End":"05:08.000","Text":"Then the middle 1 gets a minus."},{"Start":"05:08.000 ","End":"05:09.560","Text":"The last 1 is plus again,"},{"Start":"05:09.560 ","End":"05:18.245","Text":"we get minus something j and then plus something k. The same idea for j,"},{"Start":"05:18.245 ","End":"05:22.055","Text":"I erase this and this and I\u0027ve got 5, 6,1, 6."},{"Start":"05:22.055 ","End":"05:25.220","Text":"Like so and for the k,"},{"Start":"05:25.220 ","End":"05:31.015","Text":"I get these 4; 5, 3, 6, 6."},{"Start":"05:31.015 ","End":"05:35.985","Text":"This comes out to be, let\u0027s see."},{"Start":"05:35.985 ","End":"05:41.045","Text":"Determinant of a 2-by-2 is this diagonal minus this diagonal,"},{"Start":"05:41.045 ","End":"05:43.310","Text":"product minus product,"},{"Start":"05:43.310 ","End":"05:48.240","Text":"18 minus 6 is 12,"},{"Start":"05:49.900 ","End":"05:55.470","Text":"30 minus 6 is 24,"},{"Start":"05:56.080 ","End":"06:05.190","Text":"and 30 minus 18 is 12 again, that\u0027s 12i."},{"Start":"06:06.170 ","End":"06:13.630","Text":"We could leave that as an answer or I could also use angular notation,"},{"Start":"06:13.630 ","End":"06:18.805","Text":"angular bracket 12 minus 24, 12."},{"Start":"06:18.805 ","End":"06:21.535","Text":"This is a perfectly good answer."},{"Start":"06:21.535 ","End":"06:24.040","Text":"I like to tidy things up a bit."},{"Start":"06:24.040 ","End":"06:26.095","Text":"If I divide it by 12,"},{"Start":"06:26.095 ","End":"06:28.210","Text":"it\u0027s still going to be a perpendicular,"},{"Start":"06:28.210 ","End":"06:30.700","Text":"scalar times a vector keeps its direction."},{"Start":"06:30.700 ","End":"06:34.595","Text":"I could also take as an answer,"},{"Start":"06:34.595 ","End":"06:39.320","Text":"1, minus 2, 1 would be good,"},{"Start":"06:39.320 ","End":"06:45.925","Text":"but this is perfectly fine if you don\u0027t want to make it neater."},{"Start":"06:45.925 ","End":"06:51.390","Text":"This or this and that\u0027s all."}],"ID":10655},{"Watched":false,"Name":"Exercise 3","Duration":"9m ","ChapterTopicVideoID":10312,"CourseChapterTopicPlaylistID":12291,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"This exercise has 2 parts,"},{"Start":"00:02.310 ","End":"00:04.065","Text":"and in each of the part,"},{"Start":"00:04.065 ","End":"00:07.695","Text":"we\u0027re given 3 vectors in"},{"Start":"00:07.695 ","End":"00:14.295","Text":"3D and we have to decide if they all lie in the same plane or not."},{"Start":"00:14.295 ","End":"00:18.045","Text":"I want to remind you of the strategy we use here."},{"Start":"00:18.045 ","End":"00:24.750","Text":"In general, 3 vectors will form what is called a parallelepiped."},{"Start":"00:24.750 ","End":"00:27.630","Text":"It could look something like this."},{"Start":"00:27.630 ","End":"00:33.495","Text":"It\u0027s like the 3 dimensional analog of a parallelogram."},{"Start":"00:33.495 ","End":"00:38.670","Text":"Now, if these 3 vectors happen to be in the same plane,"},{"Start":"00:38.670 ","End":"00:41.270","Text":"then the volume of this thing is 0."},{"Start":"00:41.270 ","End":"00:43.760","Text":"Otherwise it\u0027s not going to be 0."},{"Start":"00:43.760 ","End":"00:49.160","Text":"We have a formula that gives us the volume of this."},{"Start":"00:49.160 ","End":"00:51.530","Text":"It uses the cross and the dot product."},{"Start":"00:51.530 ","End":"00:59.070","Text":"The volume of this parallelepiped is equal to"},{"Start":"00:59.120 ","End":"01:09.410","Text":"a.b cross c. You could actually take these in any order,"},{"Start":"01:09.410 ","End":"01:13.055","Text":"but you first have to do the cross product then you get a vector,"},{"Start":"01:13.055 ","End":"01:15.875","Text":"and then a vector dot with a vector gives you a scalar."},{"Start":"01:15.875 ","End":"01:18.305","Text":"Actually this could come out negative,"},{"Start":"01:18.305 ","End":"01:24.080","Text":"so we also put an absolute value sign around this."},{"Start":"01:24.080 ","End":"01:28.465","Text":"All we have to do is check if this is equal to 0."},{"Start":"01:28.465 ","End":"01:35.080","Text":"The other thing is, that there\u0027s a formula for this with determinants."},{"Start":"01:36.380 ","End":"01:40.150","Text":"Well, we\u0027ve talked about this in the theory part,"},{"Start":"01:40.150 ","End":"01:41.560","Text":"so let me just write it."},{"Start":"01:41.560 ","End":"01:44.150","Text":"That for part 1,"},{"Start":"01:44.150 ","End":"01:51.300","Text":"that a.b cross c"},{"Start":"01:51.300 ","End":"01:55.740","Text":"is equal to the determinant."},{"Start":"01:55.740 ","End":"01:58.260","Text":"You just take the components of a,"},{"Start":"01:58.260 ","End":"02:01.785","Text":"3, minus 2, 5."},{"Start":"02:01.785 ","End":"02:06.675","Text":"Then for b, 6, minus 4, 7,"},{"Start":"02:06.675 ","End":"02:13.040","Text":"and for c, 1, 0,1."},{"Start":"02:13.040 ","End":"02:14.795","Text":"We just want to see if this is 0 or not."},{"Start":"02:14.795 ","End":"02:16.610","Text":"If you actually want the volume,"},{"Start":"02:16.610 ","End":"02:18.500","Text":"the end you have take absolute value,"},{"Start":"02:18.500 ","End":"02:21.780","Text":"and if the results comes out negative throughout the minus."},{"Start":"02:21.780 ","End":"02:24.270","Text":"Let\u0027s see what this is."},{"Start":"02:24.270 ","End":"02:27.080","Text":"If you haven\u0027t studied determinants,"},{"Start":"02:27.080 ","End":"02:28.850","Text":"you can always do it the long way."},{"Start":"02:28.850 ","End":"02:33.110","Text":"Figure out what is b cross c using another formula and then take the dot product."},{"Start":"02:33.110 ","End":"02:35.795","Text":"I\u0027ll assume you know a bit about determinants,"},{"Start":"02:35.795 ","End":"02:39.460","Text":"another way of expanding using cofactors,"},{"Start":"02:39.460 ","End":"02:43.410","Text":"and you expand according to the most convenient row or column."},{"Start":"02:43.410 ","End":"02:49.775","Text":"The 3rd row looks very good because it has a 0 in it and the numbers are small to 1s."},{"Start":"02:49.775 ","End":"02:56.780","Text":"What this will equal will be,"},{"Start":"02:56.780 ","End":"03:03.140","Text":"we take the 1 times the determinant"},{"Start":"03:03.140 ","End":"03:08.805","Text":"of what\u0027s left if you delete it\u0027s row and column."},{"Start":"03:08.805 ","End":"03:11.825","Text":"It\u0027s times minus 2,"},{"Start":"03:11.825 ","End":"03:15.455","Text":"5, minus 4, 7,"},{"Start":"03:15.455 ","End":"03:20.875","Text":"and then minus 0 times,"},{"Start":"03:20.875 ","End":"03:22.530","Text":"well it doesn\u0027t matter what it is,"},{"Start":"03:22.530 ","End":"03:24.890","Text":"something, something, something, something."},{"Start":"03:24.890 ","End":"03:27.050","Text":"Because it\u0027s a 0, I don\u0027t care."},{"Start":"03:27.050 ","End":"03:29.345","Text":"It\u0027s actually 3, 6, and 5, 7,"},{"Start":"03:29.345 ","End":"03:33.330","Text":"and then plus 1 times,"},{"Start":"03:33.330 ","End":"03:36.470","Text":"I\u0027m running out of space,"},{"Start":"03:36.470 ","End":"03:37.490","Text":"I\u0027ll move it over here."},{"Start":"03:37.490 ","End":"03:39.725","Text":"I move down here."},{"Start":"03:39.725 ","End":"03:45.645","Text":"This 1 times the determinant of 3,"},{"Start":"03:45.645 ","End":"03:49.790","Text":"minus 2, 6, minus 4."},{"Start":"03:49.790 ","End":"03:53.540","Text":"There\u0027s a rule for knowing when to start with"},{"Start":"03:53.540 ","End":"03:57.590","Text":"a plus or a minus and it always alternates plus, minus, plus."},{"Start":"03:57.590 ","End":"04:04.420","Text":"Won\u0027t get into technical details assuming you\u0027ve studied determinants."},{"Start":"04:04.420 ","End":"04:07.005","Text":"Then let\u0027s see what this equals."},{"Start":"04:07.005 ","End":"04:12.270","Text":"1 times minus 14,"},{"Start":"04:12.270 ","End":"04:16.995","Text":"minus 20 is minus 34."},{"Start":"04:16.995 ","End":"04:20.730","Text":"I didn\u0027t need the 1, I\u0027ll just put minus 34."},{"Start":"04:20.730 ","End":"04:25.935","Text":"This 1 it\u0027s just minus 0 because it doesn\u0027t matter what this is."},{"Start":"04:25.935 ","End":"04:27.675","Text":"Here plus 1 times,"},{"Start":"04:27.675 ","End":"04:34.140","Text":"I\u0027ll just need this minus 12."},{"Start":"04:34.140 ","End":"04:35.855","Text":"I think I made a mistake here."},{"Start":"04:35.855 ","End":"04:39.440","Text":"It was a minus, minus a minus. Wait a minute."},{"Start":"04:39.440 ","End":"04:42.305","Text":"It was minus 14, minus,"},{"Start":"04:42.305 ","End":"04:47.345","Text":"minus 20, so it\u0027s minus 14 plus 20."},{"Start":"04:47.345 ","End":"04:49.720","Text":"This is 6, I\u0027m sorry."},{"Start":"04:49.720 ","End":"04:52.260","Text":"Yeah, 6."},{"Start":"04:52.260 ","End":"04:59.675","Text":"Now here again we have a negative take away a negative have minus 12,"},{"Start":"04:59.675 ","End":"05:04.955","Text":"minus, minus 12."},{"Start":"05:04.955 ","End":"05:08.645","Text":"That\u0027s actually 0,"},{"Start":"05:08.645 ","End":"05:12.970","Text":"and so the answer is 6,"},{"Start":"05:12.970 ","End":"05:20.610","Text":"but the main point is that this 6 is not equal to 0."},{"Start":"05:20.610 ","End":"05:24.840","Text":"We can say that in this case, that a, b,"},{"Start":"05:24.840 ","End":"05:30.465","Text":"and c are not in the same plane."},{"Start":"05:30.465 ","End":"05:33.525","Text":"I\u0027ll just write that, not in the same plane."},{"Start":"05:33.525 ","End":"05:36.045","Text":"That\u0027s number 1."},{"Start":"05:36.045 ","End":"05:39.655","Text":"Now, let\u0027s take number 2."},{"Start":"05:39.655 ","End":"05:46.085","Text":"I got rid of the picture that way I can have room to do number 2 over here."},{"Start":"05:46.085 ","End":"05:49.775","Text":"The same idea, different letters."},{"Start":"05:49.775 ","End":"05:56.130","Text":"What I need is u.v cross w."},{"Start":"05:56.130 ","End":"06:01.930","Text":"Let\u0027s see what determinant this gives us."},{"Start":"06:01.930 ","End":"06:04.770","Text":"Just copy the coordinates,"},{"Start":"06:04.770 ","End":"06:06.795","Text":"I\u0027ve got, 1,"},{"Start":"06:06.795 ","End":"06:10.200","Text":"4, minus 7,"},{"Start":"06:10.200 ","End":"06:11.700","Text":"and then 2,"},{"Start":"06:11.700 ","End":"06:14.620","Text":"minus 1, 4,"},{"Start":"06:15.380 ","End":"06:19.140","Text":"and then you don\u0027t have any i here,"},{"Start":"06:19.140 ","End":"06:21.300","Text":"so that\u0027s a 0 here."},{"Start":"06:21.300 ","End":"06:26.070","Text":"We\u0027ve got minus 9, 18."},{"Start":"06:26.070 ","End":"06:32.550","Text":"Again, I\u0027m going to use this method of cofactors."},{"Start":"06:32.590 ","End":"06:39.840","Text":"I\u0027ll take the first column or the last row."},{"Start":"06:39.840 ","End":"06:41.420","Text":"They each have a 0, which is helpful,"},{"Start":"06:41.420 ","End":"06:43.025","Text":"but these numbers are smaller."},{"Start":"06:43.025 ","End":"06:45.920","Text":"I\u0027m going to expand by the first column."},{"Start":"06:45.920 ","End":"06:53.840","Text":"I\u0027ve got 1 times the determinant of this,"},{"Start":"06:53.840 ","End":"06:57.645","Text":"which is minus 1, 4,"},{"Start":"06:57.645 ","End":"07:04.650","Text":"minus 9,18, and then it alternates the sign."},{"Start":"07:04.650 ","End":"07:05.730","Text":"This is a minus,"},{"Start":"07:05.730 ","End":"07:08.055","Text":"and then 2 times,"},{"Start":"07:08.055 ","End":"07:09.840","Text":"what\u0027s left is this, this, this,"},{"Start":"07:09.840 ","End":"07:12.240","Text":"this, which is 4,"},{"Start":"07:12.240 ","End":"07:15.990","Text":"minus 7, minus 9,"},{"Start":"07:15.990 ","End":"07:22.275","Text":"18, and then plus 0 times whatever."},{"Start":"07:22.275 ","End":"07:26.020","Text":"It doesn\u0027t matter because it\u0027s a 0."},{"Start":"07:27.320 ","End":"07:30.630","Text":"Maybe for the practice, I will put the numbers in."},{"Start":"07:30.630 ","End":"07:36.300","Text":"It\u0027s 4, minus 7, minus 1, 4."},{"Start":"07:36.300 ","End":"07:39.795","Text":"Even though we know it doesn\u0027t matter, but okay."},{"Start":"07:39.795 ","End":"07:41.575","Text":"This is equal to,"},{"Start":"07:41.575 ","End":"07:46.550","Text":"now here I have minus 18,"},{"Start":"07:46.550 ","End":"07:50.580","Text":"minus, minus 36."},{"Start":"07:50.580 ","End":"07:55.455","Text":"That\u0027s minus 18 plus 36, that\u0027s 18."},{"Start":"07:55.455 ","End":"07:58.785","Text":"Now this determinant is,"},{"Start":"07:58.785 ","End":"08:03.945","Text":"let\u0027s see, 4 times 18 is 72."},{"Start":"08:03.945 ","End":"08:08.085","Text":"This times this is plus 63."},{"Start":"08:08.085 ","End":"08:13.740","Text":"72 minus 63 is 9."},{"Start":"08:13.740 ","End":"08:18.285","Text":"This is minus 2 times 9,"},{"Start":"08:18.285 ","End":"08:20.795","Text":"and here plus 0."},{"Start":"08:20.795 ","End":"08:23.645","Text":"It doesn\u0027t matter, but I might as well do it."},{"Start":"08:23.645 ","End":"08:33.060","Text":"This will be 16 minus 7 is 9."},{"Start":"08:33.060 ","End":"08:35.100","Text":"Anyway, this is 0."},{"Start":"08:35.100 ","End":"08:38.980","Text":"I\u0027ve got 18 minus 18, so it\u0027s 0."},{"Start":"08:38.980 ","End":"08:42.140","Text":"In this case we do have a 0."},{"Start":"08:42.140 ","End":"08:48.615","Text":"These 3 vectors are in the same plane,"},{"Start":"08:48.615 ","End":"08:54.260","Text":"or maybe emphasize, are in the same plane."},{"Start":"08:54.260 ","End":"08:57.065","Text":"Part 1 not, Part 2,"},{"Start":"08:57.065 ","End":"09:01.050","Text":"yeah, they are. We\u0027re done."}],"ID":10656},{"Watched":false,"Name":"Exercise 4","Duration":"8m 12s","ChapterTopicVideoID":27753,"CourseChapterTopicPlaylistID":12291,"HasSubtitles":false,"VideoComments":[],"Subtitles":[],"ID":29045}],"Thumbnail":null,"ID":12291},{"Name":"The 3D Coordinates System","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System","Duration":"19m 22s","ChapterTopicVideoID":10315,"CourseChapterTopicPlaylistID":12292,"HasSubtitles":true,"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":"00:09.330","Text":"which is the plane,"},{"Start":"00:09.330 ","End":"00:11.475","Text":"and x and y."},{"Start":"00:11.475 ","End":"00:14.850","Text":"Now we\u0027re going to talk about k, 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":"00:25.485","Text":"I\u0027ll give it the name, P,"},{"Start":"00:25.485 ","End":"00:28.830","Text":"and 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":"00:41.655","Text":"This here, the plane containing these 2 axes is called the xy plane,"},{"Start":"00:41.655 ","End":"00:50.480","Text":"and the plane containing the z and the y-axis is called the yz"},{"Start":"00:50.480 ","End":"01:01.065","Text":"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":"01:17.695","Text":"it just complicates the picture but 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":"01:27.320","Text":"One way to do it is to take this point,"},{"Start":"01:27.320 ","End":"01:31.160","Text":"and I\u0027ll call it Q and just drop perpendiculars in"},{"Start":"01:31.160 ","End":"01:38.050","Text":"the xy plane and let\u0027s say this is the point where it hits that,"},{"Start":"01:38.050 ","End":"01:40.780","Text":"will be called a."},{"Start":"01:40.780 ","End":"01:43.750","Text":"The distance from here to here and from here to here,"},{"Start":"01:43.750 ","End":"01:45.320","Text":"it will be b,"},{"Start":"01:45.320 ","End":"01:48.110","Text":"and the height of this would be c,"},{"Start":"01:48.110 ","End":"01:51.715","Text":"or you could just take something with the same height on here,"},{"Start":"01:51.715 ","End":"01:53.190","Text":"hard to say exactly,"},{"Start":"01:53.190 ","End":"01:55.585","Text":"where probably here,"},{"Start":"01:55.585 ","End":"02:05.210","Text":"so 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":"02:18.840","Text":"I\u0027ll call this one R and this one S,"},{"Start":"02:18.840 ","End":"02:20.430","Text":"after P and Q,"},{"Start":"02:20.430 ","End":"02:22.020","Text":"come R and S,"},{"Start":"02:22.020 ","End":"02:29.385","Text":"and this would be the point where on the zy plane the x is 0,"},{"Start":"02:29.385 ","End":"02:31.110","Text":"so a here is 0,"},{"Start":"02:31.110 ","End":"02:33.705","Text":"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":"02:50.405","Text":"On the xy plane,"},{"Start":"02:50.405 ","End":"02:51.650","Text":"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":"03:31.200","Text":"Another point of notation, in general,"},{"Start":"03:31.200 ","End":"03:35.730","Text":"when we want to talk about 3D or any number of dimensions,"},{"Start":"03:35.730 ","End":"03:41.490","Text":"R, and I got this R with the funny line here,"},{"Start":"03:41.490 ","End":"03:43.290","Text":"represents the real numbers,"},{"Start":"03:43.290 ","End":"03:45.910","Text":"or that\u0027s 1 dimension."},{"Start":"03:47.330 ","End":"03:52.440","Text":"When we write R squared,"},{"Start":"03:52.440 ","End":"03:55.785","Text":"that\u0027s 2 dimensions of the plane."},{"Start":"03:55.785 ","End":"03:58.560","Text":"If I write R^3,"},{"Start":"03:58.560 ","End":"04:03.300","Text":"that would be our 3D space,"},{"Start":"04:03.300 ","End":"04:07.305","Text":"3 real numbers, a, b, and c,"},{"Start":"04:07.305 ","End":"04:12.320","Text":"and in general, 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":"04:24.920","Text":"This is 1D,"},{"Start":"04:24.920 ","End":"04:26.900","Text":"which is just a line,"},{"Start":"04:26.900 ","End":"04:28.555","Text":"R^2 is 2D,"},{"Start":"04:28.555 ","End":"04:30.540","Text":"this is line,"},{"Start":"04:30.540 ","End":"04:32.490","Text":"this would be a plane,"},{"Start":"04:32.490 ","End":"04:34.800","Text":"this would be space,"},{"Start":"04:34.800 ","End":"04:40.020","Text":"and R^n is more abstract n-dimensional space."},{"Start":"04:40.020 ","End":"04:44.280","Text":"That\u0027s 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":"04:51.050","Text":"they can be generalized, not always,"},{"Start":"04:51.050 ","End":"04:56.165","Text":"but they often can be generalized to 3D and 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":"05:02.465","Text":"let\u0027s say I have point P_1,"},{"Start":"05:02.465 ","End":"05:05.660","Text":"which is say, a_1,"},{"Start":"05:05.660 ","End":"05:09.600","Text":"b_1, c_1, or x_1, y_1, z_1,"},{"Start":"05:09.600 ","End":"05:13.380","Text":"whatever, and we have another point, P_2,"},{"Start":"05:13.380 ","End":"05:18.355","Text":"which is a_2, b_2, c_2,"},{"Start":"05:18.355 ","End":"05:25.565","Text":"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":"05:32.910","Text":"but I can make a good guess if I take the 2 dimensional case,"},{"Start":"05:33.020 ","End":"05:39.345","Text":"I\u0027ll use different color so we know that this is the 2D and this is the 3D."},{"Start":"05:39.345 ","End":"05:42.810","Text":"In 2D a point would just have 2 coordinates,"},{"Start":"05:42.810 ","End":"05:45.240","Text":"say a_1, b_1,"},{"Start":"05:45.240 ","End":"05:46.770","Text":"and the other point,"},{"Start":"05:46.770 ","End":"05:50.610","Text":"let\u0027s say I had 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":"06:08.655","Text":"let\u0027s say a_2 minus a_1 squared,"},{"Start":"06:08.655 ","End":"06:12.180","Text":"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":"06:25.230","Text":"What I would do is,"},{"Start":"06:25.230 ","End":"06:28.960","Text":"I would generalize this"},{"Start":"06:29.900 ","End":"06:36.150","Text":"by adding c_2 minus c_1 squared,"},{"Start":"06:36.150 ","End":"06:39.890","Text":"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":"06:47.870","Text":"The proof for the 2D case was based on"},{"Start":"06:47.870 ","End":"06:52.730","Text":"Pythagoras\u0027s theorem and 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":"07:12.470 ","End":"07:17.010","Text":"A circle is given by a center,"},{"Start":"07:17.010 ","End":"07:24.030","Text":"and say that the center is say, h,"},{"Start":"07:24.030 ","End":"07:28.530","Text":"k and the radius,"},{"Start":"07:28.530 ","End":"07:29.985","Text":"we need to know,"},{"Start":"07:29.985 ","End":"07:33.160","Text":"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":"08:12.715","Text":"but in 3D,"},{"Start":"08:12.715 ","End":"08:17.455","Text":"the points equidistant from a given point actually form a sphere."},{"Start":"08:17.455 ","End":"08:22.675","Text":"In 3D, let\u0027s say you want another equation of a sphere,"},{"Start":"08:22.675 ","End":"08:31.255","Text":"and let\u0027s say we know its center is h, k, l,"},{"Start":"08:31.255 ","End":"08:39.715","Text":"and will also have a radius of r. But this time the equation,"},{"Start":"08:39.715 ","End":"08:42.160","Text":"it\u0027s generalized from this."},{"Start":"08:42.160 ","End":"08:47.590","Text":"It\u0027s x minus h squared plus y minus k squared,"},{"Start":"08:47.590 ","End":"08:56.020","Text":"and 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":"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":"09:12.144","Text":"l is r only 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":"09:32.830","Text":"I\u0027d like to take an example of an equation,"},{"Start":"09:32.830 ","End":"09:37.480","Text":"and the equation will be x equals 3"},{"Start":"09:37.480 ","End":"09:40.375","Text":"but I want to interpret it in 1D,"},{"Start":"09:40.375 ","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, here we have x, y, z."},{"Start":"09:47.050 ","End":"09:49.585","Text":"In 1 dimension,"},{"Start":"09:49.585 ","End":"09:53.395","Text":"x equals 3 on the number line is just a point."},{"Start":"09:53.395 ","End":"09:55.930","Text":"Here\u0027s what it looks like in 1 dimension,"},{"Start":"09:55.930 ","End":"10:01.015","Text":"in 1D, 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":"10:11.455","Text":"If x equals 3, y is not restricted 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":"10:20.530","Text":"The same equation in 3D means that the x coordinate is 3,"},{"Start":"10:20.530 ","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":"10:40.900","Text":"What you do is you just take a plane which is orthogonal,"},{"Start":"10:40.900 ","End":"10:44.710","Text":"perpendicular to this x-axis,"},{"Start":"10:44.710 ","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":"10:54.190","Text":"and y and z can be anything."},{"Start":"10:54.190 ","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 the 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":"11:27.475","Text":"Because everywhere on this plane we have something of the form any x and y but no z."},{"Start":"11:27.475 ","End":"11:32.050","Text":"Similarly, if I take the x-z plane,"},{"Start":"11:32.050 ","End":"11:36.160","Text":"then its equation actually it\u0027s the missing 1."},{"Start":"11:36.160 ","End":"11:37.585","Text":"What you can do is look what\u0027s missing,"},{"Start":"11:37.585 ","End":"11:38.920","Text":"what\u0027s missing is y,"},{"Start":"11:38.920 ","End":"11:45.895","Text":"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":"11:50.785","Text":"but this is where y is 0."},{"Start":"11:50.785 ","End":"11:52.270","Text":"It\u0027s this plane here."},{"Start":"11:52.270 ","End":"11:54.835","Text":"I won\u0027t shade it, it might look messy."},{"Start":"11:54.835 ","End":"11:56.530","Text":"Well, it looks something like this,"},{"Start":"11:56.530 ","End":"11:59.920","Text":"but including this line and this line and the last 1,"},{"Start":"11:59.920 ","End":"12:03.135","Text":"which is the y-z plane,"},{"Start":"12:03.135 ","End":"12:05.760","Text":"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":"12:15.025","Text":"We have 3 equations of the coordinate planes."},{"Start":"12:15.025 ","End":"12:18.010","Text":"I\u0027ve shown you how an equation of a plane could be"},{"Start":"12:18.010 ","End":"12:21.460","Text":"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":"12:39.610","Text":"The question will be to"},{"Start":"12:39.610 ","End":"12:46.285","Text":"graph the equation y equals 2x minus 3"},{"Start":"12:46.285 ","End":"12:49.960","Text":"but in 2 scenarios in 2D, in other words,"},{"Start":"12:49.960 ","End":"12:57.025","Text":"in the case of R2 and in 3D in other words, in R3."},{"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 to find 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 3/2 one-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 and up and down."},{"Start":"13:51.460 ","End":"13:54.589","Text":"It would look something like this."},{"Start":"13:54.720 ","End":"14:04.270","Text":"Here. 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":"14:31.135","Text":"Now this concept of drawing a graph in 2D and then without z and then adding the z in,"},{"Start":"14:31.135 ","End":"14:34.060","Text":"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":"14:42.460","Text":"is that if a line will take a circle,"},{"Start":"14:42.460 ","End":"14:45.940","Text":"so 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":"14:56.950","Text":"Let\u0027s take x squared plus y squared equals 4."},{"Start":"14:56.950 ","End":"14:59.785","Text":"Now 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":"15:07.260 ","End":"15:09.760","Text":"The center is 0,"},{"Start":"15:09.760 ","End":"15:11.440","Text":"0 and the radius,"},{"Start":"15:11.440 ","End":"15:19.070","Text":"well 4 is 2 squared 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":"16:07.100","Text":"In fact, this whole part here would be the part that\u0027s in the x-y plane,"},{"Start":"16:07.100 ","End":"16:08.150","Text":"that would be the original circle"},{"Start":"16:08.150 ","End":"16:12.810","Text":"but we get a lot more because that is unrestricted z, sorry."},{"Start":"16:13.480 ","End":"16:18.770","Text":"Now so far, we\u0027ve typically seen that in 3D we get a surface."},{"Start":"16:18.770 ","End":"16:22.970","Text":"We had some planes and now we have a cylinder."},{"Start":"16:22.970 ","End":"16:26.180","Text":"Usually, this is the case that typically"},{"Start":"16:26.180 ","End":"16:29.695","Text":"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":"16:34.430","Text":"but a plane and a cylinder are surfaces"},{"Start":"16:34.430 ","End":"16:39.315","Text":"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":"16:54.625","Text":"1, 2, 3?"},{"Start":"16:54.625 ","End":"16:58.300","Text":"I don\u0027t know. 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":"17:12.460","Text":"but you get the idea and suppose this height was at z equals 4."},{"Start":"17:12.460 ","End":"17:17.475","Text":"I wanted the equation of just this circle in 3D,"},{"Start":"17:17.475 ","End":"17:22.840","Text":"1 way to do it would be to say x squared plus y"},{"Start":"17:22.840 ","End":"17:30.270","Text":"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":"17:44.409","Text":"well, I\u0027ve got a trick that beats that cheating."},{"Start":"17:44.409 ","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":"18:06.175","Text":"Plus z minus 4 squared equals 0."},{"Start":"18:06.175 ","End":"18:07.700","Text":"Now it\u0027s 1 equation."},{"Start":"18:07.700 ","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":"18:19.420","Text":"but I want to not use and,"},{"Start":"18:19.420 ","End":"18:20.725","Text":"and I just want to have 1 equation."},{"Start":"18:20.725 ","End":"18:24.970","Text":"I could write a squared plus b squared equals 0."},{"Start":"18:24.970 ","End":"18:27.545","Text":"You see, a and b,"},{"Start":"18:27.545 ","End":"18:29.470","Text":"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":"18:35.575","Text":"The only way the sum of 2,"},{"Start":"18:35.575 ","End":"18:40.165","Text":"0 are positives could be 0 is if they\u0027re both 0."},{"Start":"18:40.165 ","End":"18:45.805","Text":"This is actually both ways so 2 equations could become 1 equation."},{"Start":"18:45.805 ","End":"18:51.905","Text":"I just use that trick here by just putting the 4 over to the other side and making it 0."},{"Start":"18:51.905 ","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":"19:00.250","Text":"and it\u0027s parallel to the x-y plane,"},{"Start":"19:00.250 ","End":"19:03.790","Text":"and it\u0027s at height z equals 4."},{"Start":"19:03.790 ","End":"19:05.020","Text":"It\u0027s this circle here,"},{"Start":"19:05.020 ","End":"19:07.730","Text":"which I drew badly,"},{"Start":"19:07.890 ","End":"19:10.630","Text":"and there I straightened out a bit."},{"Start":"19:10.630 ","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":10658},{"Watched":false,"Name":"Exercise 1","Duration":"1m 32s","ChapterTopicVideoID":10316,"CourseChapterTopicPlaylistID":12292,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this exercise, we\u0027re given a point in 3D space,"},{"Start":"00:03.840 ","End":"00:10.395","Text":"and we want 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 x y plane,"},{"Start":"00:17.595 ","End":"00:19.784","Text":"that\u0027s where z is 0."},{"Start":"00:19.784 ","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 x y plane."},{"Start":"00:28.965 ","End":"00:34.070","Text":"In our case, the projection onto"},{"Start":"00:34.070 ","End":"00:40.065","Text":"the x y plane will just be 4,"},{"Start":"00:40.065 ","End":"00:44.295","Text":"7, 0, just setting z is 0."},{"Start":"00:44.295 ","End":"00:54.469","Text":"Then, similarly, the projection onto the x z plane is obtained by letting y equal 0."},{"Start":"00:54.469 ","End":"01:01.545","Text":"In this case, we\u0027ll have our original point 4 here minus 5,"},{"Start":"01:01.545 ","End":"01:03.225","Text":"except that in the middle."},{"Start":"01:03.225 ","End":"01:05.955","Text":"For the y we put 0."},{"Start":"01:05.955 ","End":"01:14.620","Text":"Let\u0027s see what\u0027s missing."},{"Start":"01:14.620 ","End":"01:17.820","Text":"The y z plane,"},{"Start":"01:17.860 ","End":"01:20.130","Text":"that\u0027s this 1 here,"},{"Start":"01:20.130 ","End":"01:22.860","Text":"so that\u0027s where we let the x equals 0."},{"Start":"01:22.860 ","End":"01:28.450","Text":"We\u0027ve got 0 and then 7 minus 5."},{"Start":"01:28.450 ","End":"01:30.020","Text":"And that\u0027s all,"},{"Start":"01:30.020 ","End":"01:32.250","Text":"so we\u0027re done here."}],"ID":10659},{"Watched":false,"Name":"Exercise 2","Duration":"3m 40s","ChapterTopicVideoID":10317,"CourseChapterTopicPlaylistID":12292,"HasSubtitles":true,"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":"00:18.480","Text":"Now, the distance you can get just by taking"},{"Start":"00:18.480 ","End":"00:22.110","Text":"the projection of this point onto this plane that will"},{"Start":"00:22.110 ","End":"00:26.295","Text":"give the closest point on the xy-plane to this."},{"Start":"00:26.295 ","End":"00:30.780","Text":"Because it\u0027s going to be perpendicular,"},{"Start":"00:30.780 ","End":"00:36.240","Text":"the line from the point to the projection,"},{"Start":"00:36.240 ","End":"00:40.455","Text":"I brought the picture from the previous clip, it might help."},{"Start":"00:40.455 ","End":"00:42.450","Text":"Here\u0027s the point P,"},{"Start":"00:42.450 ","End":"00:45.290","Text":"and if I want to know the distance to the xy-plane,"},{"Start":"00:45.290 ","End":"00:50.880","Text":"I just take the distance from here to here"},{"Start":"00:50.880 ","End":"00:53.550","Text":"because the projection has a property that"},{"Start":"00:53.550 ","End":"00:57.200","Text":"it\u0027s 90 degrees and that\u0027s what the distance means."},{"Start":"00:57.200 ","End":"01:00.199","Text":"The distance is the perpendicular distance."},{"Start":"01:00.199 ","End":"01:06.575","Text":"We need the distance from"},{"Start":"01:06.575 ","End":"01:15.285","Text":"this original 0.47 minus 5 to the projection,"},{"Start":"01:15.285 ","End":"01:20.785","Text":"and the projection we get just by setting the z equals 0,"},{"Start":"01:20.785 ","End":"01:24.930","Text":"4, 7, 0, that\u0027s this point here,"},{"Start":"01:24.930 ","End":"01:29.725","Text":"it\u0027s labeled Q in this diagram."},{"Start":"01:29.725 ","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":"01:51.635","Text":"all we need is the distance from minus 5 to 0,"},{"Start":"01:51.635 ","End":"01:55.340","Text":"so we subtract and take the absolute value,"},{"Start":"01:55.340 ","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":"02:06.355","Text":"Now in b,"},{"Start":"02:06.355 ","End":"02:13.020","Text":"the closer point means the one with the lesser distance."},{"Start":"02:14.470 ","End":"02:19.175","Text":"This one is the same point as this point."},{"Start":"02:19.175 ","End":"02:23.760","Text":"We know the distance from 4,"},{"Start":"02:23.760 ","End":"02:29.220","Text":"7, negative 5 to the xy-plane, maybe I\u0027ll write that."},{"Start":"02:29.220 ","End":"02:31.365","Text":"Quick copy-paste here."},{"Start":"02:31.365 ","End":"02:38.340","Text":"Distance to the xy-plane is 5."},{"Start":"02:38.340 ","End":"02:42.770","Text":"Then similarly, to get the distance of the other point,"},{"Start":"02:42.770 ","End":"02:48.045","Text":"was is it 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":"02:52.470","Text":"call some other letter, I don\u0027t know P,"},{"Start":"02:52.470 ","End":"02:55.575","Text":"Q, R, call it T or something."},{"Start":"02:55.575 ","End":"03:01.080","Text":"Distance of this point is where we do the same trick."},{"Start":"03:01.080 ","End":"03:03.230","Text":"We first of all find the projection."},{"Start":"03:03.230 ","End":"03:12.465","Text":"The projection of this onto the xy-plane is 5 minus 6, 0."},{"Start":"03:12.465 ","End":"03:14.370","Text":"Then the distance,"},{"Start":"03:14.370 ","End":"03:16.010","Text":"because these 2 are the same,"},{"Start":"03:16.010 ","End":"03:21.340","Text":"is just the distance from 7 to 0, so is 7."},{"Start":"03:21.340 ","End":"03:27.075","Text":"Now, because 5 is less than 7,"},{"Start":"03:27.075 ","End":"03:33.180","Text":"the answer is that this point P"},{"Start":"03:33.180 ","End":"03:39.250","Text":"is the closer one, and that\u0027s it."}],"ID":10660},{"Watched":false,"Name":"Exercise 3","Duration":"4m 26s","ChapterTopicVideoID":10318,"CourseChapterTopicPlaylistID":12292,"HasSubtitles":true,"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":"00:14.670","Text":"Previously, we\u0027ve had a distance of a point from the x,"},{"Start":"00:14.670 ","End":"00:17.830","Text":"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, its 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":"00:48.620","Text":"This is in the x, y plane,"},{"Start":"00:48.620 ","End":"00:50.930","Text":"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":"01:06.150","Text":"It\u0027s associated with P,"},{"Start":"01:06.150 ","End":"01:07.500","Text":"it\u0027s a projection of P on to"},{"Start":"01:07.500 ","End":"01:14.125","Text":"the z-axis and what I\u0027m saying is this height is the same as this height."},{"Start":"01:14.125 ","End":"01:18.800","Text":"This is the same as this and perhaps,"},{"Start":"01:18.800 ","End":"01:22.550","Text":"this is a long-winded approach to stating"},{"Start":"01:22.550 ","End":"01:24.890","Text":"the obvious is that this point,"},{"Start":"01:24.890 ","End":"01:32.730","Text":"its coordinates will be 0, 0, z."},{"Start":"01:33.790 ","End":"01:39.585","Text":"In our case, in our particular P,"},{"Start":"01:39.585 ","End":"01:45.885","Text":"that P bar would be just the same."},{"Start":"01:45.885 ","End":"01:47.490","Text":"The same Z is this,"},{"Start":"01:47.490 ","End":"01:50.830","Text":"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":"02:25.455","Text":"squared plus negative 5 minus negative 5 squared,"},{"Start":"02:25.455 ","End":"02:29.580","Text":"and this part is 0."},{"Start":"02:29.580 ","End":"02:38.115","Text":"I just get the square root of 4 squared plus 7 squared."},{"Start":"02:38.115 ","End":"02:42.395","Text":"This is the square root of, let\u0027s see,"},{"Start":"02:42.395 ","End":"02:48.940","Text":"49 and 16 is 65."},{"Start":"02:48.940 ","End":"02:53.735","Text":"Now, I\u0027m not going to do the whole thing from scratch with Q."},{"Start":"02:53.735 ","End":"02:58.070","Text":"There\u0027s a Q and then there\u0027s the projection,"},{"Start":"02:58.070 ","End":"03:03.110","Text":"just 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":"03:23.759","Text":"Now, if I did the distance between Q and Q bar,"},{"Start":"03:27.200 ","End":"03:32.970","Text":"basically, it\u0027s just this squared because it\u0027s this minus 0 squared."},{"Start":"03:33.040 ","End":"03:38.865","Text":"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":"03:42.875","Text":"just like it was here."},{"Start":"03:42.875 ","End":"03:54.335","Text":"We don\u0027t need that and 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":"04:06.555","Text":"Now, obviously, this is less than this"},{"Start":"04:06.555 ","End":"04:15.820","Text":"and so the closer 1 is the Q."},{"Start":"04:16.190 ","End":"04:22.369","Text":"This Q is the closer 1 to the z-axis"},{"Start":"04:22.369 ","End":"04:24.020","Text":"because this is smaller than this."},{"Start":"04:24.020 ","End":"04:26.880","Text":"That\u0027s it."}],"ID":10661},{"Watched":false,"Name":"Exercise 4","Duration":"4m 40s","ChapterTopicVideoID":10314,"CourseChapterTopicPlaylistID":12292,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"The purpose of this exercise is just to show you that the same equation in x and"},{"Start":"00:05.790 ","End":"00:11.520","Text":"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":"real numbers squared,"},{"Start":"00:15.825 ","End":"00:18.705","Text":"or it could be in R^3,"},{"Start":"00:18.705 ","End":"00:20.190","Text":"and in each case,"},{"Start":"00:20.190 ","End":"00:22.560","Text":"it would represent a different shape,"},{"Start":"00:22.560 ","End":"00:27.555","Text":"get an extra dimension in each of part b,"},{"Start":"00:27.555 ","End":"00:30.360","Text":"so in each case they\u0027re giving you an equation in xy,"},{"Start":"00:30.360 ","End":"00:34.155","Text":"and we want to interpret it in 2D and in 3D."},{"Start":"00:34.155 ","End":"00:36.210","Text":"Don\u0027t want to get too technical,"},{"Start":"00:36.210 ","End":"00:39.580","Text":"just wanted you to write, 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":"00:49.160","Text":"This would be a line I could even sketch it if I let x"},{"Start":"00:49.160 ","End":"00:53.405","Text":"equal 0, well why don\u0027t I in fact,"},{"Start":"00:53.405 ","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 3y is 6,"},{"Start":"01:00.000 ","End":"01:02.760","Text":"so y is 2 and the other way if y is 0,"},{"Start":"01:02.760 ","End":"01:04.020","Text":"x is 3,"},{"Start":"01:04.020 ","End":"01:09.470","Text":"so if I just do something rough like this,"},{"Start":"01:09.470 ","End":"01:11.809","Text":"we said that when x is 0,"},{"Start":"01:11.809 ","End":"01:14.595","Text":"then y was 2,"},{"Start":"01:14.595 ","End":"01:17.295","Text":"and when y was 0, x was 3,"},{"Start":"01:17.295 ","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":"01:33.320","Text":"Now, if I was looking at this as an equation in 3 dimensions,"},{"Start":"01:33.320 ","End":"01:37.070","Text":"x, y, z, and z just happens to be missing."},{"Start":"01:37.070 ","End":"01:40.400","Text":"What we do is we just add a third dimension."},{"Start":"01:40.400 ","End":"01:42.260","Text":"I can\u0027t really sketch it here,"},{"Start":"01:42.260 ","End":"01:45.050","Text":"but through each of these points on the line,"},{"Start":"01:45.050 ","End":"01:50.180","Text":"we\u0027ll take a vertical line perpendicular to the plane that we\u0027re seeing,"},{"Start":"01:50.180 ","End":"01:54.020","Text":"and so what we actually get is a plane."},{"Start":"01:54.020 ","End":"02:00.395","Text":"Here I just wanted you to identify this as the equation of a line."},{"Start":"02:00.395 ","End":"02:02.680","Text":"But in 3-dimensions,"},{"Start":"02:02.680 ","End":"02:05.450","Text":"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":"02:22.130","Text":"which is a vertical line through the origin, and that\u0027s all."},{"Start":"02:22.130 ","End":"02:32.480","Text":"In Part 2. Just few studied a little bit of implicit equations and circles."},{"Start":"02:32.480 ","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":"02:43.390","Text":"we even know the center, it\u0027s 1,0."},{"Start":"02:43.390 ","End":"02:46.150","Text":"The same might be 1,"},{"Start":"02:46.150 ","End":"02:49.700","Text":"and this would be 0, where y,"},{"Start":"02:49.700 ","End":"02:51.510","Text":"and then x,"},{"Start":"02:51.510 ","End":"02:54.060","Text":"and then 9 is 3 squared,"},{"Start":"02:54.060 ","End":"02:57.750","Text":"so the radius would be 3."},{"Start":"02:57.750 ","End":"03:00.730","Text":"If I go 3 in this direction,"},{"Start":"03:00.730 ","End":"03:02.710","Text":"that will take me to 4,"},{"Start":"03:02.710 ","End":"03:04.720","Text":"and if I take 3 in this direction,"},{"Start":"03:04.720 ","End":"03:07.360","Text":"it will take me to minus 2,"},{"Start":"03:07.360 ","End":"03:15.985","Text":"and then we get some kind of a circle which it really doesn\u0027t matter."},{"Start":"03:15.985 ","End":"03:19.670","Text":"You 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":"03:33.175","Text":"Anyway, the idea is just to identify that this is a circle in 2D."},{"Start":"03:33.175 ","End":"03:37.960","Text":"But if I add the z dimension through each of these points,"},{"Start":"03:37.960 ","End":"03:44.005","Text":"I can draw a straight line extending infinitely in both directions."},{"Start":"03:44.005 ","End":"03:50.150","Text":"What we\u0027ll get, and we\u0027ve seen this before is a cylinder."},{"Start":"03:53.900 ","End":"03:56.130","Text":"As I said, the whole idea"},{"Start":"03:56.130 ","End":"04:01.360","Text":"here is just to emphasize that if you see an equation in x and y,"},{"Start":"04:01.360 ","End":"04:06.560","Text":"you don\u0027t know if it\u0027s in 2D or in 3D,"},{"Start":"04:06.560 ","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 plain 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":"04:21.320","Text":"These are both equations in x, y,"},{"Start":"04:21.320 ","End":"04:24.215","Text":"and z, 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":"04:32.840","Text":"If you wanted to, you could always write plus 0,"},{"Start":"04:32.840 ","End":"04:36.650","Text":"z and then force z to be in here or here."},{"Start":"04:36.650 ","End":"04:39.570","Text":"Okay, That\u0027s all I wanted to say."}],"ID":10662}],"Thumbnail":null,"ID":12292},{"Name":"Equations of Lines","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Equations of Lines","Duration":"7m 23s","ChapterTopicVideoID":10322,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"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":"00:12.825","Text":"I\u0027m going to assume that you\u0027ve already studied parametric equations,"},{"Start":"00:12.825 ","End":"00:17.235","Text":"at least in 2D, which is what we\u0027re going to start off with."},{"Start":"00:17.235 ","End":"00:20.279","Text":"In 2D, we often gave 2 functions."},{"Start":"00:20.279 ","End":"00:22.590","Text":"We would say that x is some function of t,"},{"Start":"00:22.590 ","End":"00:27.270","Text":"or just say x is a function of t and we give another function y in"},{"Start":"00:27.270 ","End":"00:32.295","Text":"terms of t. Then we also may be restricted t to a certain range."},{"Start":"00:32.295 ","End":"00:35.895","Text":"Maybe t was between 0 and 2 Pi or whatever."},{"Start":"00:35.895 ","End":"00:42.250","Text":"T might\u0027ve been between something and something or less than or whatever."},{"Start":"00:43.040 ","End":"00:46.720","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":"00:52.230 ","End":"00:57.070","Text":"I brought an example of an ellipse even though the subject is going to be lines,"},{"Start":"00:57.070 ","End":"00:59.515","Text":"so this is a good example."},{"Start":"00:59.515 ","End":"01:03.935","Text":"In this case, what we get is not this general,"},{"Start":"01:03.935 ","End":"01:07.780","Text":"we have specifically x equals"},{"Start":"01:07.780 ","End":"01:12.910","Text":"6 cosine t. The 6 is actually related to"},{"Start":"01:12.910 ","End":"01:18.100","Text":"the 6 here and y is 3 sine t,"},{"Start":"01:18.100 ","End":"01:20.500","Text":"and that\u0027s related to the 3 here."},{"Start":"01:20.500 ","End":"01:23.170","Text":"Often we put curly braces."},{"Start":"01:23.170 ","End":"01:33.105","Text":"Then t, it happens here goes between 0 and 2 Pi or 360 degrees,"},{"Start":"01:33.105 ","End":"01:35.745","Text":"but I\u0027m taking it in radians."},{"Start":"01:35.745 ","End":"01:42.155","Text":"Now, this is all a prelude to something called vector functions."},{"Start":"01:42.155 ","End":"01:44.690","Text":"I wanted to do this in the vector notation."},{"Start":"01:44.690 ","End":"01:52.680","Text":"Now, I also hope you remember the concept of position vector."},{"Start":"01:52.680 ","End":"01:53.945","Text":"If you don\u0027t remember it,"},{"Start":"01:53.945 ","End":"01:58.535","Text":"perhaps you might want to go back to the chapter on vectors."},{"Start":"01:58.535 ","End":"02:05.310","Text":"In any event, the vector that connects the point 0,"},{"Start":"02:05.310 ","End":"02:12.135","Text":"0, the origin, to a general point x, y,"},{"Start":"02:12.135 ","End":"02:16.450","Text":"the position vector is what we get when we"},{"Start":"02:16.450 ","End":"02:22.290","Text":"subtract the xs and we get x and subtract the ys,"},{"Start":"02:22.290 ","End":"02:23.820","Text":"y minus 0 is y."},{"Start":"02:23.820 ","End":"02:26.965","Text":"The angular brackets means this is a vector notation."},{"Start":"02:26.965 ","End":"02:32.085","Text":"This is the position vector from the origin to the point x, y."},{"Start":"02:32.085 ","End":"02:41.805","Text":"Often we call this the letter r. Instead of a pair of numbers,"},{"Start":"02:41.805 ","End":"02:44.565","Text":"x, y, we have a vector."},{"Start":"02:44.565 ","End":"02:50.040","Text":"In fact, we can also use r in a function notation."},{"Start":"02:50.040 ","End":"02:51.920","Text":"What I wrote above,"},{"Start":"02:51.920 ","End":"03:00.830","Text":"I could write as r as a function of t is 6 cosine t,"},{"Start":"03:00.830 ","End":"03:10.520","Text":"3 sine t. Here t is restricted between 0 and 2 Pi."},{"Start":"03:10.520 ","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":"03:34.895","Text":"Especially if we\u0027re given,"},{"Start":"03:34.895 ","End":"03:37.770","Text":"say, 2 points on the line."},{"Start":"03:38.180 ","End":"03:46.535","Text":"Here I have a diagram which I\u0027m going to use for the equation of a line in 2D."},{"Start":"03:46.535 ","End":"03:48.290","Text":"This is the line,"},{"Start":"03:48.290 ","End":"03:51.630","Text":"I\u0027ll just call it the line."},{"Start":"03:51.800 ","End":"03:56.000","Text":"Here I have a specific point on the line, p naught,"},{"Start":"03:56.000 ","End":"04:00.255","Text":"let\u0027s give it coordinates x naught, y naught."},{"Start":"04:00.255 ","End":"04:02.400","Text":"Then we have another point,"},{"Start":"04:02.400 ","End":"04:05.495","Text":"this will be a more general point, x, y."},{"Start":"04:05.495 ","End":"04:09.215","Text":"We have position vectors to P and to P naught,"},{"Start":"04:09.215 ","End":"04:13.210","Text":"call them r vector and r naught vector."},{"Start":"04:13.210 ","End":"04:16.370","Text":"Included the I and j came with the picture,"},{"Start":"04:16.370 ","End":"04:18.020","Text":"the standard basis vectors."},{"Start":"04:18.020 ","End":"04:24.680","Text":"I want to know how to write the equation of this line in parametric form,"},{"Start":"04:24.680 ","End":"04:27.870","Text":"in vector parametric form."},{"Start":"04:27.890 ","End":"04:30.360","Text":"What I can say is,"},{"Start":"04:30.360 ","End":"04:33.255","Text":"yeah, this vector, this red 1 here,"},{"Start":"04:33.255 ","End":"04:40.040","Text":"I\u0027ll call it V. This is just any direction vector for the line."},{"Start":"04:40.040 ","End":"04:43.145","Text":"Of course, a vector would be drawn usually through the origin,"},{"Start":"04:43.145 ","End":"04:44.585","Text":"it doesn\u0027t matter where you place it,"},{"Start":"04:44.585 ","End":"04:50.265","Text":"but there\u0027s going to be a vector non-0 and parallel to the line, any direction vector."},{"Start":"04:50.265 ","End":"04:53.720","Text":"What I want to say is that I\u0027m going to develop"},{"Start":"04:53.720 ","End":"04:57.290","Text":"the formula by saying that I can get to this point,"},{"Start":"04:57.290 ","End":"04:58.520","Text":"the general point x,"},{"Start":"04:58.520 ","End":"05:05.180","Text":"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 P naught to P."},{"Start":"05:08.960 ","End":"05:13.410","Text":"Remember the triangle rule for addition of vectors."},{"Start":"05:13.410 ","End":"05:14.895","Text":"What I\u0027m saying is,"},{"Start":"05:14.895 ","End":"05:21.350","Text":"is that the vector r is equal to"},{"Start":"05:21.350 ","End":"05:30.205","Text":"the vector r naught plus the position vector from P naught to P,"},{"Start":"05:30.205 ","End":"05:36.590","Text":"I\u0027ll just call it for the moment p naught p. Now,"},{"Start":"05:36.590 ","End":"05:39.640","Text":"this vector P naught P,"},{"Start":"05:39.640 ","End":"05:41.745","Text":"on the other hand,"},{"Start":"05:41.745 ","End":"05:45.405","Text":"the vector P naught P,"},{"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":"05:54.755","Text":"Conversely, any multiple of this position vector,"},{"Start":"05:54.755 ","End":"05:58.025","Text":"if I place it from P naught will be on the line."},{"Start":"05:58.025 ","End":"06:02.660","Text":"This is some constant that the parameter"},{"Start":"06:02.660 ","End":"06:09.200","Text":"t times the position vector V. If I put that in here,"},{"Start":"06:09.200 ","End":"06:17.045","Text":"I get that r is equal to r naught plus"},{"Start":"06:17.045 ","End":"06:20.975","Text":"t times v. If I have"},{"Start":"06:20.975 ","End":"06:26.560","Text":"a point on the line in vector form and a direction vector for the line,"},{"Start":"06:26.560 ","End":"06:29.055","Text":"then this is the equation of the line."},{"Start":"06:29.055 ","End":"06:32.960","Text":"What we do is we say that because r is x,"},{"Start":"06:32.960 ","End":"06:38.330","Text":"y, I can write this equivalently as x,"},{"Start":"06:38.330 ","End":"06:42.490","Text":"y is equal to x naught,"},{"Start":"06:42.490 ","End":"06:47.580","Text":"y naught plus t times,"},{"Start":"06:47.580 ","End":"06:51.570","Text":"and let\u0027s say this vector v is,"},{"Start":"06:51.570 ","End":"06:54.945","Text":"let\u0027s say a, b."},{"Start":"06:54.945 ","End":"07:01.165","Text":"Here I can write a, b."},{"Start":"07:01.165 ","End":"07:07.715","Text":"Either of these, either this form or this form,"},{"Start":"07:07.715 ","End":"07:13.940","Text":"will give the equation of a line in parametric form in 2D."},{"Start":"07:13.940 ","End":"07:18.290","Text":"Every value of t gives me a different point on the line,"},{"Start":"07:18.290 ","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":10663},{"Watched":false,"Name":"The 3D Coordinate System - Equations of Lines (continued)","Duration":"15m 19s","ChapterTopicVideoID":10321,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"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":"00:10.120","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":"00:17.870","Text":"but I want to stay with 2D just because it\u0027s easier,"},{"Start":"00:17.870 ","End":"00:19.170","Text":"and then we\u0027ll generalize."},{"Start":"00:19.170 ","End":"00:22.125","Text":"There is something called a parametric form also,"},{"Start":"00:22.125 ","End":"00:25.120","Text":"and 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":"00:32.340","Text":"I think I\u0027d rather put a variable t here just to emphasize"},{"Start":"00:32.340 ","End":"00:37.620","Text":"that the position vector is a function of a parameter t. This is called the vector form,"},{"Start":"00:37.620 ","End":"00:40.195","Text":"both of these are called the vector form."},{"Start":"00:40.195 ","End":"00:45.700","Text":"The parametric form, is if I just write this 1 component wise,"},{"Start":"00:45.700 ","End":"00:51.130","Text":"then I get that x equals x naught plus ta,"},{"Start":"00:51.130 ","End":"00:56.530","Text":"and y equals y naught plus tb."},{"Start":"00:56.530 ","End":"01:02.830","Text":"This is called the parametric form."},{"Start":"01:06.020 ","End":"01:10.210","Text":"There is a third 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":"01:19.525","Text":"If I isolate t from each of the equations here,"},{"Start":"01:19.525 ","End":"01:25.720","Text":"I can say that t is equal to x"},{"Start":"01:25.720 ","End":"01:32.640","Text":"minus x naught over a."},{"Start":"01:32.640 ","End":"01:42.615","Text":"From here, I get the t equals y minus y naught over b,"},{"Start":"01:42.615 ","End":"01:47.510","Text":"and 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":"02:15.410","Text":"but it\u0027s the same v,"},{"Start":"02:15.410 ","End":"02:18.140","Text":"only this time it is not a,"},{"Start":"02:18.140 ","End":"02:21.450","Text":"b, it\u0027s a, b,"},{"Start":"02:21.450 ","End":"02:25.380","Text":"c. The r naught,"},{"Start":"02:25.380 ","End":"02:31.370","Text":"instead of being x naught, y naught is x naught, y naught,"},{"Start":"02:31.370 ","End":"02:35.330","Text":"z naught, and r is x,"},{"Start":"02:35.330 ","End":"02:38.820","Text":"y, c, and so on."},{"Start":"02:38.820 ","End":"02:47.650","Text":"I\u0027ll just write the analogs of these 2D formulas in 3D."},{"Start":"02:47.650 ","End":"02:49.580","Text":"Let\u0027s look at them."},{"Start":"02:49.580 ","End":"02:52.210","Text":"I\u0027ll start with the vector form,"},{"Start":"02:52.210 ","End":"02:54.725","Text":"I\u0027ll use different color for 3D."},{"Start":"02:54.725 ","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 the 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":"03:20.615","Text":"these parts are identical,"},{"Start":"03:20.615 ","End":"03:22.760","Text":"but when I write it out in components,"},{"Start":"03:22.760 ","End":"03:24.070","Text":"the same thing,"},{"Start":"03:24.070 ","End":"03:28.175","Text":"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":"03:36.810","Text":"z naught, plus t and not a, b but a,"},{"Start":"03:36.810 ","End":"03:44.325","Text":"b, c. I labeled these vector form,"},{"Start":"03:44.325 ","End":"03:46.545","Text":"either or both of these."},{"Start":"03:46.545 ","End":"03:49.560","Text":"Now let\u0027s get to the parametric form."},{"Start":"03:49.560 ","End":"03:53.790","Text":"Also, we just break this up component-wise,"},{"Start":"03:53.790 ","End":"03:56.370","Text":"maybe put some curly braces,"},{"Start":"03:56.370 ","End":"03:59.780","Text":"only this time we\u0027re going to have x,"},{"Start":"03:59.780 ","End":"04:08.420","Text":"y, 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":"04:26.260 ","End":"04:30.920","Text":"here I labeled this parametric form."},{"Start":"04:30.920 ","End":"04:33.580","Text":"I need to scroll down,"},{"Start":"04:33.580 ","End":"04:38.990","Text":"and what we\u0027re heading towards is something called the symmetric equations of the line."},{"Start":"04:38.990 ","End":"04:40.390","Text":"That\u0027s the third form."},{"Start":"04:40.390 ","End":"04:43.375","Text":"I\u0027m just going to extend that,"},{"Start":"04:43.375 ","End":"04:44.840","Text":"and I will write,"},{"Start":"04:44.840 ","End":"04:46.930","Text":"and this is what we get when we isolate t,"},{"Start":"04:46.930 ","End":"04:49.220","Text":"t equals reach these 3 things."},{"Start":"04:49.220 ","End":"04:58.220","Text":"We get that x minus x naught over a equals y minus y naught"},{"Start":"04:58.220 ","End":"05:07.655","Text":"over b equals z minus z naught over c. This is not really an equation."},{"Start":"05:07.655 ","End":"05:10.025","Text":"Here it\u0027s 2 equations,"},{"Start":"05:10.025 ","End":"05:13.310","Text":"so it\u0027s a set of equations."},{"Start":"05:13.310 ","End":"05:17.180","Text":"There is a small technical snag with this form"},{"Start":"05:17.180 ","End":"05:20.590","Text":"because 1 of these could be 0, the position vector."},{"Start":"05:20.590 ","End":"05:24.275","Text":"They can all be 0, but 1 or even 2 of them could be 0."},{"Start":"05:24.275 ","End":"05:26.000","Text":"What we do is that,"},{"Start":"05:26.000 ","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":"05:33.290","Text":"If denominator is 0,"},{"Start":"05:33.290 ","End":"05:35.060","Text":"we force the numerator to be 0."},{"Start":"05:35.060 ","End":"05:36.770","Text":"That means that y is a constant,"},{"Start":"05:36.770 ","End":"05:39.080","Text":"y naught, just like with the parameters."},{"Start":"05:39.080 ","End":"05:42.110","Text":"If p is 0, then y would be y naught,"},{"Start":"05:42.110 ","End":"05:43.985","Text":"so we take it with that understanding."},{"Start":"05:43.985 ","End":"05:51.545","Text":"That turns out to be a line that\u0027s parallel to the x, z plane."},{"Start":"05:51.545 ","End":"05:53.300","Text":"Actually, if 2 of them are 0,"},{"Start":"05:53.300 ","End":"05:55.070","Text":"so 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":"06:00.855","Text":"but then there\u0027s no equation,"},{"Start":"06:00.855 ","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":"06:11.870","Text":"If you do get them,"},{"Start":"06:11.870 ","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":"06:30.235","Text":"which is 2 minus 13."},{"Start":"06:30.235 ","End":"06:32.000","Text":"Another point, I don\u0027t know,"},{"Start":"06:32.000 ","End":"06:39.265","Text":"Q will equal 1, 4 minus 3."},{"Start":"06:39.265 ","End":"06:49.010","Text":"My question is to find the equations of the line passing through these 2 points?"},{"Start":"06:49.010 ","End":"06:51.290","Text":"I want all 3 forms, the vector form,"},{"Start":"06:51.290 ","End":"06:55.770","Text":"parametric form, and the symmetric equations."},{"Start":"06:56.060 ","End":"07:02.280","Text":"What I can do is for r naught,"},{"Start":"07:02.280 ","End":"07:05.640","Text":"I could take just the position vector of p,"},{"Start":"07:05.640 ","End":"07:07.335","Text":"this could be p naught."},{"Start":"07:07.335 ","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":"07:17.595","Text":"so 2, minus 1, 3."},{"Start":"07:17.595 ","End":"07:21.310","Text":"Now I need also a direction vector."},{"Start":"07:21.310 ","End":"07:25.260","Text":"I need a position vector of a point on the line and the direction vector."},{"Start":"07:25.260 ","End":"07:26.640","Text":"For the direction vector,"},{"Start":"07:26.640 ","End":"07:32.840","Text":"the most obvious thing to do is to take the vector that joins P to Q."},{"Start":"07:32.840 ","End":"07:35.555","Text":"The tail is at P and the head is at 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":"07:48.635","Text":"4 minus minus 1 is 5,"},{"Start":"07:48.635 ","End":"07:50.115","Text":"and minus 3,"},{"Start":"07:50.115 ","End":"07:53.840","Text":"minus 3 is minus 6."},{"Start":"07:53.840 ","End":"07:55.830","Text":"Now I have r naught and"},{"Start":"07:55.830 ","End":"08:05.320","Text":"v. Both of these"},{"Start":"08:05.320 ","End":"08:07.370","Text":"are vector form,"},{"Start":"08:07.370 ","End":"08:11.665","Text":"I\u0027ll use that x, y,"},{"Start":"08:11.665 ","End":"08:16.975","Text":"z is equal to 2 minus 1,"},{"Start":"08:16.975 ","End":"08:26.740","Text":"3 plus t 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":"08:36.745","Text":"The parametric form, is just take some curly braces"},{"Start":"08:36.745 ","End":"08:43.855","Text":"and say x equals y equals z equals,"},{"Start":"08:43.855 ","End":"08:52.925","Text":"and then it comes out to 2 minus t minus"},{"Start":"08:52.925 ","End":"08:59.350","Text":"1 plus 5t and"},{"Start":"08:59.350 ","End":"09:04.270","Text":"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, the x naught,"},{"Start":"09:16.140 ","End":"09:21.185","Text":"x minus the 2 over the minus"},{"Start":"09:21.185 ","End":"09:27.880","Text":"1 is equal to y minus this."},{"Start":"09:27.880 ","End":"09:35.285","Text":"It\u0027s y plus 1 over the 5,"},{"Start":"09:35.285 ","End":"09:45.260","Text":"and then z minus 3 over minus 6."},{"Start":"09:45.840 ","End":"09:48.965","Text":"That answers the questions."},{"Start":"09:48.965 ","End":"09:54.500","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":"09:59.510","Text":"I think we\u0027ll do 1 more example,"},{"Start":"09:59.510 ","End":"10:01.480","Text":"I\u0027ll erase this 1."},{"Start":"10:01.480 ","End":"10:08.290","Text":"In this question, we\u0027re given a line and I\u0027m going to give it in parametric form,"},{"Start":"10:08.290 ","End":"10:10.860","Text":"but I won\u0027t use the braces."},{"Start":"10:10.860 ","End":"10:16.970","Text":"I\u0027ll just write x equals 10 plus 3t,"},{"Start":"10:16.970 ","End":"10:20.575","Text":"and y equals 12t,"},{"Start":"10:20.575 ","End":"10:27.850","Text":"and z equals 3 minus t. That\u0027s 1 line."},{"Start":"10:27.850 ","End":"10:34.130","Text":"Now I want, there\u0027s another line and this is the one we\u0027re going to look for."},{"Start":"10:34.130 ","End":"10:37.955","Text":"What we know about this one is that,"},{"Start":"10:37.955 ","End":"10:41.795","Text":"well, it\u0027s parallel to the above line,"},{"Start":"10:41.795 ","End":"10:50.750","Text":"to this line and it passes through a given point."},{"Start":"10:50.750 ","End":"11:00.364","Text":"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":"11:10.630","Text":"The question is, does it cross the x, z plane pass through it?"},{"Start":"11:10.940 ","End":"11:16.270","Text":"If so, then we give the coordinates of the point."},{"Start":"11:16.270 ","End":"11:20.555","Text":"The middle coordinate will be 0, 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":"11:45.305","Text":"let\u0027s call it v of this line."},{"Start":"11:45.305 ","End":"11:49.540","Text":"Just take the coefficients of the t. The a,"},{"Start":"11:49.540 ","End":"11:52.310","Text":"b, and c here are the direction vectors."},{"Start":"11:52.310 ","End":"11:59.180","Text":"In our case, we take 3,"},{"Start":"11:59.510 ","End":"12:04.964","Text":"12, minus 1 will be a direction vector."},{"Start":"12:04.964 ","End":"12:09.855","Text":"I also know a position vector because I have a point on the line."},{"Start":"12:09.855 ","End":"12:20.604","Text":"I can take my r naught to be 0 minus 3, 8."},{"Start":"12:20.604 ","End":"12:30.310","Text":"I already have the parametric form of the line where r or let\u0027s write it as x, y,"},{"Start":"12:30.310 ","End":"12:34.035","Text":"z. I know that x, y,"},{"Start":"12:34.035 ","End":"12:43.060","Text":"z is equal to 0 minus 3,"},{"Start":"12:43.060 ","End":"12:51.040","Text":"8 plus t times"},{"Start":"12:51.060 ","End":"12:58.970","Text":"the same direction vector 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 one row."},{"Start":"13:09.885 ","End":"13:18.320","Text":"X equals is 0 plus 3t,"},{"Start":"13:18.320 ","End":"13:23.860","Text":"y equals minus 3 plus 12t."},{"Start":"13:26.250 ","End":"13:30.490","Text":"Z equals 8 minus"},{"Start":"13:30.490 ","End":"13:38.850","Text":"t. Now I have the parametric equation and the vector equation of this line."},{"Start":"13:38.850 ","End":"13:40.489","Text":"Now what about the question?"},{"Start":"13:40.489 ","End":"13:42.725","Text":"Does it cross the x z plane?"},{"Start":"13:42.725 ","End":"13:45.185","Text":"Or what\u0027s special about the x z plane."},{"Start":"13:45.185 ","End":"13:46.600","Text":"In the x z plane,"},{"Start":"13:46.600 ","End":"13:50.375","Text":"we know that y equals 0."},{"Start":"13:50.375 ","End":"13:53.895","Text":"That\u0027s the equation of the x-z plane, we\u0027ve seen it before."},{"Start":"13:53.895 ","End":"13:57.560","Text":"The other letter y that\u0027s missing is 0 there."},{"Start":"13:57.560 ","End":"14:05.550","Text":"All I have to do is compare this to this and I get an equation"},{"Start":"14:05.550 ","End":"14:13.950","Text":"that minus 3 plus 12t equals 0."},{"Start":"14:13.950 ","End":"14:15.335","Text":"If I solve that,"},{"Start":"14:15.335 ","End":"14:18.020","Text":"I get 12t equals 3."},{"Start":"14:18.020 ","End":"14:20.835","Text":"T equals 3 over 12."},{"Start":"14:20.835 ","End":"14:23.430","Text":"3 over 12 is a quarter."},{"Start":"14:23.430 ","End":"14:29.365","Text":"That\u0027s just the t. That answers the question, the first one."},{"Start":"14:29.365 ","End":"14:31.925","Text":"Yes, there is a solution."},{"Start":"14:31.925 ","End":"14:36.825","Text":"We have to put t equals 1/4 into here."},{"Start":"14:36.825 ","End":"14:43.750","Text":"The where would be this plus 1/4."},{"Start":"14:43.750 ","End":"14:46.325","Text":"Well, I can do it here. Put 1/4 in here."},{"Start":"14:46.325 ","End":"14:52.740","Text":"X is going to be equal to 3/4 if t is a 1/4,"},{"Start":"14:52.740 ","End":"14:55.085","Text":"y will equal 0."},{"Start":"14:55.085 ","End":"14:59.610","Text":"Of course, I mean that\u0027s the whole point of passing through the x-z plane."},{"Start":"14:59.610 ","End":"15:05.670","Text":"Z will equal 8 minus a 1/4,"},{"Start":"15:05.670 ","End":"15:10.485","Text":"so that would be 7 and 3/4."},{"Start":"15:10.485 ","End":"15:13.030","Text":"That\u0027s the answer to the where."},{"Start":"15:13.030 ","End":"15:20.170","Text":"This exercise is solved and were done with lines in 3D."}],"ID":10664},{"Watched":false,"Name":"Exercise 1","Duration":"4m 50s","ChapterTopicVideoID":10323,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"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":"00:11.565","Text":"We want to all 3 forms of the line that we\u0027ve learned,"},{"Start":"00:11.565 ","End":"00:16.240","Text":"the vector form, 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":"00:23.675","Text":"In general, the vector form of a line would be"},{"Start":"00:23.675 ","End":"00:28.580","Text":"that the general point r on the line is equal"},{"Start":"00:28.580 ","End":"00:33.020","Text":"to some position vector of a point on"},{"Start":"00:33.020 ","End":"00:39.680","Text":"the line plus parameter t times the direction vector of the line,"},{"Start":"00:39.680 ","End":"00:43.160","Text":"meaning a vector which is parallel to the line."},{"Start":"00:43.160 ","End":"00:48.290","Text":"Now, we can easily get both r naught and v,"},{"Start":"00:48.290 ","End":"00:54.920","Text":"because I could take r naught to be the position vector of either one of these,"},{"Start":"00:54.920 ","End":"00:56.960","Text":"and this one\u0027s the first one."},{"Start":"00:56.960 ","End":"00:58.505","Text":"I\u0027ll take this one."},{"Start":"00:58.505 ","End":"01:03.760","Text":"So r naught I can take as minus 10, 4, 0."},{"Start":"01:03.760 ","End":"01:06.290","Text":"We\u0027re using bracket notation for vectors,"},{"Start":"01:06.290 ","End":"01:08.960","Text":"not the i, j, k here."},{"Start":"01:08.960 ","End":"01:11.345","Text":"As the vector v,"},{"Start":"01:11.345 ","End":"01:14.480","Text":"is any number of vectors parallel to the line,"},{"Start":"01:14.480 ","End":"01:19.970","Text":"but the easiest is to take the vector from here to here."},{"Start":"01:19.970 ","End":"01:22.745","Text":"It\u0027s like the displacement vector."},{"Start":"01:22.745 ","End":"01:25.040","Text":"It would be a direction vector."},{"Start":"01:25.040 ","End":"01:31.830","Text":"V, I could take and subtract the second minus the first."},{"Start":"01:33.370 ","End":"01:37.620","Text":"I\u0027ll just write it that way as 1 minus minus 10,"},{"Start":"01:37.620 ","End":"01:41.960","Text":"and then afterwards we\u0027ll do the computation minus 4,"},{"Start":"01:41.960 ","End":"01:48.250","Text":"minus 4, and 2 minus 0."},{"Start":"01:48.250 ","End":"01:55.910","Text":"This comes out to be 1 minus minus 10 is 11."},{"Start":"01:55.910 ","End":"01:58.850","Text":"This comes out minus 8,"},{"Start":"01:58.850 ","End":"02:00.980","Text":"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":"02:22.160","Text":"4, 0 plus t times 11,"},{"Start":"02:22.160 ","End":"02:26.645","Text":"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":"02:36.520","Text":"From the vector it\u0027s easy to get to the parametric because"},{"Start":"02:36.520 ","End":"02:41.445","Text":"basically you just say that r is x, y, z."},{"Start":"02:41.445 ","End":"02:50.135","Text":"The parametric we can get from here component-wise by saying x equals y equals z equals."},{"Start":"02:50.135 ","End":"02:52.975","Text":"Then we take first component,"},{"Start":"02:52.975 ","End":"02:58.720","Text":"which would be minus 10 plus 11t."},{"Start":"02:58.720 ","End":"03:01.430","Text":"In fact, you know what?"},{"Start":"03:02.280 ","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":"03:21.020 ","End":"03:25.095","Text":"and then 0 plus 2t,"},{"Start":"03:25.095 ","End":"03:27.225","Text":"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 and z equals 2t."},{"Start":"03:44.150 ","End":"03:51.055","Text":"This is the parametric form of the line."},{"Start":"03:51.055 ","End":"03:52.840","Text":"T is the parameter."},{"Start":"03:52.840 ","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":"04:12.755","Text":"I can say from the first one that t is x plus 10 over 11."},{"Start":"04:12.755 ","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":"04:24.030 ","End":"04:32.725","Text":"T from the last one is just z over 2."},{"Start":"04:32.725 ","End":"04:36.530","Text":"Then I don\u0027t need this."},{"Start":"04:36.800 ","End":"04:41.420","Text":"This would be the symmetric equation of"},{"Start":"04:41.420 ","End":"04:49.430","Text":"the line. We\u0027re done."}],"ID":10665},{"Watched":false,"Name":"Exercise 2","Duration":"4m 13s","ChapterTopicVideoID":10324,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"In this exercise, we have a line or a given point"},{"Start":"00:05.970 ","End":"00:11.940","Text":"on the line 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":"00:22.460","Text":"Easiest to start with the vector form of the line."},{"Start":"00:22.460 ","End":"00:24.740","Text":"Now, for our line,"},{"Start":"00:24.740 ","End":"00:28.040","Text":"what we need is a point on the line and a direction vector."},{"Start":"00:28.040 ","End":"00:33.330","Text":"Because then we can say that the general point r"},{"Start":"00:33.330 ","End":"00:38.525","Text":"on the line is the specific given point on the line,"},{"Start":"00:38.525 ","End":"00:44.335","Text":"plus parameter t times a direction vector of the line."},{"Start":"00:44.335 ","End":"00:48.725","Text":"Now, for r naught, we are 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 brackets just 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":"01:11.890","Text":"Direction vector means a vector parallel to the line."},{"Start":"01:11.890 ","End":"01:15.310","Text":"Now, 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":"01:27.010","Text":"If A is parallel to B and B is parallel to C then A is parallel to C."},{"Start":"01:27.010 ","End":"01:32.410","Text":"Now we know that when we have an equation like this in the parametric form,"},{"Start":"01:32.410 ","End":"01:37.045","Text":"that the coefficients of the t are a direction vector of the line."},{"Start":"01:37.045 ","End":"01:38.695","Text":"I\u0027m just looking here,"},{"Start":"01:38.695 ","End":"01:41.215","Text":"here, and here,"},{"Start":"01:41.215 ","End":"01:49.830","Text":"and I\u0027ve got my v to be 4, 3, minus 5."},{"Start":"01:49.830 ","End":"01:51.770","Text":"Now, if I plug-in here,"},{"Start":"01:51.770 ","End":"01:57.245","Text":"I\u0027ve got my vector equation of the line that r is equal to,"},{"Start":"01:57.245 ","End":"02:00.140","Text":"copying from here, minus 10,"},{"Start":"02:00.140 ","End":"02:08.030","Text":"4, 0, plus t times the direction vector of this line hands of our line,"},{"Start":"02:08.030 ","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 4_t,"},{"Start":"02:21.770 ","End":"02:26.140","Text":"and the next component, 4 plus 3_t."},{"Start":"02:27.710 ","End":"02:29.880","Text":"What\u0027s the next 1?"},{"Start":"02:29.880 ","End":"02:34.150","Text":"0 minus 5_t, so just minus 5_t."},{"Start":"02:34.790 ","End":"02:39.220","Text":"That\u0027s the vector form."},{"Start":"02:39.220 ","End":"02:41.705","Text":"Now for the parametric,"},{"Start":"02:41.705 ","End":"02:43.880","Text":"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":"02:50.370","Text":"sometimes when I have room I like to do it with a curly brace,"},{"Start":"02:50.540 ","End":"02:52.680","Text":"and 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":"02:58.930","Text":"minus 10 plus 4_t,"},{"Start":"02:58.930 ","End":"03:03.685","Text":"4 plus 3_t, minus 5_t."},{"Start":"03:03.685 ","End":"03:07.050","Text":"That\u0027s the parametric."},{"Start":"03:07.050 ","End":"03:09.060","Text":"By the symmetric,"},{"Start":"03:09.060 ","End":"03:14.340","Text":"what we do is we isolate t from each of these."},{"Start":"03:14.340 ","End":"03:23.235","Text":"From here we would have t equals x plus 10 over 4."},{"Start":"03:23.235 ","End":"03:31.995","Text":"On this 1, we would get that t equals y minus 4 over 3,"},{"Start":"03:31.995 ","End":"03:39.540","Text":"and from the last 1 we would get that t equals z over minus 5."},{"Start":"03:39.540 ","End":"03:41.665","Text":"Then if you compare the t\u0027s,"},{"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":"03:55.440","Text":"Then that\u0027s going to equal the t from here,"},{"Start":"03:55.440 ","End":"03:59.940","Text":"which is y minus 4 over 3,"},{"Start":"03:59.940 ","End":"04:02.310","Text":"and this is going to equal,"},{"Start":"04:02.310 ","End":"04:07.335","Text":"t from here is just z over minus 5."},{"Start":"04:07.335 ","End":"04:12.970","Text":"That gives us the symmetric and that\u0027s it."}],"ID":10666},{"Watched":false,"Name":"Exercise 3","Duration":"6m 49s","ChapterTopicVideoID":10325,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we have 2 lines, l_1 and l_2."},{"Start":"00:04.950 ","End":"00:09.360","Text":"L_1 is given by 2 points that it goes through and"},{"Start":"00:09.360 ","End":"00:14.805","Text":"l_2 is given in the form of the vector equation of the line."},{"Start":"00:14.805 ","End":"00:22.240","Text":"We want to know whether these 2 lines are parallel or perpendicular or neither."},{"Start":"00:22.730 ","End":"00:27.510","Text":"The most useful thing we can do for either the parallel or the"},{"Start":"00:27.510 ","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":"00:44.160","Text":"a direction vector cause there\u0027s infinitely number."},{"Start":"00:44.160 ","End":"00:48.980","Text":"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":"00:59.795","Text":"the displacement vector that would take me from this point to this point."},{"Start":"00:59.795 ","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":"1 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":"01:32.955","Text":"Now, a direction vector for line 2,"},{"Start":"01:32.955 ","End":"01:35.065","Text":"call it V_2,"},{"Start":"01:35.065 ","End":"01:39.170","Text":"can easily be seen from the vector form by taking"},{"Start":"01:39.170 ","End":"01:43.830","Text":"the coefficients of the t in each of the components."},{"Start":"01:44.510 ","End":"01:49.080","Text":"We don\u0027t have a t here so the coefficient it\u0027s like 0,"},{"Start":"01:49.080 ","End":"01:52.560","Text":"it\u0027s like 5 plus 0t, so 0."},{"Start":"01:52.560 ","End":"01:58.245","Text":"Here minus 9, and here minus 4."},{"Start":"01:58.245 ","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":"02:30.540","Text":"How do I know if this one and this one are parallel?"},{"Start":"02:30.540 ","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":"02:47.785","Text":"One of them has to be a non-zero constant,"},{"Start":"02:47.785 ","End":"02:53.520","Text":"say V_2 has got to be some non-zero constant times V_1."},{"Start":"02:54.070 ","End":"02:57.080","Text":"The question is whether there is such"},{"Start":"02:57.080 ","End":"03:03.050","Text":"a K and I claim there\u0027s no such K. Whichever side you put the K on,"},{"Start":"03:03.050 ","End":"03:06.185","Text":"if you went for V_1 is K times V_2."},{"Start":"03:06.185 ","End":"03:08.510","Text":"The reason is,"},{"Start":"03:08.510 ","End":"03:11.850","Text":"because look, I have a 0 here."},{"Start":"03:11.990 ","End":"03:21.525","Text":"The only way V_2 could be K times V_1 is if 0 is K times minus 2."},{"Start":"03:21.525 ","End":"03:24.680","Text":"We\u0027d have to have, maybe I will write this down."},{"Start":"03:24.680 ","End":"03:26.330","Text":"Looking at the first component,"},{"Start":"03:26.330 ","End":"03:32.675","Text":"we get 0 is K times minus 2 and that would mean that K is 0."},{"Start":"03:32.675 ","End":"03:40.695","Text":"If K is 0, then V_2 would be the 0 vector, which it isn\u0027t."},{"Start":"03:40.695 ","End":"03:44.405","Text":"The answer is no, there is no such K,"},{"Start":"03:44.405 ","End":"03:50.730","Text":"doesn\u0027t exist so 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":"04:01.000","Text":"then you\u0027d have minus 2 is K times 0,"},{"Start":"04:01.000 ","End":"04:03.315","Text":"not going to work either."},{"Start":"04:03.315 ","End":"04:07.290","Text":"Either way it\u0027s no good, so not parallel."},{"Start":"04:07.290 ","End":"04:10.995","Text":"Next, let\u0027s go for perpendicular."},{"Start":"04:10.995 ","End":"04:14.850","Text":"Now l_1 and l_2,"},{"Start":"04:14.850 ","End":"04:17.930","Text":"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":"04:35.115","Text":"They intersect and they have to intersect at 90 degrees."},{"Start":"04:35.115 ","End":"04:38.550","Text":"Working in radians, it\u0027s Pi over 2."},{"Start":"04:38.550 ","End":"04:43.070","Text":"90 degrees to each other means the dot product is 0."},{"Start":"04:43.070 ","End":"04:46.885","Text":"Let\u0027s see, let\u0027s try the second condition."},{"Start":"04:46.885 ","End":"04:50.000","Text":"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\u0027s 90 degrees,"},{"Start":"04:55.069 ","End":"05:04.340","Text":"that means they\u0027re"},{"Start":"05:04.340 ","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":"05:23.990","Text":"In short, we have to check that V_ dot V_2,"},{"Start":"05:23.990 ","End":"05:31.285","Text":"is it equal to 0 because that\u0027s the condition for perpendicularity with vectors."},{"Start":"05:31.285 ","End":"05:34.145","Text":"If these are parallel to these,"},{"Start":"05:34.145 ","End":"05:39.995","Text":"and these are at 90 degrees and these are at 90 degrees so 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":"05:54.210","Text":"14 dot-product with 0 minus 9 minus 4."},{"Start":"05:54.210 ","End":"05:57.700","Text":"This is equal to this times this is 0,"},{"Start":"05:57.700 ","End":"06:00.265","Text":"this times this is 9."},{"Start":"06:00.265 ","End":"06:11.520","Text":"9 minus 56, which was not 0 at any rate."},{"Start":"06:11.520 ","End":"06:13.125","Text":"What is it equal to?"},{"Start":"06:13.125 ","End":"06:17.445","Text":"Minus 47, I believe."},{"Start":"06:17.445 ","End":"06:20.355","Text":"Anyway, it\u0027s not 0."},{"Start":"06:20.355 ","End":"06:25.255","Text":"Not perpendicular, so 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":"06:28.915","Text":"then it wouldn\u0027t be enough."},{"Start":"06:28.915 ","End":"06:30.970","Text":"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":"06:36.555","Text":"so they\u0027re not 90 degrees, so we\u0027re done,"},{"Start":"06:36.555 ","End":"06:41.180","Text":"not parallel, not perpendicular."},{"Start":"06:41.180 ","End":"06:49.450","Text":"Abbreviate that to PERP and so the answer is neither, and we\u0027re done."}],"ID":10667},{"Watched":false,"Name":"Exercise 4","Duration":"7m 9s","ChapterTopicVideoID":10326,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"In this exercise, we\u0027re given 2 lines, slightly different forms."},{"Start":"00:05.490 ","End":"00:12.120","Text":"L_1 is given in parametric form and l_2 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":"00:18.770 ","End":"00:25.610","Text":"L_1 could be written as x equals y equals z equals,"},{"Start":"00:25.610 ","End":"00:27.655","Text":"I\u0027ll fill it in a moment."},{"Start":"00:27.655 ","End":"00:34.865","Text":"L_2, we can also write x equals y equals z equals."},{"Start":"00:34.865 ","End":"00:37.265","Text":"In the first one just copying,"},{"Start":"00:37.265 ","End":"00:41.455","Text":"minus 7 plus 12t from here,"},{"Start":"00:41.455 ","End":"00:47.160","Text":"3 minus t, and then 14 plus 8t."},{"Start":"00:47.160 ","End":"00:50.400","Text":"From the second one, just component-wise,"},{"Start":"00:50.400 ","End":"00:57.520","Text":"8 plus t, 5 plus 6t, 4 minus 2t."},{"Start":"00:58.460 ","End":"01:01.270","Text":"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":"01:12.530","Text":"z, and a different value of t possibly here that will give us the same x, y, z."},{"Start":"01:12.530 ","End":"01:18.195","Text":"The point is that it doesn\u0027t have to be the same t and it usually will not be."},{"Start":"01:18.195 ","End":"01:21.300","Text":"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":"01:28.050","Text":"U, and I plug in here,"},{"Start":"01:28.050 ","End":"01:29.900","Text":"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":"01:56.975","Text":"Plug-in u here, I\u0027ve got 3 minus u and plug in u here, 14 plus 8u."},{"Start":"01:56.975 ","End":"01:58.800","Text":"Here I\u0027ll put in another value,"},{"Start":"01:58.800 ","End":"02:02.010","Text":"v, so 8 plus v,"},{"Start":"02:02.010 ","End":"02:08.710","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 in 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":"02:47.830","Text":"what I suggest is to choose 2 of these 3 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":"03:01.265","Text":"Solve for u and v and see if the u and v that I found fit the third equation also."},{"Start":"03:01.265 ","End":"03:06.120","Text":"Then we\u0027ll say yes, we have a solution or no we don\u0027t."},{"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":"03:29.900","Text":"Let me just see if there\u0027s anything easy to do."},{"Start":"03:29.900 ","End":"03:32.360","Text":"I think I\u0027ll isolate v from"},{"Start":"03:32.360 ","End":"03:37.970","Text":"the first equation and get that v equals bring the 8 to the other side,"},{"Start":"03:37.970 ","End":"03:40.770","Text":"so I have 12u minus 7,"},{"Start":"03:40.770 ","End":"03:43.955","Text":"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":"04:14.910","Text":"3 minus u equals 5 plus 6 times 12 is 72u, minus 90."},{"Start":"04:14.910 ","End":"04:18.875","Text":"I\u0027ll put the u\u0027s on the right."},{"Start":"04:18.875 ","End":"04:20.900","Text":"But then I\u0027ll switch sides."},{"Start":"04:20.900 ","End":"04:27.950","Text":"I\u0027ll get here, 73u and the numbers on the left,"},{"Start":"04:27.950 ","End":"04:30.320","Text":"which will then go on the right,"},{"Start":"04:30.320 ","End":"04:40.480","Text":"3 plus 90 is 93 minus 5 is 88."},{"Start":"04:40.480 ","End":"04:51.490","Text":"We get that u is equal to 88 over 73."},{"Start":"04:51.490 ","End":"04:58.535","Text":"Numbers often come out messy and I cooked it up so it come out nice."},{"Start":"04:58.535 ","End":"05:01.250","Text":"Yeah, 88 over 73."},{"Start":"05:01.250 ","End":"05:03.020","Text":"Now let\u0027s continue."},{"Start":"05:03.020 ","End":"05:04.895","Text":"We have u,"},{"Start":"05:04.895 ","End":"05:13.980","Text":"so we can plug u in this value into here and then we\u0027ll get that"},{"Start":"05:13.980 ","End":"05:19.425","Text":"v equals 12 times"},{"Start":"05:19.425 ","End":"05:26.235","Text":"88 over 73 minus 15."},{"Start":"05:26.235 ","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":"05:42.345","Text":"I have now u and I have v. What we have to do,"},{"Start":"05:42.345 ","End":"05:46.080","Text":"let me just highlight what\u0027s the important,"},{"Start":"05:46.080 ","End":"05:51.675","Text":"that\u0027s u and v is this,"},{"Start":"05:51.675 ","End":"05:59.670","Text":"and 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":"06:14.750","Text":"Now we\u0027re verifying,"},{"Start":"06:14.750 ","End":"06:17.645","Text":"we\u0027re checking, 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 1,726 over 73"},{"Start":"06:37.010 ","End":"06:44.880","Text":"and this comes out to be 370 over 73."},{"Start":"06:44.880 ","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":"07:00.220","Text":"Do l and l_2 intersect?"},{"Start":"07:00.220 ","End":"07:02.845","Text":"The answer is no."},{"Start":"07:02.845 ","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":10668},{"Watched":false,"Name":"Exercise 5","Duration":"8m 38s","ChapterTopicVideoID":10319,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"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":"00:23.310","Text":"t. I\u0027d prefer to have l_1 in this form also,"},{"Start":"00:23.310 ","End":"00:28.020","Text":"and then we can answer the question 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":"00:36.810","Text":"What I can do for l_1 would be to write it as the usual formula."},{"Start":"00:36.810 ","End":"00:39.225","Text":"We take 1 of the points,"},{"Start":"00:39.225 ","End":"00:40.400","Text":"say the first 1,"},{"Start":"00:40.400 ","End":"00:44.180","Text":"its position vector is minus 5, 0,"},{"Start":"00:44.180 ","End":"00:48.590","Text":"2 plus t times direction vector,"},{"Start":"00:48.590 ","End":"00:52.985","Text":"which I can get by subtracting 1 from the other set of second minus the first"},{"Start":"00:52.985 ","End":"00:58.065","Text":"13 less minus 5 is 18 minus 2,"},{"Start":"00:58.065 ","End":"01:01.180","Text":"1 minus 2, minus 1."},{"Start":"01:01.460 ","End":"01:08.120","Text":"If I put it in parametric form, it\u0027s more convenient,"},{"Start":"01:08.120 ","End":"01:17.180","Text":"I can say x equals y equals z equals component-wise minus 5 plus 18t"},{"Start":"01:17.180 ","End":"01:27.390","Text":"and then 0 minus 2t and 2 minus t. That\u0027s l_1."},{"Start":"01:27.390 ","End":"01:34.574","Text":"Now l_2, from here,"},{"Start":"01:34.574 ","End":"01:36.870","Text":"just write it in x,"},{"Start":"01:36.870 ","End":"01:39.130","Text":"y, z form."},{"Start":"01:39.130 ","End":"01:44.180","Text":"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":"02:05.795","Text":"Now, if they intersect the sum value of t here and sum value of t here,"},{"Start":"02:05.795 ","End":"02:08.630","Text":"that will give us the same x, y, z,"},{"Start":"02:08.630 ","End":"02:11.590","Text":"but they\u0027re going to be different values of t."},{"Start":"02:11.590 ","End":"02:14.960","Text":"The t for here and the t for here may not be the same."},{"Start":"02:14.960 ","End":"02:18.380","Text":"So let me use 2 other letters say,"},{"Start":"02:18.380 ","End":"02:25.220","Text":"u and v. If I let t equals u here and t equals v here,"},{"Start":"02:25.220 ","End":"02:29.000","Text":"what I\u0027m looking for,"},{"Start":"02:29.000 ","End":"02:31.395","Text":"get the same point."},{"Start":"02:31.395 ","End":"02:35.045","Text":"If I do that, I\u0027ll get the following equations."},{"Start":"02:35.045 ","End":"02:37.190","Text":"If plugin u here,"},{"Start":"02:37.190 ","End":"02:41.465","Text":"I\u0027ve got minus 5 plus 18u,"},{"Start":"02:41.465 ","End":"02:44.014","Text":"and if I plug in v here,"},{"Start":"02:44.014 ","End":"02:48.365","Text":"well, I think the plug-in with x that will equal 3."},{"Start":"02:48.365 ","End":"02:54.810","Text":"Here, I\u0027ll get minus 2u equals,"},{"Start":"02:54.810 ","End":"03:00.785","Text":"and here minus 1 minus v. The last 1 for z,"},{"Start":"03:00.785 ","End":"03:13.200","Text":"I\u0027ll get 2 plus 4v."},{"Start":"03:13.200 ","End":"03:14.520","Text":"Equation for x, for y,"},{"Start":"03:14.520 ","End":"03:18.165","Text":"and for z and I don\u0027t know if it has a solution,"},{"Start":"03:18.165 ","End":"03:26.215","Text":"because very often if you have more equations than unknowns,"},{"Start":"03:26.215 ","End":"03:28.975","Text":"you\u0027re more likely not to have a solution."},{"Start":"03:28.975 ","End":"03:31.435","Text":"But we don\u0027t know there could be a solution."},{"Start":"03:31.435 ","End":"03:34.950","Text":"What we\u0027re going to do is, well,"},{"Start":"03:34.950 ","End":"03:38.440","Text":"1 way is to take just 2 of these 3 equations."},{"Start":"03:38.440 ","End":"03:40.585","Text":"Let\u0027s say take the first 2,"},{"Start":"03:40.585 ","End":"03:43.090","Text":"then I have 2 equations and 2 unknowns,"},{"Start":"03:43.090 ","End":"03:49.055","Text":"solve it for u and v and then check if it satisfies the third 1,"},{"Start":"03:49.055 ","End":"03:51.100","Text":"the values of u and v that we get."},{"Start":"03:51.100 ","End":"03:53.600","Text":"I\u0027ll write these 2 over here."},{"Start":"03:53.600 ","End":"04:00.810","Text":"I\u0027ve got just copying minus 5 plus 18u equals 3 and"},{"Start":"04:00.810 ","End":"04:08.450","Text":"minus 2u equals minus 1 minus v. Looking at this,"},{"Start":"04:08.450 ","End":"04:12.740","Text":"what I think we could do would be to extract v from the second 1,"},{"Start":"04:12.740 ","End":"04:16.920","Text":"and then substitute in the first."},{"Start":"04:16.920 ","End":"04:21.660","Text":"Actually, there is no V in the first."},{"Start":"04:21.660 ","End":"04:23.580","Text":"So on second thoughts,"},{"Start":"04:23.580 ","End":"04:27.410","Text":"we\u0027ll just solve the first 1 for u because v is missing from it."},{"Start":"04:27.410 ","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":"04:50.780","Text":"Then I\u0027ll plug the value of u from here into here."},{"Start":"04:50.780 ","End":"04:53.825","Text":"I\u0027ll get minus 2u,"},{"Start":"04:53.825 ","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":"and 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. V will be minus 1/9."},{"Start":"05:25.940 ","End":"05:31.969","Text":"We found u and v for the first 2 equations,"},{"Start":"05:31.969 ","End":"05:35.170","Text":"and now we need to check with the last equation."},{"Start":"05:35.170 ","End":"05:39.070","Text":"I\u0027ll take this u and put it into here."},{"Start":"05:39.070 ","End":"05:41.970","Text":"I\u0027ll take v from here,"},{"Start":"05:41.970 ","End":"05:43.965","Text":"and put it into here,"},{"Start":"05:43.965 ","End":"05:46.035","Text":"and let\u0027s see what we get."},{"Start":"05:46.035 ","End":"05:54.950","Text":"2 minus u is 4/9 equals but question mark that\u0027s what we\u0027re going to check."},{"Start":"05:54.950 ","End":"05:57.110","Text":"We know the first 2 are satisfied."},{"Start":"05:57.110 ","End":"06:06.480","Text":"We don\u0027t know about the third equals 2 plus 4 and v is minus 1/9."},{"Start":"06:06.980 ","End":"06:13.615","Text":"Yes, they are equal because here it\u0027s minus 4/9 and here it\u0027s minus 4/9."},{"Start":"06:13.615 ","End":"06:19.885","Text":"I mean, the answer would be 1 and 5/9."},{"Start":"06:19.885 ","End":"06:23.200","Text":"We\u0027re both sides, but we can already see that they\u0027re equal."},{"Start":"06:23.200 ","End":"06:26.570","Text":"Yes, it is equal."},{"Start":"06:26.730 ","End":"06:32.240","Text":"They intersect. 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 maybe 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 a plugin to l_2,"},{"Start":"06:57.025 ","End":"07:00.995","Text":"what I need is to let t equal the v,"},{"Start":"07:00.995 ","End":"07:07.200","Text":"which is minus 1/9."},{"Start":"07:07.200 ","End":"07:11.055","Text":"Let me just copy this down here."},{"Start":"07:11.055 ","End":"07:17.900","Text":"There\u0027s a copy-paste and now I just have to plug into all these,"},{"Start":"07:19.340 ","End":"07:25.540","Text":"t equals minus 1/9 and"},{"Start":"07:25.540 ","End":"07:31.605","Text":"so we\u0027ll get x equals 3 because nothing to substitute."},{"Start":"07:31.605 ","End":"07:40.190","Text":"Y equals minus 1 minus t is minus 1 plus 1/9 is minus"},{"Start":"07:40.190 ","End":"07:50.750","Text":"8/9 and z equals 2 minus 4/9,"},{"Start":"07:50.750 ","End":"08:00.615","Text":"which will be 1 and 5/9."},{"Start":"08:00.615 ","End":"08:04.790","Text":"That will give us the answer that the intersection point is"},{"Start":"08:04.790 ","End":"08:12.000","Text":"3 minus 8/9, 1 5/9."},{"Start":"08:13.460 ","End":"08:16.115","Text":"They\u0027ll highlight that."},{"Start":"08:16.115 ","End":"08:18.950","Text":"I\u0027ll just mention that of course,"},{"Start":"08:18.950 ","End":"08:21.350","Text":"you could have, instead of using l_2,"},{"Start":"08:21.350 ","End":"08:23.270","Text":"we could\u0027ve used l_1."},{"Start":"08:23.270 ","End":"08:29.375","Text":"If you substitute t equals 4/9 into here,"},{"Start":"08:29.375 ","End":"08:34.220","Text":"you should get the 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":10669},{"Watched":false,"Name":"Exercise 6","Duration":"3m 15s","ChapterTopicVideoID":10320,"CourseChapterTopicPlaylistID":12293,"HasSubtitles":true,"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":"00:28.860","Text":"Now, the xy plane is characterized by the equation z equals 0."},{"Start":"00:28.860 ","End":"00:32.370","Text":"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 equals 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":"01:06.140","Text":"the line will intersect the xy plane and I even know the z coordinate."},{"Start":"01:06.140 ","End":"01:11.015","Text":"But all I have to do since we want to know where,"},{"Start":"01:11.015 ","End":"01:15.420","Text":"I just need to complete the x and the y part."},{"Start":"01:17.360 ","End":"01:24.260","Text":"The x of the point of intersection will"},{"Start":"01:24.260 ","End":"01:32.460","Text":"be minus 7 plus 12 and then t is minus 2."},{"Start":"01:33.410 ","End":"01:38.630","Text":"Then y, there\u0027s no t in it,"},{"Start":"01:38.630 ","End":"01:42.055","Text":"will be 3 and z will equal,"},{"Start":"01:42.055 ","End":"01:45.870","Text":"we already know that that\u0027s got to be zero."},{"Start":"01:45.870 ","End":"01:48.000","Text":"We\u0027ve got already 2 out of 3,"},{"Start":"01:48.000 ","End":"01:50.160","Text":"all I need is the x now."},{"Start":"01:50.160 ","End":"01:55.860","Text":"Minus 7, minus 24."},{"Start":"01:55.860 ","End":"02:02.250","Text":"This is minus 31."},{"Start":"02:02.250 ","End":"02:07.845","Text":"The intersection will be minus 31,"},{"Start":"02:07.845 ","End":"02:14.400","Text":"3, 0 and that\u0027s part A."},{"Start":"02:14.400 ","End":"02:17.115","Text":"Now for part B,"},{"Start":"02:17.115 ","End":"02:19.215","Text":"so in my b here,"},{"Start":"02:19.215 ","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":"02:27.245","Text":"Now we have the xz plane,"},{"Start":"02:27.245 ","End":"02:30.845","Text":"which is characterized by y equals 0."},{"Start":"02:30.845 ","End":"02:39.235","Text":"We have to find the value of t that will make the y here 0 but y is 3."},{"Start":"02:39.235 ","End":"02:42.240","Text":"It doesn\u0027t even depend on t,"},{"Start":"02:42.240 ","End":"02:43.755","Text":"it doesn\u0027t matter what t is,"},{"Start":"02:43.755 ","End":"02:44.970","Text":"y is 3,"},{"Start":"02:44.970 ","End":"02:51.600","Text":"and 3 will not equal 0 ever regardless of what t is."},{"Start":"02:51.600 ","End":"02:56.625","Text":"The answer is no."},{"Start":"02:56.625 ","End":"03:00.825","Text":"The l does not intersect."},{"Start":"03:00.825 ","End":"03:05.160","Text":"There is no intersection,"},{"Start":"03:05.160 ","End":"03:10.595","Text":"if there is no intersection we can\u0027t find where it is because there isn\u0027t one."},{"Start":"03:10.595 ","End":"03:14.970","Text":"That\u0027s all there is to it. We\u0027re done."}],"ID":10670}],"Thumbnail":null,"ID":12293},{"Name":"Equations of Planes","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Equations of Planes","Duration":"19m 37s","ChapterTopicVideoID":10327,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:02.849","Text":"This is what we just finished,"},{"Start":"00:02.849 ","End":"00:04.830","Text":"3D equations of lines,"},{"Start":"00:04.830 ","End":"00:07.155","Text":"so 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":"00:21.740 ","End":"00:29.189","Text":"that was 1, and we needed a direction vector for the line."},{"Start":"00:29.950 ","End":"00:36.965","Text":"Because the line is defined mainly, or 1 of the most significant things is its direction."},{"Start":"00:36.965 ","End":"00:39.920","Text":"Once we have the direction, and everything else will be parallel and"},{"Start":"00:39.920 ","End":"00:43.130","Text":"once we have the point on the line,"},{"Start":"00:43.130 ","End":"00:44.900","Text":"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":"00:56.020","Text":"We need a point on the plane."},{"Start":"00:56.020 ","End":"01:01.340","Text":"But the question is, what\u0027s going to take the place of the direction?"},{"Start":"01:01.340 ","End":"01:03.620","Text":"It\u0027s also going to be a vector."},{"Start":"01:03.620 ","End":"01:08.910","Text":"But a plane is actually determined by something called a normal,"},{"Start":"01:08.910 ","End":"01:14.720","Text":"and I\u0027ll get into that in a moment though we have learned it in the chapter on vectors."},{"Start":"01:14.720 ","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":"01:24.590","Text":"but here we have a direction for a line, 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":"01:38.495","Text":"Here we have a diagram."},{"Start":"01:38.495 ","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":"01:47.520","Text":"This is what we\u0027re given P_0 and well,"},{"Start":"01:47.520 ","End":"01:50.010","Text":"it could be x_0, y_0,"},{"Start":"01:50.010 ","End":"01:54.550","Text":"z_0, or ABC or whatever, and here\u0027s the normal vector."},{"Start":"01:54.550 ","End":"01:58.030","Text":"Normal vector means it\u0027s perpendicular to the whole plane."},{"Start":"01:58.030 ","End":"02:05.815","Text":"It\u0027s perpendicular to any vector in the plane when aligning the plane if you want."},{"Start":"02:05.815 ","End":"02:08.380","Text":"The fact that this is perpendicular to"},{"Start":"02:08.380 ","End":"02:13.760","Text":"any vector that\u0027s parallel to the plane is what gives us the equation."},{"Start":"02:13.760 ","End":"02:16.800","Text":"If we take any point P,"},{"Start":"02:16.800 ","End":"02:20.370","Text":"preferably not P_0 itself,"},{"Start":"02:20.370 ","End":"02:25.600","Text":"then P minus P_0 will be a perpendicular vector to n. On the other hand,"},{"Start":"02:25.600 ","End":"02:28.345","Text":"if this is vector r, and this is vector r_0,"},{"Start":"02:28.345 ","End":"02:30.715","Text":"then this vector P_0,"},{"Start":"02:30.715 ","End":"02:32.925","Text":"P will be r minus r_0."},{"Start":"02:32.925 ","End":"02:42.070","Text":"In short, the equation of the plane will be that this normal vector,"},{"Start":"02:42.070 ","End":"02:45.330","Text":"so I\u0027ll call the normal vector here n,"},{"Start":"02:45.330 ","End":"02:48.470","Text":"and I\u0027ll call the 1 for the line v. I\u0027ll just want to"},{"Start":"02:48.470 ","End":"02:52.660","Text":"contrast, and compare the 2 equations."},{"Start":"02:52.660 ","End":"02:58.995","Text":"We have the n dot product with r minus"},{"Start":"02:58.995 ","End":"03:05.580","Text":"r_0 is equal to 0, and is this vector r minus r_0,"},{"Start":"03:05.580 ","End":"03:09.260","Text":"is this vector perpendicular it means equal 0, because it works at"},{"Start":"03:09.260 ","End":"03:13.820","Text":"P_0 itself, because then this minus this is 0."},{"Start":"03:13.820 ","End":"03:19.000","Text":"Of course, it\u0027s still 0. Just compare this."},{"Start":"03:19.130 ","End":"03:27.485","Text":"I\u0027m just for comparison or write the equation of a line is parametric, and it\u0027s"},{"Start":"03:27.485 ","End":"03:36.380","Text":"r is equal to r_0 plus parameter times direction vector."},{"Start":"03:36.380 ","End":"03:43.290","Text":"Here\u0027s normal vector times r minus the r of a particular point equals 0."},{"Start":"03:43.290 ","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":"03:55.255","Text":"we get n dot r minus n. r_0, and we can bring it over to the other side."},{"Start":"03:55.255 ","End":"04:05.315","Text":"It becomes n. r equals n. r_0."},{"Start":"04:05.315 ","End":"04:07.600","Text":"Anyway, that\u0027s just a variation of"},{"Start":"04:07.600 ","End":"04:09.700","Text":"this, and this is the important thing we\u0027re looking for,"},{"Start":"04:09.700 ","End":"04:11.680","Text":"so I\u0027ll highlight it."},{"Start":"04:11.680 ","End":"04:14.904","Text":"Here we go, this is the formula."},{"Start":"04:14.904 ","End":"04:16.510","Text":"Now just like with a line,"},{"Start":"04:16.510 ","End":"04:18.480","Text":"we also had an x y z version,"},{"Start":"04:18.480 ","End":"04:21.960","Text":"so we\u0027ll have an x y z version for the plane also."},{"Start":"04:21.960 ","End":"04:27.090","Text":"R is the vector x y z,"},{"Start":"04:27.090 ","End":"04:30.340","Text":"r_0 is the vector x_0, y_0,"},{"Start":"04:30.340 ","End":"04:38.245","Text":"z_0, and I need also the n, and let\u0027s just assume that that\u0027s a, b,"},{"Start":"04:38.245 ","End":"04:43.590","Text":"c. What we get from this equation is that a,"},{"Start":"04:43.590 ","End":"04:47.740","Text":"b, c, which is the n dot with."},{"Start":"04:47.740 ","End":"04:52.905","Text":"Now r minus r_0 is going to be x minus x_0,"},{"Start":"04:52.905 ","End":"04:55.605","Text":"y minus y_0,"},{"Start":"04:55.605 ","End":"05:00.720","Text":"z minus z_0 is equal to"},{"Start":"05:00.720 ","End":"05:08.690","Text":"0, and this gives us the equation a times x minus x_0,"},{"Start":"05:08.690 ","End":"05:14.690","Text":"just by the definition of dot product plus b times y"},{"Start":"05:14.690 ","End":"05:23.190","Text":"minus y_0 plus c times z minus z_0 equals 0."},{"Start":"05:23.190 ","End":"05:28.670","Text":"This is the equivalent of the formula above,"},{"Start":"05:28.670 ","End":"05:31.370","Text":"just with x y z."},{"Start":"05:31.370 ","End":"05:34.400","Text":"There is another variation of this."},{"Start":"05:34.400 ","End":"05:37.220","Text":"If we bring all the constants to the other side,"},{"Start":"05:37.220 ","End":"05:40.745","Text":"like ax_0, by_0, cz_0."},{"Start":"05:40.745 ","End":"05:43.835","Text":"They\u0027re all with a minus, on the other side they become a plus,"},{"Start":"05:43.835 ","End":"05:46.220","Text":"and I put them all together."},{"Start":"05:46.220 ","End":"05:55.850","Text":"Then we get a variant that ax plus by plus cz equals d, where as I said,"},{"Start":"05:55.850 ","End":"06:05.210","Text":"d is equal to ax_0 plus by_0 plus cz_0, and it"},{"Start":"06:05.210 ","End":"06:09.090","Text":"happens not by coincidence to also equal, and"},{"Start":"06:09.090 ","End":"06:15.615","Text":"note the dot product of these 2 is ax_0 plus by_0, cz_0."},{"Start":"06:15.615 ","End":"06:20.465","Text":"Really this is just another variation of this,"},{"Start":"06:20.465 ","End":"06:26.610","Text":"but with this abbreviated to d. That\u0027s"},{"Start":"06:26.610 ","End":"06:35.764","Text":"a plane in 3D, and in a way it\u0027s an analogous to a line in 2D,"},{"Start":"06:35.764 ","End":"06:41.075","Text":"we had ax plus by equals c with a symmetric equation of a line."},{"Start":"06:41.075 ","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":"06:57.930","Text":"this 1 here is called the vector equation of the plane,"},{"Start":"06:57.930 ","End":"07:05.470","Text":"and this 1 here,"},{"Start":"07:05.470 ","End":"07:13.585","Text":"well I suppose you could refer to this 1 here is called the scalar equation of the plane."},{"Start":"07:13.585 ","End":"07:16.145","Text":"Just to give them names."},{"Start":"07:16.145 ","End":"07:18.090","Text":"I\u0027m about to do an example,"},{"Start":"07:18.090 ","End":"07:19.480","Text":"this 1 more comment."},{"Start":"07:19.480 ","End":"07:21.760","Text":"If you\u0027re given the plane in this form with something"},{"Start":"07:21.760 ","End":"07:24.265","Text":"x plus something y plus something z equals d,"},{"Start":"07:24.265 ","End":"07:26.110","Text":"then these 3 numbers,"},{"Start":"07:26.110 ","End":"07:28.945","Text":"the a, b, and c,"},{"Start":"07:28.945 ","End":"07:34.540","Text":"will actually be a normal to the plane."},{"Start":"07:34.540 ","End":"07:36.250","Text":"If we\u0027re given an equation in this form,"},{"Start":"07:36.250 ","End":"07:38.125","Text":"we can find the normal."},{"Start":"07:38.125 ","End":"07:43.580","Text":"I\u0027m going to start the examples on a fresh page, and I\u0027ll carry these formulas with me."},{"Start":"07:43.580 ","End":"07:50.120","Text":"Now it\u0027s time for a couple of examples, and I just kept whatever I needed from"},{"Start":"07:50.120 ","End":"07:52.835","Text":"the previous page that"},{"Start":"07:52.835 ","End":"07:58.000","Text":"basically the best way to find the equation of a plane is to find 2 things,"},{"Start":"07:58.000 ","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":"08:04.280","Text":"Then we can either have the vector equation, or"},{"Start":"08:04.280 ","End":"08:07.775","Text":"the scalar equation, and 1 of the examples I wanted"},{"Start":"08:07.775 ","End":"08:13.940","Text":"to bring is how to find the equation of a plane if we\u0027re given 3 points in the plane."},{"Start":"08:13.940 ","End":"08:15.740","Text":"Now it\u0027s well-known that in general,"},{"Start":"08:15.740 ","End":"08:19.265","Text":"3 points on the plane completely determine the plane."},{"Start":"08:19.265 ","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":"08:28.310","Text":"But if they\u0027re not on the same line, then any 3 points"},{"Start":"08:28.310 ","End":"08:31.205","Text":"determine the plane, and we\u0027ll take an example of this."},{"Start":"08:31.205 ","End":"08:33.144","Text":"I\u0027ll give you 3 points."},{"Start":"08:33.144 ","End":"08:35.040","Text":"I\u0027ll take point P,"},{"Start":"08:35.040 ","End":"08:37.050","Text":"which is 1, 1,"},{"Start":"08:37.050 ","End":"08:40.710","Text":"1, and I\u0027ll take point Q,"},{"Start":"08:40.710 ","End":"08:44.445","Text":"which is minus 1,"},{"Start":"08:44.445 ","End":"08:47.820","Text":"1, 0, and R,"},{"Start":"08:47.820 ","End":"08:50.130","Text":"which is 2, 0,"},{"Start":"08:50.130 ","End":"08:56.750","Text":"3, and we want to find the equation of the plane that goes through these 3 points."},{"Start":"08:56.750 ","End":"09:01.295","Text":"We\u0027ll find out soon enough if these points are co-linear, and then we won\u0027t get a plane."},{"Start":"09:01.295 ","End":"09:04.835","Text":"But meanwhile, we have the first part done to find a point."},{"Start":"09:04.835 ","End":"09:06.140","Text":"We have a choice of 3,"},{"Start":"09:06.140 ","End":"09:07.475","Text":"any 1 of these will do,"},{"Start":"09:07.475 ","End":"09:10.775","Text":"and 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":"09:15.990","Text":"Here\u0027s the way this thing works."},{"Start":"09:15.990 ","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":"09:23.020","Text":"Suppose I had a third point."},{"Start":"09:23.020 ","End":"09:28.115","Text":"Here\u0027s a third point in the plane, and it doesn\u0027t look like they\u0027re in the same line,"},{"Start":"09:28.115 ","End":"09:31.070","Text":"it doesn\u0027t matter what its name is."},{"Start":"09:31.070 ","End":"09:33.935","Text":"What I do is I join,"},{"Start":"09:33.935 ","End":"09:37.205","Text":"let\u0027s say P_0 to this point."},{"Start":"09:37.205 ","End":"09:43.594","Text":"Here we are, and now we have another vector that\u0027s in the plane."},{"Start":"09:43.594 ","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":"09:54.275","Text":"but you can think of it as in the plane, or it\u0027s parallel to the plane."},{"Start":"09:54.275 ","End":"09:57.500","Text":"Now the normal, since it\u0027s normal to the plane,"},{"Start":"09:57.500 ","End":"10:00.580","Text":"it will be normal to these 2 vectors."},{"Start":"10:00.580 ","End":"10:04.100","Text":"To find a normal, in other words,"},{"Start":"10:04.100 ","End":"10:07.565","Text":"to find something that\u0027s perpendicular, or orthogonal to both of these,"},{"Start":"10:07.565 ","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":"10:18.379","Text":"In general, if the cross-product turns out to be 0,"},{"Start":"10:18.379 ","End":"10:22.594","Text":"then it means that these 3 points were co-linear."},{"Start":"10:22.594 ","End":"10:24.665","Text":"We\u0027ll soon enough find out."},{"Start":"10:24.665 ","End":"10:27.250","Text":"Let\u0027s take that here."},{"Start":"10:27.250 ","End":"10:31.890","Text":"Maybe P_0 is the first 1, and P is what\u0027s here,"},{"Start":"10:31.890 ","End":"10:34.365","Text":"Q and this 1 is R, it don\u0027t matter."},{"Start":"10:34.365 ","End":"10:38.060","Text":"I take any 2 vectors in the plane that are not parallel,"},{"Start":"10:38.060 ","End":"10:43.890","Text":"so let\u0027s say I\u0027ll take PQ and PQ,"},{"Start":"10:43.890 ","End":"10:48.290","Text":"the vector is I take the coordinates of this minus this,"},{"Start":"10:48.290 ","End":"10:51.860","Text":"so I get minus 1 minus 1 is minus 2,"},{"Start":"10:51.860 ","End":"10:59.105","Text":"0 minus 1 and there\u0027s another 1 let\u0027s take PR, there\u0027s many possibilities."},{"Start":"10:59.105 ","End":"11:04.530","Text":"I just need the 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":"11:17.350","Text":"PR is from here, to here, to subtract 1 from each of these, and I\u0027ll get 1 minus 1, 2."},{"Start":"11:17.350 ","End":"11:19.955","Text":"That\u0027s like this, and this, and now to get a normal,"},{"Start":"11:19.955 ","End":"11:22.775","Text":"something orthogonal to both I take the cross-product."},{"Start":"11:22.775 ","End":"11:24.965","Text":"I\u0027ll do minus 2,"},{"Start":"11:24.965 ","End":"11:28.860","Text":"0, minus 1, cross."},{"Start":"11:28.860 ","End":"11:33.310","Text":"1 minus 1, 2."},{"Start":"11:33.310 ","End":"11:36.730","Text":"The answer comes out minus 1, 3, 2,"},{"Start":"11:36.730 ","End":"11:40.060","Text":"and I\u0027m not going to go into the computation, because you"},{"Start":"11:40.060 ","End":"11:43.540","Text":"can go to the section on vectors, and the cross-product."},{"Start":"11:43.540 ","End":"11:48.460","Text":"Also, there\u0027s many techniques I taught a way using a determinant and a way without it."},{"Start":"11:48.460 ","End":"11:51.160","Text":"1 is easier and you may not have learned the other."},{"Start":"11:51.160 ","End":"11:53.740","Text":"Well, just go refer to the chapter,"},{"Start":"11:53.740 ","End":"11:55.315","Text":"I\u0027m just quoting the answer."},{"Start":"11:55.315 ","End":"11:59.005","Text":"This, I\u0027m taking as my normal vector."},{"Start":"11:59.005 ","End":"12:05.170","Text":"For the point, I\u0027m going to choose this,"},{"Start":"12:05.170 ","End":"12:09.830","Text":"and for the normal vector, I\u0027ve got this."},{"Start":"12:12.630 ","End":"12:16.465","Text":"Let\u0027s get the scalar equation."},{"Start":"12:16.465 ","End":"12:20.485","Text":"Just for reference in the formula,"},{"Start":"12:20.485 ","End":"12:23.920","Text":"this would be the x naught,"},{"Start":"12:23.920 ","End":"12:26.950","Text":"y naught, z naught,"},{"Start":"12:26.950 ","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":"12:42.925","Text":"which is minus 1 times x minus 1 plus b times"},{"Start":"12:42.925 ","End":"12:53.040","Text":"y minus 1 plus 2 times z minus 1 equals 0."},{"Start":"12:53.040 ","End":"12:55.740","Text":"There\u0027s a variation on this, like I mentioned."},{"Start":"12:55.740 ","End":"12:57.420","Text":"If you just want to take the x, y, and z,"},{"Start":"12:57.420 ","End":"13:04.705","Text":"we got minus x plus 3y plus 2z."},{"Start":"13:04.705 ","End":"13:07.000","Text":"All the constants go on the other side,"},{"Start":"13:07.000 ","End":"13:13.840","Text":"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":"13:21.040","Text":"This is maybe a nicer way of getting the equation of the plane."},{"Start":"13:21.040 ","End":"13:24.490","Text":"If this was on an exam and you had time,"},{"Start":"13:24.490 ","End":"13:28.465","Text":"I would plug in these 3 points and see that they all work."},{"Start":"13:28.465 ","End":"13:31.585","Text":"Let\u0027s, for example, take the second 1."},{"Start":"13:31.585 ","End":"13:36.700","Text":"If I take x is minus 1 and y is 1 and z is 0,"},{"Start":"13:36.700 ","End":"13:44.245","Text":"this becomes plus 1 and this becomes plus 3, plus 0."},{"Start":"13:44.245 ","End":"13:46.315","Text":"1 plus 3 plus 0 is 4."},{"Start":"13:46.315 ","End":"13:48.910","Text":"Yes, q is on the line."},{"Start":"13:48.910 ","End":"13:50.845","Text":"I know that P is on the line."},{"Start":"13:50.845 ","End":"13:52.270","Text":"If I substitute it here,"},{"Start":"13:52.270 ","End":"13:54.040","Text":"x, y, and z are 1,"},{"Start":"13:54.040 ","End":"13:57.760","Text":"I can see 1 minus 1 minus 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":"14:05.200","Text":"this equation really goes through these 3 points."},{"Start":"14:05.200 ","End":"14:09.160","Text":"1 more example before we\u0027re done."},{"Start":"14:09.160 ","End":"14:10.840","Text":"In the next example,"},{"Start":"14:10.840 ","End":"14:17.990","Text":"I\u0027m going to give you the equation of a plane and then the equation of a line."},{"Start":"14:18.030 ","End":"14:21.039","Text":"Then I\u0027m going to ask a question."},{"Start":"14:21.039 ","End":"14:27.715","Text":"The plane will be minus x plus 2z equals 10."},{"Start":"14:27.715 ","End":"14:32.935","Text":"Note that the y is missing, but that\u0027s okay."},{"Start":"14:32.935 ","End":"14:37.870","Text":"The line, I\u0027ll write it over here."},{"Start":"14:37.870 ","End":"14:39.520","Text":"I\u0027ll give it in the form of r,"},{"Start":"14:39.520 ","End":"14:41.770","Text":"which is r of t, but I won\u0027t bother with that."},{"Start":"14:41.770 ","End":"14:42.850","Text":"It\u0027s x, y, z,"},{"Start":"14:42.850 ","End":"14:48.055","Text":"r is equal to say, 5,"},{"Start":"14:48.055 ","End":"14:50.785","Text":"2 minus t,"},{"Start":"14:50.785 ","End":"14:55.300","Text":"10 plus 4 t. Now, the question."},{"Start":"14:55.300 ","End":"15:00.070","Text":"I have the plane on the line and I want to know if relative to each other,"},{"Start":"15:00.070 ","End":"15:05.095","Text":"if they\u0027re parallel, that\u0027s 1 possibility."},{"Start":"15:05.095 ","End":"15:08.290","Text":"If they are orthogonal,"},{"Start":"15:08.290 ","End":"15:12.355","Text":"that\u0027s another possibility, meaning perpendicular."},{"Start":"15:12.355 ","End":"15:17.515","Text":"Third possibility, neither, none of the above."},{"Start":"15:17.515 ","End":"15:23.935","Text":"We have to decide, and it looks like a difficult question."},{"Start":"15:23.935 ","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":"15:30.070","Text":"There are 2 vectors that are very important to"},{"Start":"15:30.070 ","End":"15:32.290","Text":"me and that\u0027s all I need to solve this problem."},{"Start":"15:32.290 ","End":"15:37.210","Text":"I need a normal vector to the plane and a direction vector of the line."},{"Start":"15:37.210 ","End":"15:38.905","Text":"We learned how to do this."},{"Start":"15:38.905 ","End":"15:42.370","Text":"Remember, it\u0027s like we have a plus 0y here."},{"Start":"15:42.370 ","End":"15:47.140","Text":"The coefficients are a normal vector,"},{"Start":"15:47.140 ","End":"15:56.469","Text":"so I can take a vector n. The normal to the plane will be 1 possibility is minus 1,"},{"Start":"15:56.469 ","End":"16:00.730","Text":"0 for the missing y, and 2."},{"Start":"16:00.730 ","End":"16:05.080","Text":"The other thing I need is the direction vector of the line."},{"Start":"16:05.080 ","End":"16:10.960","Text":"We already learned that that\u0027s the coefficients of t. Well,"},{"Start":"16:10.960 ","End":"16:13.030","Text":"there is no, it\u0027s 0."},{"Start":"16:13.030 ","End":"16:14.875","Text":"Here, it\u0027s minus 1,"},{"Start":"16:14.875 ","End":"16:17.690","Text":"and here, it\u0027s 4."},{"Start":"16:17.910 ","End":"16:25.885","Text":"Now, how can these help me to find out if the plane in line are parallel or orthogonal?"},{"Start":"16:25.885 ","End":"16:28.435","Text":"Well, let\u0027s take the parallel."},{"Start":"16:28.435 ","End":"16:31.570","Text":"If this vector is parallel to the plane,"},{"Start":"16:31.570 ","End":"16:35.155","Text":"it means that I could put it inside the plane,"},{"Start":"16:35.155 ","End":"16:38.080","Text":"any vector inside the plane or parallel to"},{"Start":"16:38.080 ","End":"16:42.020","Text":"the plane has got to be orthogonal to the normal."},{"Start":"16:42.020 ","End":"16:48.240","Text":"The normal is orthogonal to any vector that\u0027s parallel to the plane."},{"Start":"16:48.240 ","End":"16:54.755","Text":"What I have to check is that if these 2 are perpendicular or orthogonal,"},{"Start":"16:54.755 ","End":"16:59.830","Text":"then the line and the plane will be parallel."},{"Start":"16:59.830 ","End":"17:02.920","Text":"Let\u0027s check that. Remember,"},{"Start":"17:02.920 ","End":"17:09.700","Text":"the test for 2 vectors being orthogonal is if their dot product is 0."},{"Start":"17:09.700 ","End":"17:15.205","Text":"I have to check what is n dot v and ask,"},{"Start":"17:15.205 ","End":"17:16.990","Text":"is this equal to 0?"},{"Start":"17:16.990 ","End":"17:20.905","Text":"If so, these 2 vectors are orthogonal,"},{"Start":"17:20.905 ","End":"17:25.015","Text":"but that will mean that the plane and the line are parallel."},{"Start":"17:25.015 ","End":"17:27.010","Text":"Parallel is orthogonal and later,"},{"Start":"17:27.010 ","End":"17:29.365","Text":"it turns out that orthogonal is parallel."},{"Start":"17:29.365 ","End":"17:31.630","Text":"Actually, reverse."},{"Start":"17:31.630 ","End":"17:38.260","Text":"Because the normal is perpendicular to the plane, it reverses everything."},{"Start":"17:38.260 ","End":"17:47.365","Text":"Let\u0027s see, n dot 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":"17:53.410","Text":"0 times minus 1 is 0,"},{"Start":"17:53.410 ","End":"17:56.860","Text":"and 2 times 4 is 8."},{"Start":"17:56.860 ","End":"18:01.690","Text":"At any rate, it is not 0."},{"Start":"18:01.690 ","End":"18:05.500","Text":"We\u0027ve ruled out the parallel."},{"Start":"18:05.500 ","End":"18:09.400","Text":"No. Now, orthogonal."},{"Start":"18:09.400 ","End":"18:15.070","Text":"If a line is going to be orthogonal to the plane,"},{"Start":"18:15.070 ","End":"18:19.554","Text":"perpendicular, it\u0027s got to be parallel to the normal vector."},{"Start":"18:19.554 ","End":"18:21.700","Text":"Let\u0027s really think about it."},{"Start":"18:21.700 ","End":"18:24.460","Text":"If it\u0027s parallel to a perpendicular to the plane,"},{"Start":"18:24.460 ","End":"18:27.415","Text":"and it\u0027s also perpendicular to the plane are orthogonal."},{"Start":"18:27.415 ","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":"18:43.210","Text":"Similar but using the cross-product."},{"Start":"18:43.210 ","End":"18:49.450","Text":"Parallel would mean that n cross v was equal to 0,"},{"Start":"18:49.450 ","End":"18:51.730","Text":"but this time, the 0 vector."},{"Start":"18:51.730 ","End":"18:54.820","Text":"Let\u0027s check the cross-product."},{"Start":"18:54.820 ","End":"18:58.150","Text":"Minus 1, 0,"},{"Start":"18:58.150 ","End":"19:01.900","Text":"2 cross with 0,"},{"Start":"19:01.900 ","End":"19:05.755","Text":"minus 1, 4 equals,"},{"Start":"19:05.755 ","End":"19:08.770","Text":"it\u0027s 2, 4, 1."},{"Start":"19:08.770 ","End":"19:12.505","Text":"I\u0027m not doing the computation for you like I didn\u0027t do in the previous exercise,"},{"Start":"19:12.505 ","End":"19:14.335","Text":"don\u0027t want to waste time with that."},{"Start":"19:14.335 ","End":"19:18.895","Text":"At any rate, this is not the 0 vector."},{"Start":"19:18.895 ","End":"19:24.775","Text":"We\u0027ve also answered that orthogonal, no."},{"Start":"19:24.775 ","End":"19:26.365","Text":"Also, yeah,"},{"Start":"19:26.365 ","End":"19:30.055","Text":"it\u0027s neither parallel nor orthogonal."},{"Start":"19:30.055 ","End":"19:37.010","Text":"It\u0027s the end of this example and we\u0027re done with the 3D equation of planes."}],"ID":10671},{"Watched":false,"Name":"The 3D Coordinate System - Equations of Planes (continued)","Duration":"11m 29s","ChapterTopicVideoID":10337,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"We\u0027re continuing with the 3D coordinate system."},{"Start":"00:03.690 ","End":"00:05.520","Text":"In the previous clip,"},{"Start":"00:05.520 ","End":"00:09.270","Text":"we learned about planes and their equations,"},{"Start":"00:09.270 ","End":"00:16.710","Text":"and 1 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":"00:35.300","Text":"d is minus big D and I put everything on the left."},{"Start":"00:35.300 ","End":"00:39.905","Text":"Now this is basically a linear equation with 3 variables."},{"Start":"00:39.905 ","End":"00:43.325","Text":"Everything appears most with degree 1."},{"Start":"00:43.325 ","End":"00:47.675","Text":"Now, if I allow degree 2 terms,"},{"Start":"00:47.675 ","End":"00:51.650","Text":"then we will get something known as quadric surfaces."},{"Start":"00:51.650 ","End":"00:56.410","Text":"Quadric like quadratic, meaning degree 2."},{"Start":"00:56.410 ","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":"01:16.080","Text":"Then we have all the mixed terms like xy."},{"Start":"01:16.080 ","End":"01:19.180","Text":"It\u0027s also a degree 2 term."},{"Start":"01:19.180 ","End":"01:22.675","Text":"We\u0027re going to get xz,"},{"Start":"01:22.675 ","End":"01:28.890","Text":"and also another letter with the yz,"},{"Start":"01:28.890 ","End":"01:30.985","Text":"and then of course,"},{"Start":"01:30.985 ","End":"01:35.350","Text":"we also have the linear terms a,"},{"Start":"01:35.350 ","End":"01:39.805","Text":"b, c, d, e, f, g, x,"},{"Start":"01:39.805 ","End":"01:41.710","Text":"I will get rid of this,"},{"Start":"01:41.710 ","End":"01:46.545","Text":"plus hy, let\u0027s see, g, h,"},{"Start":"01:46.545 ","End":"01:55.470","Text":"i, and z after i comes j 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":"02:05.485","Text":"but this is the general definition of a quadric surface."},{"Start":"02:05.485 ","End":"02:08.990","Text":"I guess it has to have at least 1 of the degree 2 terms,"},{"Start":"02:08.990 ","End":"02:10.445","Text":"which is from here to here."},{"Start":"02:10.445 ","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":"02:16.490","Text":"that will be the plane which is also a surface,"},{"Start":"02:16.490 ","End":"02:18.240","Text":"but it\u0027s not quadric."},{"Start":"02:18.240 ","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":"02:30.170","Text":"We\u0027re going to study an ellipsoid,"},{"Start":"02:30.170 ","End":"02:33.485","Text":"next we\u0027ll study a cone,"},{"Start":"02:33.485 ","End":"02:36.920","Text":"and then the cylinder."},{"Start":"02:36.920 ","End":"02:39.515","Text":"They\u0027re all surfaces you\u0027ll notice,"},{"Start":"02:39.515 ","End":"02:44.150","Text":"and then we\u0027re going to have something called a hyperboloid."},{"Start":"02:44.150 ","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 there\u0027s kind"},{"Start":"02:54.170 ","End":"03:01.170","Text":"that\u0027s called 2 sheets or 2 sheeted hyperboloid."},{"Start":"03:01.170 ","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":"03:19.450","Text":"There is going to be the elliptic variety and there\u0027s going to be the hyperbolic variety."},{"Start":"03:19.450 ","End":"03:24.185","Text":"I would say hyperbolic paraboloid or elliptic paraboloid."},{"Start":"03:24.185 ","End":"03:25.925","Text":"That\u0027s the agenda."},{"Start":"03:25.925 ","End":"03:29.540","Text":"We\u0027re going to start with the ellipsoid."},{"Start":"03:29.540 ","End":"03:32.950","Text":"The equation of the ellipsoid is as follows."},{"Start":"03:32.950 ","End":"03:40.340","Text":"x squared over a squared plus y squared over b"},{"Start":"03:40.340 ","End":"03:48.285","Text":"squared plus z squared over c squared equals 1."},{"Start":"03:48.285 ","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":"04:01.840","Text":"which I can bring over to the other side,"},{"Start":"04:01.840 ","End":"04:03.875","Text":"then this really is of this form,"},{"Start":"04:03.875 ","End":"04:06.775","Text":"and it\u0027s basically what we get."},{"Start":"04:06.775 ","End":"04:11.020","Text":"If we don\u0027t have all this linear,"},{"Start":"04:11.020 ","End":"04:13.030","Text":"the mixed and linear terms of missing,"},{"Start":"04:13.030 ","End":"04:15.895","Text":"we just have these terms and this 1,"},{"Start":"04:15.895 ","End":"04:21.265","Text":"and these are all positive and J is negative on the other side, it\u0027s positive."},{"Start":"04:21.265 ","End":"04:23.840","Text":"Then we get an ellipsoid."},{"Start":"04:24.950 ","End":"04:30.320","Text":"I\u0027ll show you a picture of what it looks like."},{"Start":"04:30.350 ","End":"04:35.600","Text":"I\u0027d like to compare it with the 2D equivalent,"},{"Start":"04:35.600 ","End":"04:39.110","Text":"which is an ellipse. Here\u0027s an ellipse."},{"Start":"04:39.110 ","End":"04:48.105","Text":"I\u0027m just assuming that this is part of the y-axis and this is part of the x-axis,"},{"Start":"04:48.105 ","End":"04:52.430","Text":"and let\u0027s say this is the origin."},{"Start":"04:52.430 ","End":"04:55.025","Text":"This is the point to where x is a."},{"Start":"04:55.025 ","End":"04:57.755","Text":"This is the point where y is b,"},{"Start":"04:57.755 ","End":"05:00.305","Text":"and then we get the equation,"},{"Start":"05:00.305 ","End":"05:08.870","Text":"x squared over a squared plus y squared over b squared equals 1."},{"Start":"05:08.870 ","End":"05:18.955","Text":"There is a geometric meaning to a and b. a is like half of the long diameter,"},{"Start":"05:18.955 ","End":"05:21.950","Text":"and it\u0027s called the major axis, the minor axis,"},{"Start":"05:21.950 ","End":"05:23.810","Text":"and if b was bigger than a,"},{"Start":"05:23.810 ","End":"05:26.360","Text":"then it would be a more vertical."},{"Start":"05:26.360 ","End":"05:30.030","Text":"Anyway, we have a meaning for a and b,"},{"Start":"05:30.030 ","End":"05:33.680","Text":"and the same thing applies here, that the a, b,"},{"Start":"05:33.680 ","End":"05:36.230","Text":"and c, it\u0027s harder to see in 3D,"},{"Start":"05:36.230 ","End":"05:42.170","Text":"but a would be the point at which it cuts the x-axis."},{"Start":"05:42.170 ","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":"06:01.850","Text":"and this here would be where z is equal to c. That\u0027s really it."},{"Start":"06:01.850 ","End":"06:05.285","Text":"This assumes that it\u0027s centered at the origin, like here."},{"Start":"06:05.285 ","End":"06:07.220","Text":"There is a variation on this."},{"Start":"06:07.220 ","End":"06:09.510","Text":"In all what follows,"},{"Start":"06:09.510 ","End":"06:12.709","Text":"we\u0027re going to assume that things are centered at the origin."},{"Start":"06:12.709 ","End":"06:16.865","Text":"If you want to shift things from the origin to a new point,"},{"Start":"06:16.865 ","End":"06:20.430","Text":"for example, I\u0027ve thought of 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":"06:28.385","Text":"hk, let\u0027s say,"},{"Start":"06:28.385 ","End":"06:31.535","Text":"then I would get the equation."},{"Start":"06:31.535 ","End":"06:35.000","Text":"We\u0027d replace the x by x minus h,"},{"Start":"06:35.000 ","End":"06:37.160","Text":"we\u0027d replaced the y by y minus k,"},{"Start":"06:37.160 ","End":"06:38.885","Text":"and essentially the same thing."},{"Start":"06:38.885 ","End":"06:43.730","Text":"This also works in 3D if we didn\u0027t want"},{"Start":"06:43.730 ","End":"06:48.845","Text":"the center at the origin and we wanted it at the point,"},{"Start":"06:48.845 ","End":"06:51.770","Text":"let\u0027s say, I don\u0027t know x_0,"},{"Start":"06:51.770 ","End":"06:55.620","Text":"y_0, z_0,"},{"Start":"06:55.620 ","End":"06:58.655","Text":"then we\u0027d get an adapted form of this."},{"Start":"06:58.655 ","End":"07:01.400","Text":"I just squeeze it in here."},{"Start":"07:01.400 ","End":"07:04.220","Text":"Again, you just replace x by x minus x naught,"},{"Start":"07:04.220 ","End":"07:07.430","Text":"y by y minus y naught and everything else holds."},{"Start":"07:07.430 ","End":"07:10.730","Text":"In the future, after the ellipsoid,"},{"Start":"07:10.730 ","End":"07:15.350","Text":"I won\u0027t be talking about transferring the surface,"},{"Start":"07:15.350 ","End":"07:18.440","Text":"so as it\u0027s not centered at the origin but centered somewhere else,"},{"Start":"07:18.440 ","End":"07:20.120","Text":"we just do the same trick,"},{"Start":"07:20.120 ","End":"07:26.645","Text":"replacing x, y, and z by x minus x naught and so on in general."},{"Start":"07:26.645 ","End":"07:29.990","Text":"Before we move on, I want to mention a special case."},{"Start":"07:29.990 ","End":"07:36.110","Text":"Again, I\u0027ll take the analogy in 2D that the ellipse can actually also be a circle."},{"Start":"07:36.110 ","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":"07:49.280","Text":"If a equals b, let\u0027s call that r. Then we multiply both sides by r squared."},{"Start":"07:49.280 ","End":"07:55.130","Text":"Then we get x squared plus y squared equals r squared,"},{"Start":"07:55.130 ","End":"07:59.330","Text":"and that\u0027s a circle of radius r. Analogously,"},{"Start":"07:59.330 ","End":"08:04.550","Text":"here, if we take this equation and also let"},{"Start":"08:04.550 ","End":"08:10.335","Text":"a equals b equals c and call that r,"},{"Start":"08:10.335 ","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":"08:36.795","Text":"Let\u0027s move on to the next 1."},{"Start":"08:36.795 ","End":"08:41.185","Text":"Now we come to the cone and I\u0027ll start straight away with a diagram."},{"Start":"08:41.185 ","End":"08:44.450","Text":"Here\u0027s what the cone looks like."},{"Start":"08:44.450 ","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":"08:54.545","Text":"First of all, there\u0027s 2 parts to it,"},{"Start":"08:54.545 ","End":"08:59.285","Text":"there\u0027s tip to tip and also it\u0027s not finite."},{"Start":"08:59.285 ","End":"09:01.489","Text":"Lines go on infinitely,"},{"Start":"09:01.489 ","End":"09:05.245","Text":"so the cone is double infinite if you like."},{"Start":"09:05.245 ","End":"09:09.560","Text":"Also this particular cone,"},{"Start":"09:09.560 ","End":"09:12.740","Text":"we say opens up in the z direction."},{"Start":"09:12.740 ","End":"09:14.350","Text":"It\u0027s like swallowed the z-axis."},{"Start":"09:14.350 ","End":"09:19.500","Text":"The z-axis is the axis of symmetry also, if you like."},{"Start":"09:19.880 ","End":"09:27.095","Text":"There are others which open up along the x-axis or the y-axis,"},{"Start":"09:27.095 ","End":"09:29.765","Text":"and I\u0027ll say something in a moment about them."},{"Start":"09:29.765 ","End":"09:31.850","Text":"Also, there\u0027s no good 2D analogy,"},{"Start":"09:31.850 ","End":"09:33.950","Text":"so I won\u0027t bring a 2D analogy,"},{"Start":"09:33.950 ","End":"09:35.250","Text":"but I will give you the equation."},{"Start":"09:35.250 ","End":"09:41.615","Text":"The equation I\u0027m going to give you is of general kind of cone called elliptic,"},{"Start":"09:41.615 ","End":"09:44.210","Text":"because there is also a circular cone."},{"Start":"09:44.210 ","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":"09:56.015","Text":"ellipse and show you how it could become a circle."},{"Start":"09:56.015 ","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, the z is the exception because x and y are on 1 side,"},{"Start":"10:21.230 ","End":"10:24.685","Text":"z is on the other side and it opens up in the z direction."},{"Start":"10:24.685 ","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":"10:45.740","Text":"That we can change the order of the variables or the names or the positions"},{"Start":"10:45.740 ","End":"10:51.860","Text":"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":"11:16.450 ","End":"11:26.225","Text":"Let\u0027s see, that\u0027s about it for elliptic cones and circular cones."},{"Start":"11:26.225 ","End":"11:29.190","Text":"Let\u0027s move on."}],"ID":10672},{"Watched":false,"Name":"Exercise 1","Duration":"6m 15s","ChapterTopicVideoID":10328,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we have a plane described by"},{"Start":"00:04.410 ","End":"00:09.689","Text":"3 points on the plane but we want to find the equation of the plane."},{"Start":"00:09.689 ","End":"00:11.580","Text":"To find the equation of a 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":"00:42.540","Text":"take a vector that\u0027s perpendicular to those using the cross-product."},{"Start":"00:42.540 ","End":"00:46.310","Text":"Now, there\u0027s any number of combinations I could use."},{"Start":"00:46.310 ","End":"00:54.130","Text":"What I\u0027m going to do is I\u0027m going to compute the vectors PQ and PR."},{"Start":"00:54.130 ","End":"00:57.545","Text":"Or you could have made different choices."},{"Start":"00:57.545 ","End":"00:59.870","Text":"You might get a slightly different answer,"},{"Start":"00:59.870 ","End":"01:01.775","Text":"which would also be correct."},{"Start":"01:01.775 ","End":"01:07.085","Text":"Anyway, PQ is,"},{"Start":"01:07.085 ","End":"01:12.905","Text":"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 see 1 minus 0 is 1,"},{"Start":"01:21.895 ","End":"01:28.275","Text":"0 minus 1, minus 1, 1 minus 1 is 0."},{"Start":"01:28.275 ","End":"01:30.465","Text":"That\u0027s the PQ vector."},{"Start":"01:30.465 ","End":"01:35.820","Text":"The other 1, the PR, the same thing."},{"Start":"01:35.820 ","End":"01:42.615","Text":"Say method 1 minus 0 is 1, minus 3,"},{"Start":"01:42.615 ","End":"01:46.020","Text":"minus 1 is minus 4,"},{"Start":"01:46.020 ","End":"01:48.465","Text":"and minus 1,"},{"Start":"01:48.465 ","End":"01:52.080","Text":"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":"02:11.010","Text":"then it will be, say, the word orthogonal."},{"Start":"02:11.010 ","End":"02:14.340","Text":"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 1,"},{"Start":"02:22.805 ","End":"02:28.540","Text":"cross-product with the second 1."},{"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":"02:40.375","Text":"It\u0027s just the mechanical exercise, plugging into 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":"02:44.815","Text":"this is the answer."},{"Start":"02:44.815 ","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":"02:54.895","Text":"I\u0027ll just take the first 1."},{"Start":"02:54.895 ","End":"02:57.740","Text":"The first 1, it\u0027s position vector,"},{"Start":"02:57.740 ","End":"03:06.860","Text":"we\u0027ll call it r naught is 0, 1, 1."},{"Start":"03:06.860 ","End":"03:14.720","Text":"Then the formula for the plane is that n"},{"Start":"03:14.720 ","End":"03:23.580","Text":"dot product with r minus r naught"},{"Start":"03:23.650 ","End":"03:27.155","Text":"is equal to 0."},{"Start":"03:27.155 ","End":"03:32.910","Text":"We get that 2, 2,"},{"Start":"03:33.430 ","End":"03:42.030","Text":"minus 3 dot-product with now r is x, y, z."},{"Start":"03:42.030 ","End":"03:47.260","Text":"Here we have x minus 0,"},{"Start":"03:47.260 ","End":"03:49.405","Text":"putting the 0 in for emphasis,"},{"Start":"03:49.405 ","End":"03:51.415","Text":"y minus 1,"},{"Start":"03:51.415 ","End":"03:53.005","Text":"it\u0027s the 1 from here,"},{"Start":"03:53.005 ","End":"03:55.555","Text":"and z minus 1,"},{"Start":"03:55.555 ","End":"03:59.605","Text":"is the 1 from there, equals 0."},{"Start":"03:59.605 ","End":"04:03.040","Text":"If we expand, we get"},{"Start":"04:03.040 ","End":"04:11.280","Text":"2x plus 2,"},{"Start":"04:11.280 ","End":"04:15.675","Text":"y minus 1 minus 3,"},{"Start":"04:15.675 ","End":"04:21.690","Text":"z minus 1 equals 0 and then that gives"},{"Start":"04:21.690 ","End":"04:28.720","Text":"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 folk 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":"Yeah, it works."},{"Start":"05:28.590 ","End":"05:30.075","Text":"Let\u0027s try the last 1,"},{"Start":"05:30.075 ","End":"05:32.295","Text":"1 minus 3 minus 1."},{"Start":"05:32.295 ","End":"05:42.310","Text":"I\u0027ve got minus 2 and then minus 6 is minus 8."},{"Start":"05:44.660 ","End":"05:52.785","Text":"Is that right? Sorry, 2 minus 6 is minus 4,"},{"Start":"05:52.785 ","End":"05:56.520","Text":"minus 3 times that 1 is plus 3."},{"Start":"05:56.520 ","End":"05:58.215","Text":"Minus 4 plus 3 Is minus 1."},{"Start":"05:58.215 ","End":"06:03.510","Text":"Yeah, the middle 1 also works, 1, 0, 1."},{"Start":"06:03.510 ","End":"06:07.185","Text":"I\u0027ve got 2 and I think minus 3."},{"Start":"06:07.185 ","End":"06:15.210","Text":"Yeah. That\u0027s an optional step but recommended if you have time. We\u0027re done."}],"ID":10673},{"Watched":false,"Name":"Exercise 2","Duration":"4m 16s","ChapterTopicVideoID":10329,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"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":"00:09.330","Text":"and it\u0027s also orthogonal,"},{"Start":"00:09.330 ","End":"00:13.200","Text":"perpendicular to this line,"},{"Start":"00:13.200 ","End":"00:15.300","Text":"which is given in vector form."},{"Start":"00:15.300 ","End":"00:18.930","Text":"Normally for a plane 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":"00:30.545","Text":"Well, if we take a direction vector for this line,"},{"Start":"00:30.545 ","End":"00:38.695","Text":"and the direction vector is gotten by the coefficients of the t. If I take the vector"},{"Start":"00:38.695 ","End":"00:49.550","Text":"v as 1,3,4 from the coefficients of t,"},{"Start":"00:49.550 ","End":"00:55.010","Text":"This vector is parallel to the line."},{"Start":"00:55.010 ","End":"00:58.175","Text":"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":"01:15.130","Text":"If I say that something is orthogonal to this line,"},{"Start":"01:15.130 ","End":"01:20.125","Text":"it\u0027s exactly the same as saying orthogonal to this vector."},{"Start":"01:20.125 ","End":"01:25.750","Text":"Which means that this vector will be our normal vector for the plane."},{"Start":"01:25.750 ","End":"01:28.180","Text":"I\u0027ll just take my end to be the same thing,"},{"Start":"01:28.180 ","End":"01:31.505","Text":"just using a different letter."},{"Start":"01:31.505 ","End":"01:35.630","Text":"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":"01:51.010","Text":"and I have the point 0-1."},{"Start":"01:51.010 ","End":"01:52.690","Text":"This is my n,"},{"Start":"01:52.690 ","End":"01:57.540","Text":"this is my r naught,"},{"Start":"01:57.540 ","End":"02:03.470","Text":"we called it, which is the position vector of the 0 to minus 1."},{"Start":"02:03.470 ","End":"02:08.045","Text":"When I have the position vector of a point,"},{"Start":"02:08.045 ","End":"02:15.850","Text":"and the normal then the equation is that r minus"},{"Start":"02:15.850 ","End":"02:26.090","Text":"r naught dot n is equal to 0."},{"Start":"02:26.090 ","End":"02:28.350","Text":"Now, r is vector x,"},{"Start":"02:28.350 ","End":"02:31.030","Text":"y, z in general."},{"Start":"02:37.490 ","End":"02:43.985","Text":"Maybe I\u0027ll write that. R in general is x, y,"},{"Start":"02:43.985 ","End":"02:50.240","Text":"z. R minus r naught is going to be x. I\u0027ll"},{"Start":"02:50.240 ","End":"02:56.960","Text":"write minus 0 to emphasize y minus 2,"},{"Start":"02:56.960 ","End":"02:58.985","Text":"z minus minus 1,"},{"Start":"02:58.985 ","End":"03:06.170","Text":"z plus 1 dot with the normal vector, which we got,"},{"Start":"03:06.170 ","End":"03:09.815","Text":"as I said, from the direction vector of"},{"Start":"03:09.815 ","End":"03:17.410","Text":"the line 1,3,4 is equal to 0."},{"Start":"03:17.410 ","End":"03:23.435","Text":"The dot-product, we just multiply each pair and then we add."},{"Start":"03:23.435 ","End":"03:26.585","Text":"This times this is x."},{"Start":"03:26.585 ","End":"03:31.940","Text":"This times this 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":"03:38.980","Text":"This is equal to 0."},{"Start":"03:38.980 ","End":"03:41.625","Text":"Then I can expand."},{"Start":"03:41.625 ","End":"03:45.935","Text":"Let\u0027s see I have x and I have 3y,"},{"Start":"03:45.935 ","End":"03:48.815","Text":"and from here I have 4z."},{"Start":"03:48.815 ","End":"03:54.020","Text":"Now the numbers, we have to decide if we want them on the left or the right."},{"Start":"03:54.020 ","End":"03:58.270","Text":"Some people for this that I\u0027ll put them on the right."},{"Start":"03:58.270 ","End":"04:03.995","Text":"On the left I have minus 6 plus 4,"},{"Start":"04:03.995 ","End":"04:05.990","Text":"which is minus 2."},{"Start":"04:05.990 ","End":"04:09.570","Text":"On the right it becomes 2."},{"Start":"04:10.160 ","End":"04:15.700","Text":"That is the answer and we\u0027re done."}],"ID":10674},{"Watched":false,"Name":"Exercise 3","Duration":"4m 38s","ChapterTopicVideoID":10330,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.080","Text":"In this exercise, we have to find the equation of the plane containing this point."},{"Start":"00:07.080 ","End":"00:12.135","Text":"The other condition is that it\u0027s going to be parallel to this plane."},{"Start":"00:12.135 ","End":"00:16.275","Text":"Now, the best conditions that we can get, the easiest,"},{"Start":"00:16.275 ","End":"00:21.360","Text":"for a plane is a point on the plane and a normal vector."},{"Start":"00:21.360 ","End":"00:24.780","Text":"Now, we don\u0027t have a normal vector,"},{"Start":"00:24.780 ","End":"00:27.765","Text":"we have the condition that is parallel to this plane."},{"Start":"00:27.765 ","End":"00:32.580","Text":"But that\u0027s actually very close because we know the normal vector for"},{"Start":"00:32.580 ","End":"00:38.870","Text":"this plane is just gotten from the coefficients,"},{"Start":"00:38.870 ","End":"00:43.285","Text":"4, 8, minus 2."},{"Start":"00:43.285 ","End":"00:45.850","Text":"Actually, I should say a normal vector,"},{"Start":"00:45.850 ","End":"00:48.140","Text":"not the normal vector because there\u0027s many of them,"},{"Start":"00:48.140 ","End":"00:53.660","Text":"many nonzero scalar times a normal vector is a normal vector."},{"Start":"00:53.660 ","End":"00:57.065","Text":"Anyway, this is the most obvious one."},{"Start":"00:57.065 ","End":"01:00.140","Text":"Now, if the planes are parallel normal"},{"Start":"01:00.140 ","End":"01:02.960","Text":"to 1 is going to be a normal to the other. They\u0027re parallel."},{"Start":"01:02.960 ","End":"01:05.840","Text":"So anything orthogonal to one is going to be orthogonal to the other."},{"Start":"01:05.840 ","End":"01:08.035","Text":"We have the normal vector."},{"Start":"01:08.035 ","End":"01:14.659","Text":"This is our vector n. Then we have our point on the plane."},{"Start":"01:14.659 ","End":"01:20.120","Text":"Usually, we take its position vector rather than the point itself,"},{"Start":"01:20.120 ","End":"01:25.645","Text":"and we call it r_0, 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":"01:36.110","Text":"Now, we have the normal and we have the point,"},{"Start":"01:36.110 ","End":"01:38.690","Text":"and now we just apply the equation."},{"Start":"01:38.690 ","End":"01:44.630","Text":"Let me just remind you that we use x, y,"},{"Start":"01:44.630 ","End":"01:51.770","Text":"z to represent in the formula r. The formula"},{"Start":"01:51.770 ","End":"01:59.759","Text":"is that r minus r_0 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":"02:05.130","Text":"this is dot product."},{"Start":"02:05.240 ","End":"02:10.520","Text":"What we get is if we just substitute r is this and r_0 is that,"},{"Start":"02:10.520 ","End":"02:21.310","Text":"so x minus minus 7 is x plus 7 times 4."},{"Start":"02:22.630 ","End":"02:33.870","Text":"Then the next coordinate I get is going to be y minus 3 and then times 8."},{"Start":"02:35.240 ","End":"02:37.580","Text":"Then we 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":"02:43.775","Text":"it doesn\u0027t make any real difference. Put it after."},{"Start":"02:43.775 ","End":"02:47.850","Text":"I think it\u0027s usually put before, whatever."},{"Start":"02:48.080 ","End":"02:52.875","Text":"Last one is z minus"},{"Start":"02:52.875 ","End":"02:59.770","Text":"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":"03:06.855","Text":"I\u0027ll just put a minus 2 here, equals 0."},{"Start":"03:06.855 ","End":"03:09.585","Text":"I\u0027ll collect the variables."},{"Start":"03:09.585 ","End":"03:11.280","Text":"I\u0027ve got 4x,"},{"Start":"03:11.280 ","End":"03:18.405","Text":"from here I have 8y minus 2z."},{"Start":"03:18.405 ","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":"03:31.600","Text":"I\u0027ll go with the numbers on the right. Let\u0027s see."},{"Start":"03:31.600 ","End":"03:41.035","Text":"On the left we had 28 minus 24 is 4 plus 18 is 22."},{"Start":"03:41.035 ","End":"03:43.225","Text":"But when I bring it over to the right,"},{"Start":"03:43.225 ","End":"03:47.120","Text":"it will be minus 22."},{"Start":"03:47.760 ","End":"03:53.955","Text":"Something I do sometimes just as a check,"},{"Start":"03:53.955 ","End":"03:56.040","Text":"if it\u0027s a plane with 3 points,"},{"Start":"03:56.040 ","End":"03:57.340","Text":"I can do 3 checks here."},{"Start":"03:57.340 ","End":"04:04.110","Text":"I have a point, at least I can check that this is on the plane."},{"Start":"04:04.110 ","End":"04:06.700","Text":"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":"04:14.200","Text":"This and this is plus 24."},{"Start":"04:14.200 ","End":"04:16.895","Text":"We\u0027re down to minus 4."},{"Start":"04:16.895 ","End":"04:20.700","Text":"Minus 2z is minus 18,"},{"Start":"04:20.700 ","End":"04:23.505","Text":"minus 22. This point works."},{"Start":"04:23.505 ","End":"04:26.550","Text":"We can also see that we have 4,"},{"Start":"04:26.550 ","End":"04:30.550","Text":"8, minus 2 and we have here 4, 8, minus 2."},{"Start":"04:31.520 ","End":"04:33.650","Text":"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":10675},{"Watched":false,"Name":"Exercise 4","Duration":"4m 47s","ChapterTopicVideoID":10331,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"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, it\u0027s Pi."},{"Start":"00:18.195 ","End":"00:23.100","Text":"Plane Pi_1 is given by this formula and"},{"Start":"00:23.100 ","End":"00:28.855","Text":"the second plane Pi_2 is given with this formula."},{"Start":"00:28.855 ","End":"00:34.270","Text":"I just don\u0027t think this has anything to do with Pi from the circle,"},{"Start":"00:34.270 ","End":"00:39.500","Text":"circumference over diameter, whatever."},{"Start":"00:39.500 ","End":"00:41.645","Text":"Yeah, just a Greek letter."},{"Start":"00:41.645 ","End":"00:43.595","Text":"What we want to know is,"},{"Start":"00:43.595 ","End":"00:46.110","Text":"are these 2 planes parallel?"},{"Start":"00:46.110 ","End":"00:50.230","Text":"Are they orthogonal, meaning perpendicular normal?"},{"Start":"00:50.230 ","End":"00:54.645","Text":"Neither this nor that, let\u0027s see."},{"Start":"00:54.645 ","End":"00:58.755","Text":"What are going to help us here are the normals."},{"Start":"00:58.755 ","End":"01:03.990","Text":"Let\u0027s call n_1 the normal for the first plane and we can get"},{"Start":"01:03.990 ","End":"01:09.435","Text":"that from the coefficients so that would be vector 4,"},{"Start":"01:09.435 ","End":"01:18.400","Text":"8, minus 2 and the normal vector for the second plane,"},{"Start":"01:18.400 ","End":"01:20.585","Text":"or a normal vector,"},{"Start":"01:20.585 ","End":"01:24.150","Text":"is gotten from the coefficients here."},{"Start":"01:24.150 ","End":"01:25.965","Text":"I could take 2,"},{"Start":"01:25.965 ","End":"01:30.880","Text":"1, 8. Now here\u0027s the thing."},{"Start":"01:30.880 ","End":"01:35.170","Text":"The planes 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":"01:43.780","Text":"Specifically, if the planes are parallel to think about it,"},{"Start":"01:43.780 ","End":"01:46.255","Text":"the normals are also going to be parallel,"},{"Start":"01:46.255 ","End":"01:50.415","Text":"perpendicular to parallel planes,"},{"Start":"01:50.415 ","End":"01:56.850","Text":"going to be the same and likewise orthogonal."},{"Start":"01:56.850 ","End":"01:59.470","Text":"If 2 planes are orthogonal,"},{"Start":"01:59.470 ","End":"02:05.445","Text":"then the normal vectors are also going to be orthogonal."},{"Start":"02:05.445 ","End":"02:09.745","Text":"All we have to do is check these 2 normals to see,"},{"Start":"02:09.745 ","End":"02:11.185","Text":"are they parallel?"},{"Start":"02:11.185 ","End":"02:13.240","Text":"Are they orthogonal?"},{"Start":"02:13.240 ","End":"02:16.915","Text":"For parallel, we need for 1 of them to be"},{"Start":"02:16.915 ","End":"02:21.250","Text":"a multiple of the other and these are not 0 vectors."},{"Start":"02:21.250 ","End":"02:24.640","Text":"1 of them has to be a nonzero 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":"02:37.530","Text":"not 0, times n_1 vector."},{"Start":"02:39.670 ","End":"02:45.140","Text":"Well, there\u0027s many ways to see this is not going to work out."},{"Start":"02:45.140 ","End":"02:49.880","Text":"For 1 thing, if k is positive and k times n_1,"},{"Start":"02:49.880 ","End":"02:55.005","Text":"they\u0027re all going to be plus, plus, minus."},{"Start":"02:55.005 ","End":"02:56.490","Text":"If k is negative,"},{"Start":"02:56.490 ","End":"02:58.910","Text":"I\u0027m going to get minus, minus, plus."},{"Start":"02:58.910 ","End":"03:02.200","Text":"But there\u0027s no way I\u0027m going to get plus, plus, plus,"},{"Start":"03:02.200 ","End":"03:06.230","Text":"so there is no such k. We\u0027ll 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":"03:11.920","Text":"so k would have to be a 0.5."},{"Start":"03:11.920 ","End":"03:14.700","Text":"Then we\u0027d have to have 0.5 times 8 is 1,"},{"Start":"03:14.700 ","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":"03:23.570","Text":"This does not work out and so"},{"Start":"03:23.570 ","End":"03:29.340","Text":"these 2 normal vectors are not parallel and if they\u0027re not parallel,"},{"Start":"03:29.340 ","End":"03:32.880","Text":"then neither are the planes."},{"Start":"03:32.880 ","End":"03:36.030","Text":"Let\u0027s go rule the parallel out,"},{"Start":"03:36.030 ","End":"03:38.695","Text":"now let\u0027s go for orthogonal."},{"Start":"03:38.695 ","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 dot product is 0."},{"Start":"03:50.000 ","End":"03:58.669","Text":"Let\u0027s see what is n_1 dot n_2 and see is this equal to 0?"},{"Start":"03:58.669 ","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":"04:23.205","Text":"Let\u0027s see. This is"},{"Start":"04:23.205 ","End":"04:30.360","Text":"8 plus 8 minus 16 and yes,"},{"Start":"04:30.360 ","End":"04:36.615","Text":"bingo, we have 0 so they are orthogonal."},{"Start":"04:36.615 ","End":"04:44.535","Text":"Yes, that is the answer that the 2 planes are orthogonal,"},{"Start":"04:44.535 ","End":"04:47.740","Text":"and we are done."}],"ID":10676},{"Watched":false,"Name":"Exercise 5","Duration":"11m 33s","ChapterTopicVideoID":10332,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have two planes,"},{"Start":"00:03.480 ","End":"00:06.405","Text":"Pi 1 and Pi 2."},{"Start":"00:06.405 ","End":"00:09.480","Text":"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 three points."},{"Start":"00:22.630 ","End":"00:25.400","Text":"We\u0027re going to have to work a bit harder."},{"Start":"00:25.400 ","End":"00:29.030","Text":"The question here asks us about whether"},{"Start":"00:29.030 ","End":"00:35.160","Text":"the two planes are parallel or orthogonal or neither of these."},{"Start":"00:35.930 ","End":"00:39.035","Text":"We had a similar question like this."},{"Start":"00:39.035 ","End":"00:41.450","Text":"But it\u0027s okay if you don\u0027t remember,"},{"Start":"00:41.450 ","End":"00:46.335","Text":"what we have to do is get normal vectors for each of these planes."},{"Start":"00:46.335 ","End":"00:49.220","Text":"Then ask these questions about the normal vectors,"},{"Start":"00:49.220 ","End":"00:53.375","Text":"are the normal vectors parallel or orthogonal or neither?"},{"Start":"00:53.375 ","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":"01:06.470","Text":"that\u0027s easy to get because you can get it from the coefficients of the x,y,z."},{"Start":"01:06.470 ","End":"01:11.930","Text":"A normal vector we could take as 2,"},{"Start":"01:11.930 ","End":"01:17.100","Text":"minus 3, 4."},{"Start":"01:17.100 ","End":"01:18.110","Text":"Then the question is,"},{"Start":"01:18.110 ","End":"01:24.155","Text":"how do we get a normal vector n_2 for the second plane?"},{"Start":"01:24.155 ","End":"01:26.930","Text":"We\u0027re just given hints."},{"Start":"01:26.930 ","End":"01:29.510","Text":"Here\u0027s what we can do,"},{"Start":"01:29.510 ","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":"01:43.520","Text":"we could call this one R. If I have three points in a plane,"},{"Start":"01:43.520 ","End":"01:55.160","Text":"say P,Q,R,"},{"Start":"01:55.160 ","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":"02:18.155","Text":"I could take the vector any combination, let\u0027s say PR."},{"Start":"02:18.155 ","End":"02:25.210","Text":"Now, these two vectors are parallel to the plane containing P,"},{"Start":"02:25.210 ","End":"02:33.254","Text":"Q and R. If I have two vectors that are parallel to the plane,"},{"Start":"02:33.254 ","End":"02:39.215","Text":"then their cross-product will be perpendicular to each of these."},{"Start":"02:39.215 ","End":"02:43.760","Text":"It will be a normal vector to the plane."},{"Start":"02:43.760 ","End":"02:53.360","Text":"What I have to do is take PQ and then cross that vector,"},{"Start":"02:53.360 ","End":"02:57.385","Text":"cross product with PR."},{"Start":"02:57.385 ","End":"03:03.170","Text":"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":"03:16.760","Text":"What we\u0027ll do is say PQ."},{"Start":"03:16.760 ","End":"03:18.545","Text":"We can get this,"},{"Start":"03:18.545 ","End":"03:20.840","Text":"it\u0027s a displacement vector,"},{"Start":"03:20.840 ","End":"03:24.260","Text":"sometimes called one that takes me from P to Q,"},{"Start":"03:24.260 ","End":"03:30.065","Text":"subtract components, 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":"03:38.600","Text":"That\u0027s the PQ part,"},{"Start":"03:38.600 ","End":"03:41.620","Text":"now the PR part."},{"Start":"03:41.620 ","End":"03:49.950","Text":"I\u0027ll take the components of R and subtract the components of P. I want the cross product."},{"Start":"03:49.950 ","End":"03:53.850","Text":"That will be minus 3,"},{"Start":"03:53.850 ","End":"03:58.140","Text":"minus 1 is minus 4,"},{"Start":"03:58.140 ","End":"04:05.820","Text":"minus 2,"},{"Start":"04:05.820 ","End":"04:12.165","Text":"minus 2 is minus 4,"},{"Start":"04:12.165 ","End":"04:15.600","Text":"and minus 6,"},{"Start":"04:15.600 ","End":"04:22.605","Text":"minus 2 would be minus 8."},{"Start":"04:22.605 ","End":"04:28.540","Text":"Now, like I said,"},{"Start":"04:29.060 ","End":"04:36.800","Text":"it\u0027s not a unique normal multiplying by a non-zero scalar will"},{"Start":"04:36.800 ","End":"04:41.210","Text":"also give me a parallel vector reason I\u0027m"},{"Start":"04:41.210 ","End":"04:46.250","Text":"saying this is I see all of these are divisible by 4."},{"Start":"04:46.250 ","End":"04:51.510","Text":"I also see that they\u0027re all negative and these are both nuisances."},{"Start":"04:52.840 ","End":"04:59.480","Text":"What I\u0027m going to 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":"05:05.015","Text":"I\u0027ll get 1,"},{"Start":"05:05.015 ","End":"05:09.770","Text":"1, 2 that\u0027s much nicer."},{"Start":"05:09.770 ","End":"05:16.790","Text":"It\u0027s a scalar times this so will give me a multiple of the normal,"},{"Start":"05:16.790 ","End":"05:19.049","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":"05:37.535","Text":"What we do is we take a 3 by 3 determinant on the top row we put i, j, k,"},{"Start":"05:37.535 ","End":"05:47.190","Text":"Then we put one of the first 1,0,1 and then the other one 1,1,2,"},{"Start":"05:48.560 ","End":"05:52.690","Text":"and we compute this determinant."},{"Start":"05:52.730 ","End":"05:58.370","Text":"What we get, I\u0027m assuming, you know determinants,"},{"Start":"05:58.370 ","End":"06:04.400","Text":"if not the formula that just gives you the three components."},{"Start":"06:04.400 ","End":"06:06.620","Text":"Like in the first component,"},{"Start":"06:06.620 ","End":"06:12.720","Text":"it might be this times this minus this times this."},{"Start":"06:12.720 ","End":"06:14.675","Text":"There\u0027s another way of doing it."},{"Start":"06:14.675 ","End":"06:19.265","Text":"I\u0027ll do it this way. We look at the i and"},{"Start":"06:19.265 ","End":"06:25.460","Text":"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":"06:31.010","Text":"which is this 0,"},{"Start":"06:31.010 ","End":"06:38.985","Text":"1, 1, 2, and then that\u0027s the coefficient of i."},{"Start":"06:38.985 ","End":"06:42.970","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":"07:01.245","Text":"a 2 by 2 determinant with the k. Now,"},{"Start":"07:01.245 ","End":"07:05.190","Text":"for the j, I just cross off the row and column with the j."},{"Start":"07:05.190 ","End":"07:09.120","Text":"What we\u0027re left with is 1, 1, 1, 2,"},{"Start":"07:09.120 ","End":"07:14.055","Text":"1, 1, 1, 2."},{"Start":"07:14.055 ","End":"07:22.840","Text":"For the k, we get this determinant 1, 0, 1, 1."},{"Start":"07:23.360 ","End":"07:32.360","Text":"That gives us a 2 by 2 determinant is this diagonal product minus this diagonals product."},{"Start":"07:32.360 ","End":"07:35.950","Text":"It\u0027s 0 minus 1."},{"Start":"07:35.950 ","End":"07:45.060","Text":"That\u0027s minus i. I\u0027ll go back to the angular bracket notation."},{"Start":"07:45.060 ","End":"07:47.595","Text":"Instead of writing minus i,"},{"Start":"07:47.595 ","End":"07:49.485","Text":"I will write minus 1,"},{"Start":"07:49.485 ","End":"07:53.135","Text":"here, in the next one,"},{"Start":"07:53.135 ","End":"07:56.044","Text":"we have for the j,"},{"Start":"07:56.044 ","End":"08:01.340","Text":"2 minus 1 is 1,"},{"Start":"08:01.340 ","End":"08:03.225","Text":"but there\u0027s a minus."},{"Start":"08:03.225 ","End":"08:07.950","Text":"It\u0027s minus 1 or minus j,"},{"Start":"08:07.950 ","End":"08:12.330","Text":"which is like this with the bracket notation for vectors."},{"Start":"08:12.330 ","End":"08:19.250","Text":"The last one, 1 minus 0 is 1 and there\u0027s a plus,"},{"Start":"08:19.250 ","End":"08:25.170","Text":"so it\u0027s plus k. I just put plus 1 or plain old one."},{"Start":"08:25.360 ","End":"08:31.760","Text":"Now I have the two normal vectors."},{"Start":"08:31.760 ","End":"08:39.365","Text":"This is n_1 and I\u0027ll write this again as n_2."},{"Start":"08:39.365 ","End":"08:42.710","Text":"Let me just highlight these."},{"Start":"08:42.710 ","End":"08:48.220","Text":"Here is a normal vector to the first plane"},{"Start":"08:48.220 ","End":"08:54.755","Text":"Pi_1 and here\u0027s a normal vector to the second plane Pi_2."},{"Start":"08:54.755 ","End":"08:57.845","Text":"Now we just have to ask,"},{"Start":"08:57.845 ","End":"09:00.785","Text":"are these two parallel?"},{"Start":"09:00.785 ","End":"09:06.750","Text":"If these are parallel and the planes are going to be parallel and vice versa."},{"Start":"09:07.500 ","End":"09:13.550","Text":"I like using the trick with the sign because if I"},{"Start":"09:13.550 ","End":"09:19.715","Text":"take a non-zero constant and multiply it by 1 and hope to get to the other."},{"Start":"09:19.715 ","End":"09:22.610","Text":"The middle one\u0027s 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":"09:37.430","Text":"But in no way does it fit this because I can\u0027t have a minus, minus plus."},{"Start":"09:37.430 ","End":"09:39.320","Text":"It\u0027s the middle one is the odd one out."},{"Start":"09:39.320 ","End":"09:45.020","Text":"No positive or negative scalar can multiply this to give me this."},{"Start":"09:45.020 ","End":"09:50.310","Text":"They are not parallel."},{"Start":"09:50.310 ","End":"09:52.925","Text":"The normal vectors aren\u0027t parallel,"},{"Start":"09:52.925 ","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":"10:06.560","Text":"The planes are orthogonal if and only if the normal vectors are orthogonal."},{"Start":"10:06.870 ","End":"10:15.370","Text":"The test for orthogonal is using the dot-product to see if it\u0027s 0."},{"Start":"10:15.370 ","End":"10:19.450","Text":"Is n_1 dot-product with n_2 equal,"},{"Start":"10:19.450 ","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":"10:32.370","Text":"we get 2 times minus"},{"Start":"10:32.370 ","End":"10:40.520","Text":"1 and then minus 3 times minus 1,"},{"Start":"10:40.520 ","End":"10:48.870","Text":"and then 4 times 1."},{"Start":"10:48.870 ","End":"10:51.200","Text":"What does this come out to be?"},{"Start":"10:51.200 ","End":"10:54.475","Text":"Comes out minus 2"},{"Start":"10:54.475 ","End":"11:05.090","Text":"plus 3 plus 4."},{"Start":"11:05.090 ","End":"11:08.120","Text":"This is equal to"},{"Start":"11:08.120 ","End":"11:16.435","Text":"5 and 5 is not equal to 0."},{"Start":"11:16.435 ","End":"11:23.740","Text":"We\u0027re also not orthogonal, not perpendicular."},{"Start":"11:24.740 ","End":"11:34.290","Text":"I guess that leaves us with neither or neither wherever you say it. We\u0027re done."}],"ID":10677},{"Watched":false,"Name":"Exercise 6","Duration":"3m 58s","ChapterTopicVideoID":10333,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this exercise, we\u0027re given a plane and the line, the plane,"},{"Start":"00:04.200 ","End":"00:05.835","Text":"we\u0027ll call it Pi,"},{"Start":"00:05.835 ","End":"00:07.860","Text":"is given by this equation,"},{"Start":"00:07.860 ","End":"00:09.735","Text":"typical plane equation,"},{"Start":"00:09.735 ","End":"00:13.140","Text":"and the line is given in the parametric form,"},{"Start":"00:13.140 ","End":"00:21.120","Text":"x equals y equals z equals as functions of t. The question is,"},{"Start":"00:21.120 ","End":"00:24.210","Text":"does the line intersect the plane?"},{"Start":"00:24.210 ","End":"00:28.755","Text":"If it does, 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":"00:39.060","Text":"a typical point on the line is given by 1 minus t,"},{"Start":"00:39.060 ","End":"00:42.525","Text":"3t, 1 plus t for x, y, z,"},{"Start":"00:42.525 ","End":"00:47.520","Text":"and to substitute into the plane equation,"},{"Start":"00:47.520 ","End":"00:52.445","Text":"and that gives us an equation in t. If it has a solution then good,"},{"Start":"00:52.445 ","End":"00:55.490","Text":"we have an intersection, so let\u0027s do that."},{"Start":"00:55.490 ","End":"00:58.645","Text":"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":"01:06.550","Text":"x is 1 minus t."},{"Start":"01:06.550 ","End":"01:12.465","Text":"Then I have minus y is 3t."},{"Start":"01:12.465 ","End":"01:18.585","Text":"Then 3 times z is 1 plus t,"},{"Start":"01:18.585 ","End":"01:23.229","Text":"and this has to equal 6 if it\u0027s going to be on the plane."},{"Start":"01:23.229 ","End":"01:29.930","Text":"Let\u0027s see if it has a solution for t. 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":"01:45.815","Text":"The minus 3t and the plus 3t cancel."},{"Start":"01:45.815 ","End":"01:53.150","Text":"On the left I have minus 2t and let\u0027s see,"},{"Start":"01:53.150 ","End":"01:58.730","Text":"on the right I get 6 minus 3 minus 2,"},{"Start":"01:58.730 ","End":"02:01.860","Text":"that gives me 1,"},{"Start":"02:01.860 ","End":"02:08.680","Text":"so t is equal to minus 1.5."},{"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":"02:22.700","Text":"so I have to just put this value of t into the parametric equation of the line,"},{"Start":"02:22.700 ","End":"02:27.065","Text":"so 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 0.5 is 1.5."},{"Start":"02:35.070 ","End":"02:41.085","Text":"Y is going to equal 3 times t,"},{"Start":"02:41.085 ","End":"02:46.435","Text":"which is minus 3 times 0.5 minus 1.5."},{"Start":"02:46.435 ","End":"02:50.495","Text":"I\u0027ll do it as mixed numbers fractions."},{"Start":"02:50.495 ","End":"02:54.270","Text":"Z is 1 plus t,"},{"Start":"02:54.270 ","End":"02:59.260","Text":"1 minus 0.5 is 1.5."},{"Start":"02:59.690 ","End":"03:09.570","Text":"The intersection point is the point 1.5,"},{"Start":"03:09.570 ","End":"03:16.110","Text":"minus 1.5, 1.5,"},{"Start":"03:16.110 ","End":"03:18.045","Text":"and that\u0027s the answer."},{"Start":"03:18.045 ","End":"03:22.675","Text":"But I just like to do a quick check to see,"},{"Start":"03:22.675 ","End":"03:27.325","Text":"for example, if this point really is on the plane."},{"Start":"03:27.325 ","End":"03:32.645","Text":"Don\u0027t have to do this, but I like to sometimes verify things."},{"Start":"03:32.645 ","End":"03:39.085","Text":"Twice x would be 3 minus y,"},{"Start":"03:39.085 ","End":"03:44.035","Text":"so it\u0027s minus minus 1.5 sum up to 4.5."},{"Start":"03:44.035 ","End":"03:48.840","Text":"4.5 and 3z is another 1.5."},{"Start":"03:48.840 ","End":"03:51.345","Text":"Yeah, that gives me 6,"},{"Start":"03:51.345 ","End":"03:57.860","Text":"so I\u0027m more confident that this is the right answer. That\u0027s all."}],"ID":10678},{"Watched":false,"Name":"Exercise 7","Duration":"3m ","ChapterTopicVideoID":10334,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.299","Text":"This question is similar to the previous question,"},{"Start":"00:03.299 ","End":"00:05.295","Text":"but given the plane,"},{"Start":"00:05.295 ","End":"00:07.245","Text":"we call the plane Pi,"},{"Start":"00:07.245 ","End":"00:09.330","Text":"not to be confused with the number Pi is"},{"Start":"00:09.330 ","End":"00:17.160","Text":"just a Greek letter and its equation is x minus y plus z equals 3."},{"Start":"00:17.160 ","End":"00:23.730","Text":"Then we have a line l given in vector form as"},{"Start":"00:23.730 ","End":"00:27.570","Text":"follows as a function of t. We want to"},{"Start":"00:27.570 ","End":"00:32.280","Text":"know if the line and the plane intersect and if so,"},{"Start":"00:32.280 ","End":"00:35.950","Text":"where, meaning what\u0027s the point of intersection?"},{"Start":"00:35.960 ","End":"00:45.555","Text":"What we do is we see that this is like x, y, and z?"},{"Start":"00:45.555 ","End":"00:49.190","Text":"We want to know for some t if x, y,"},{"Start":"00:49.190 ","End":"00:54.484","Text":"and z 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":"01:01.220","Text":"We\u0027ll get to figure out x minus y plus z for a point on the line,"},{"Start":"01:01.220 ","End":"01:10.260","Text":"so x is 5 plus 2t minus y,"},{"Start":"01:10.260 ","End":"01:17.250","Text":"which is 1 minus 5t and then plus z,"},{"Start":"01:17.250 ","End":"01:23.235","Text":"which is 3t has got to equal 3."},{"Start":"01:23.235 ","End":"01:25.380","Text":"If we find such t,"},{"Start":"01:25.380 ","End":"01:28.830","Text":"then we know that there is an intersection."},{"Start":"01:28.830 ","End":"01:33.930","Text":"Let\u0027s see if we can find t. Opening up."},{"Start":"01:33.930 ","End":"01:40.320","Text":"We have 5 plus 2t minus 1 plus"},{"Start":"01:40.320 ","End":"01:48.880","Text":"5t plus 3t is equal to 3."},{"Start":"01:52.190 ","End":"01:56.495","Text":"Now, I just noticed that I miss copied the question."},{"Start":"01:56.495 ","End":"02:03.350","Text":"This was actually a plus and so this here is also"},{"Start":"02:03.350 ","End":"02:11.280","Text":"a plus and that means that here is a minus."},{"Start":"02:11.280 ","End":"02:17.195","Text":"I knew something was wrong because I knew was supposed to cancel."},{"Start":"02:17.195 ","End":"02:21.995","Text":"Look 2t and 3t is 5t with minus 5t,"},{"Start":"02:21.995 ","End":"02:24.385","Text":"the ts cancel,"},{"Start":"02:24.385 ","End":"02:27.195","Text":"5 minus 1 is 4."},{"Start":"02:27.195 ","End":"02:29.115","Text":"We get the equation,"},{"Start":"02:29.115 ","End":"02:31.495","Text":"4 equals 3."},{"Start":"02:31.495 ","End":"02:34.040","Text":"Now that\u0027s impossible."},{"Start":"02:34.040 ","End":"02:39.425","Text":"That cannot be and that means that for no value of t,"},{"Start":"02:39.425 ","End":"02:42.685","Text":"will this point B on the plane."},{"Start":"02:42.685 ","End":"02:48.530","Text":"No point on the line is also on the plane so they don\u0027t intersect,"},{"Start":"02:48.530 ","End":"02:50.750","Text":"so do l and Pi intersect?"},{"Start":"02:50.750 ","End":"02:55.560","Text":"No. We don\u0027t answer the second part if so,"},{"Start":"02:55.560 ","End":"03:00.130","Text":"where could they don\u0027t intersect and that\u0027s it."}],"ID":10679},{"Watched":false,"Name":"Exercise 8","Duration":"19m 19s","ChapterTopicVideoID":10335,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"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":"pi 1 and pi 2 in the regular linear equation form."},{"Start":"00:10.755 ","End":"00:14.700","Text":"This is the equation of pi 1,"},{"Start":"00:14.700 ","End":"00:17.880","Text":"this is the equation of pi 2,"},{"Start":"00:17.880 ","End":"00:25.965","Text":"and 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":"00:33.850 ","End":"00:38.135","Text":"Actually, they could be identical, in each case,"},{"Start":"00:38.135 ","End":"00:39.620","Text":"they intersect in a plane,"},{"Start":"00:39.620 ","End":"00:41.900","Text":"but those are crazy cases."},{"Start":"00:41.900 ","End":"00:43.385","Text":"Usually, 2 planes,"},{"Start":"00:43.385 ","End":"00:45.860","Text":"unless they\u0027re parallel will intersect in a line."},{"Start":"00:45.860 ","End":"00:48.670","Text":"Let\u0027s call that line l,"},{"Start":"00:48.670 ","End":"00:51.740","Text":"and in set theory notation,"},{"Start":"00:51.740 ","End":"00:54.905","Text":"it\u0027s pi 1 intersection with pi 2."},{"Start":"00:54.905 ","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":"01:12.150","Text":"we want the vector form."},{"Start":"01:13.120 ","End":"01:18.185","Text":"Here\u0027s the trick, this line,"},{"Start":"01:18.185 ","End":"01:23.120","Text":"any line in the plane has to intersect at least 1 of"},{"Start":"01:23.120 ","End":"01:33.100","Text":"the coordinate planes, either has to hit the xy-plane, or the xz-plane, or the zy-plane."},{"Start":"01:33.100 ","End":"01:36.200","Text":"It can\u0027t miss all 3 coordinate planes."},{"Start":"01:36.200 ","End":"01:40.160","Text":"If you just think about it, they\u0027re not going to give you a proof but, no way"},{"Start":"01:40.160 ","End":"01:47.110","Text":"an infinite line cannot hit at least 1 of those 3 coordinate planes."},{"Start":"01:48.590 ","End":"01:53.490","Text":"Let\u0027s try the xy-plane."},{"Start":"01:53.490 ","End":"01:57.320","Text":"If that doesn\u0027t work, we\u0027ve got 2 more chances to see where it hits"},{"Start":"01:57.320 ","End":"02:03.180","Text":"the xz-plane, or zy-plane."},{"Start":"02:04.900 ","End":"02:11.885","Text":"My question is, where does"},{"Start":"02:11.885 ","End":"02:21.000","Text":"l cross the xy-plane?"},{"Start":"02:21.850 ","End":"02:26.550","Text":"If I don\u0027t get an answer,"},{"Start":"02:26.550 ","End":"02:35.940","Text":"it doesn\u0027t cross, then I\u0027ll try afterwards the xz-plane, and I\u0027ll try the yz-plane."},{"Start":"02:35.940 ","End":"02:39.435","Text":"It\u0027s got to cross 1 of these,"},{"Start":"02:39.435 ","End":"02:44.430","Text":"in fact, I know it\u0027s going to cross the xy-plane."},{"Start":"02:44.430 ","End":"02:48.390","Text":"I chose it that we\u0027d get first-time lucky."},{"Start":"02:48.390 ","End":"02:50.595","Text":"How do we do that?"},{"Start":"02:50.595 ","End":"02:55.750","Text":"The xy-plane is where z equals 0."},{"Start":"02:55.750 ","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, and 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":"03:45.030","Text":"Then I want to solve x and y,"},{"Start":"03:45.030 ","End":"03:50.760","Text":"and then we\u0027ll add the condition that z is 0, and we\u0027ll get the intersection."},{"Start":"03:51.340 ","End":"03:55.290","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":"04:03.410","Text":"so let\u0027s do that."},{"Start":"04:03.410 ","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":"04:12.510","Text":"will equal 7y minus"},{"Start":"04:12.510 ","End":"04:19.490","Text":"24, and then if I put that x in here,"},{"Start":"04:19.490 ","End":"04:26.600","Text":"I\u0027ll get minus 5 7y minus"},{"Start":"04:26.600 ","End":"04:34.590","Text":"24 plus 6y equals minus 3,"},{"Start":"04:34.590 ","End":"04:38.910","Text":"multiplying out, let\u0027s see if we can do it all in 1."},{"Start":"04:38.910 ","End":"04:44.820","Text":"We\u0027ve got minus 35y plus"},{"Start":"04:44.820 ","End":"04:51.590","Text":"6y so on this side we have minus 29y."},{"Start":"04:51.590 ","End":"04:55.830","Text":"I said minus 35 plus 6."},{"Start":"04:55.830 ","End":"04:58.785","Text":"Numbers go on the right,"},{"Start":"04:58.785 ","End":"05:00.630","Text":"so what I\u0027ll get,"},{"Start":"05:00.630 ","End":"05:06.535","Text":"this times this is 5 times 24 is 120,"},{"Start":"05:06.535 ","End":"05:10.700","Text":"it\u0027s a plus because of minus, minus 120,"},{"Start":"05:10.700 ","End":"05:12.425","Text":"but on the other side,"},{"Start":"05:12.425 ","End":"05:14.765","Text":"it\u0027s minus 120,"},{"Start":"05:14.765 ","End":"05:18.925","Text":"so I\u0027ve got minus 123,"},{"Start":"05:18.925 ","End":"05:23.115","Text":"and so y is equal"},{"Start":"05:23.115 ","End":"05:33.335","Text":"to 123 over 29."},{"Start":"05:33.335 ","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":"05:54.870","Text":"123 over 29 minus 24,"},{"Start":"05:54.870 ","End":"06:02.650","Text":"and I make it 165 over 29."},{"Start":"06:02.660 ","End":"06:06.370","Text":"We have a point p,"},{"Start":"06:08.660 ","End":"06:11.880","Text":"its coordinates are,"},{"Start":"06:11.880 ","End":"06:19.920","Text":"where\u0027s the x that\u0027s here, 165 over 29,"},{"Start":"06:19.920 ","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":"06:43.565","Text":"Now for the vector equation of the line 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":"07:03.500 ","End":"07:09.435","Text":"L lies in the plane pi 1."},{"Start":"07:09.435 ","End":"07:11.535","Text":"We\u0027ll get to pi 2 in a moment."},{"Start":"07:11.535 ","End":"07:13.690","Text":"L lies in pi 1,"},{"Start":"07:13.690 ","End":"07:20.800","Text":"so it\u0027s going to be perpendicular, or orthogonal to the normal to that plane."},{"Start":"07:20.800 ","End":"07:27.265","Text":"I\u0027m saying that l is the symbol for perpendicular or orthogonal,"},{"Start":"07:27.265 ","End":"07:30.610","Text":"it looks like this."},{"Start":"07:30.920 ","End":"07:39.205","Text":"The normal we\u0027re talking about for the first plane is vector minus 1,"},{"Start":"07:39.205 ","End":"07:41.990","Text":"7, minus 2,"},{"Start":"07:41.990 ","End":"07:44.795","Text":"just the coefficients of x, y, and z."},{"Start":"07:44.795 ","End":"07:50.660","Text":"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 pi 2 the second plane."},{"Start":"07:56.105 ","End":"08:00.120","Text":"L lies in that 1 also."},{"Start":"08:00.120 ","End":"08:01.510","Text":"It lies in the intersection,"},{"Start":"08:01.510 ","End":"08:02.680","Text":"it lies in each of them."},{"Start":"08:02.680 ","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":"08:32.390 ","End":"08:37.120","Text":"One way, and the most obvious way is to use the cross-product."},{"Start":"08:37.120 ","End":"08:38.785","Text":"Remember that the cross product of"},{"Start":"08:38.785 ","End":"08:45.950","Text":"two 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":"09:09.840","Text":"7, minus 2 cross,"},{"Start":"09:09.840 ","End":"09:18.830","Text":"vector cross product minus 5, 6, 3."},{"Start":"09:18.830 ","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":"09:27.055","Text":"We write i,"},{"Start":"09:27.055 ","End":"09:32.875","Text":"j, k, then we write minus 1,"},{"Start":"09:32.875 ","End":"09:37.284","Text":"7, minus 2, it\u0027s the first vector,"},{"Start":"09:37.284 ","End":"09:42.955","Text":"and then the 2nd 1 minus 5, 6, 3."},{"Start":"09:42.955 ","End":"09:49.310","Text":"Then I\u0027m going to use the method of the cofactors."},{"Start":"09:52.620 ","End":"09:54.760","Text":"For the first place,"},{"Start":"09:54.760 ","End":"09:59.410","Text":"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":"10:13.000","Text":"this 2 by 2 determinant,"},{"Start":"10:13.000 ","End":"10:17.395","Text":"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":"10:43.225","Text":"it gets a minus sign, but we also"},{"Start":"10:43.225 ","End":"10:51.010","Text":"mentally cross off the row and column, and we\u0027re left with the determinant of 4 numbers,"},{"Start":"10:51.010 ","End":"10:52.795","Text":"minus 1, minus 2,"},{"Start":"10:52.795 ","End":"10:54.595","Text":"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":"11:15.130","Text":"But, remember I said this one\u0027s a minus so it\u0027s actually plus 13."},{"Start":"11:15.130 ","End":"11:22.670","Text":"There\u0027s like a plus minus plus, when we do this method with the cofactors."},{"Start":"11:22.830 ","End":"11:25.405","Text":"We\u0027re left with the last 1,"},{"Start":"11:25.405 ","End":"11:27.070","Text":"which 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":"12:29.480","Text":"From these, it\u0027s easy to get the vector equation of the line."},{"Start":"12:31.110 ","End":"12:41.575","Text":"We get that r of t vector equals r_naught."},{"Start":"12:41.575 ","End":"12:45.310","Text":"R_naught is a particular point that would be"},{"Start":"12:45.310 ","End":"12:52.825","Text":"just the position vector of this point plus t v. In other words,"},{"Start":"12:52.825 ","End":"13:01.280","Text":"we have 165 over 29,"},{"Start":"13:01.920 ","End":"13:15.685","Text":"123 over 29, 0 plus t times"},{"Start":"13:15.685 ","End":"13:30.760","Text":"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":"13:37.210","Text":"into 1 factor,"},{"Start":"13:37.210 ","End":"13:40.540","Text":"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":"14:07.180","Text":"Notice that 5 over 29v"},{"Start":"14:07.790 ","End":"14:14.115","Text":"is 5 over 29 times this,"},{"Start":"14:14.115 ","End":"14:24.010","Text":"which would be 165 over 29, and then 5"},{"Start":"14:24.010 ","End":"14:30.130","Text":"over 29 times this would be 65 over"},{"Start":"14:30.130 ","End":"14:37.480","Text":"29 and 5 over 29 times this would be 5."},{"Start":"14:37.480 ","End":"14:42.670","Text":"What I\u0027m going to do is decompose this into 2 bits so"},{"Start":"14:42.670 ","End":"14:54.820","Text":"that this bit, I can say,"},{"Start":"14:54.820 ","End":"15:02.920","Text":"is 5 over 29 times v,"},{"Start":"15:02.920 ","End":"15:09.380","Text":"which is 33, 13, 29."},{"Start":"15:09.600 ","End":"15:13.614","Text":"The difference from here to here,"},{"Start":"15:13.614 ","End":"15:21.049","Text":"I have to add 58 over 29."},{"Start":"15:21.330 ","End":"15:23.545","Text":"I\u0027ll write 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 to see what I have to add to make them equal."},{"Start":"15:36.670 ","End":"15:46.780","Text":"The first bit is just 0, and then this minus this is 58 over"},{"Start":"15:46.780 ","End":"15:51.730","Text":"29 which is 2 and this minus"},{"Start":"15:51.730 ","End":"15:57.775","Text":"this is minus 5."},{"Start":"15:57.775 ","End":"16:08.570","Text":"Then plus t times 33, 13, 29."},{"Start":"16:10.710 ","End":"16:14.980","Text":"If I replace,"},{"Start":"16:14.980 ","End":"16:18.069","Text":"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 substitute s equals t plus 5 over 29."},{"Start":"16:30.330 ","End":"16:41.455","Text":"Just as t runs overall numbers so does s. 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":"17:04.645","Text":"I could say the equation of"},{"Start":"17:04.645 ","End":"17:09.880","Text":"the line l is"},{"Start":"17:09.880 ","End":"17:14.830","Text":"the r of s. Of course,"},{"Start":"17:14.830 ","End":"17:18.295","Text":"the particular dummy-parameter doesn\u0027t matter."},{"Start":"17:18.295 ","End":"17:22.300","Text":"It could be s, I 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":"17:38.830","Text":"The s looks a bit like a 5,"},{"Start":"17:38.830 ","End":"17:41.935","Text":"make it like an s."},{"Start":"17:41.935 ","End":"17:47.000","Text":"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":"18:12.190","Text":"I could leave this as the answer."},{"Start":"18:12.190 ","End":"18:16.420","Text":"You know what, we could just replace s with t at the end so let\u0027s write"},{"Start":"18:16.420 ","End":"18:21.460","Text":"the final answer as r of t is equal"},{"Start":"18:21.460 ","End":"18:29.200","Text":"to 33t 2 plus 13t"},{"Start":"18:29.200 ","End":"18:37.990","Text":"minus 5 plus 29t, and highlight that."},{"Start":"18:37.990 ","End":"18:40.870","Text":"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, you could have said r of t is equal"},{"Start":"18:55.330 ","End":"19:05.400","Text":"to this here, and have these ugly fractions in it."},{"Start":"19:06.770 ","End":"19:13.320","Text":"But yeah, anyway, you could, as I say,"},{"Start":"19:13.320 ","End":"19:20.700","Text":"stop here or continue, and simplify, and I think we\u0027ve had enough with this exercise."}],"ID":10680},{"Watched":false,"Name":"Exercise 9","Duration":"4m 44s","ChapterTopicVideoID":10336,"CourseChapterTopicPlaylistID":12294,"HasSubtitles":true,"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":"00:06.180","Text":"The plane is Pi, that\u0027s its name,"},{"Start":"00:06.180 ","End":"00:08.805","Text":"and it\u0027s given by this equation,"},{"Start":"00:08.805 ","End":"00:15.060","Text":"and the line is given by the vector equation of the line."},{"Start":"00:15.060 ","End":"00:19.800","Text":"What we want to know about the line and the plane is whether they\u0027re parallel,"},{"Start":"00:19.800 ","End":"00:23.139","Text":"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":"00:38.710","Text":"z as a vector 5, minus 3."},{"Start":"00:39.260 ","End":"00:50.670","Text":"Minus 6 is 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":"00:57.920","Text":"I can get a direction vector for the line."},{"Start":"00:57.920 ","End":"00:59.900","Text":"If I take V,"},{"Start":"00:59.900 ","End":"01:03.445","Text":"just the coefficients also of the t,"},{"Start":"01:03.445 ","End":"01:06.690","Text":"I can say that minus 10,"},{"Start":"01:06.690 ","End":"01:15.870","Text":"6, 12 is not a normal but"},{"Start":"01:15.870 ","End":"01:20.830","Text":"the direction vector"},{"Start":"01:23.480 ","End":"01:28.940","Text":"to the line,"},{"Start":"01:28.940 ","End":"01:33.180","Text":"so it\u0027s parallel to the line."},{"Start":"01:34.970 ","End":"01:39.140","Text":"I have a vector that\u0027s parallel to the line,"},{"Start":"01:39.140 ","End":"01:44.010","Text":"and I 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 dot 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":"02:14.010","Text":"Let\u0027s check for this V. 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":"02:23.895","Text":"minus 18 minus 72."},{"Start":"02:23.895 ","End":"02:33.710","Text":"Whatever it is, it\u0027s not 0 which means that V is not parallel to the plane,"},{"Start":"02:33.710 ","End":"02:37.920","Text":"which means that l is not parallel to the plane."},{"Start":"02:37.930 ","End":"02:43.730","Text":"V, and then hence l,"},{"Start":"02:43.730 ","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":"03:11.550","Text":"we just have to have V parallel to n. If the line is parallel to the normal,"},{"Start":"03:11.550 ","End":"03:15.005","Text":"then it\u0027s going to be perpendicular to the plane."},{"Start":"03:15.005 ","End":"03:25.350","Text":"What I want to ask, is V parallel to n?"},{"Start":"03:25.820 ","End":"03:31.265","Text":"Parallel means that 1 is going to be some scalar times the other."},{"Start":"03:31.265 ","End":"03:33.125","Text":"Now, if we look at these,"},{"Start":"03:33.125 ","End":"03:35.675","Text":"or even just looking at the first coefficient,"},{"Start":"03:35.675 ","End":"03:40.950","Text":"you see that minus 2 times 5 is minus 10,"},{"Start":"03:40.950 ","End":"03:44.010","Text":"and if I continue with this minus 2,"},{"Start":"03:44.010 ","End":"03:46.440","Text":"it will bring me from minus 3-6,"},{"Start":"03:46.440 ","End":"03:48.135","Text":"and from minus 6-12,"},{"Start":"03:48.135 ","End":"03:50.685","Text":"so the answer is yes,"},{"Start":"03:50.685 ","End":"03:54.750","Text":"because V is minus"},{"Start":"03:54.750 ","End":"04:03.330","Text":"2 times n. 1"},{"Start":"04:03.330 ","End":"04:05.340","Text":"is the scalar times the other,"},{"Start":"04:05.340 ","End":"04:13.685","Text":"and so l is perpendicular, orthogonal, normal."},{"Start":"04:13.685 ","End":"04:15.725","Text":"You can also write it like this."},{"Start":"04:15.725 ","End":"04:18.350","Text":"L is perpendicular to Pi."},{"Start":"04:18.350 ","End":"04:25.660","Text":"[inaudible] Do it orthogonal when we\u0027re using that most orthogonal."},{"Start":"04:26.750 ","End":"04:34.480","Text":"l and Pi are orthogonal."},{"Start":"04:34.760 ","End":"04:38.910","Text":"I used the word perpendicular, it\u0027s same thing."},{"Start":"04:38.910 ","End":"04:43.560","Text":"Perpendicular. That\u0027s the answer."}],"ID":10681}],"Thumbnail":null,"ID":12294},{"Name":"Quadric Surfaces","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The 3D Coordinate System - Quadric Surfaces","Duration":"14m 25s","ChapterTopicVideoID":10338,"CourseChapterTopicPlaylistID":12295,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.690","Text":"Done with the cone, now the cylinder."},{"Start":"00:03.690 ","End":"00:05.760","Text":"I guess I should have mentioned it earlier."},{"Start":"00:05.760 ","End":"00:09.105","Text":"There are actually several kinds of cylinder,"},{"Start":"00:09.105 ","End":"00:12.810","Text":"the 2 main kinds are, well,"},{"Start":"00:12.810 ","End":"00:15.460","Text":"just like with the paraboloid,"},{"Start":"00:17.510 ","End":"00:19.845","Text":"just a quick copy-paste,"},{"Start":"00:19.845 ","End":"00:22.410","Text":"there is elliptic and hyperbolic."},{"Start":"00:22.410 ","End":"00:26.360","Text":"Actually, you can get more specialized in"},{"Start":"00:26.360 ","End":"00:31.310","Text":"the elliptic as you already know special case of an ellipse is a circle."},{"Start":"00:31.310 ","End":"00:34.790","Text":"We also have a circular cylinder,"},{"Start":"00:34.790 ","End":"00:39.050","Text":"which is the usual cylinder, 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":"00:52.450","Text":"In fact, we\u0027ve been seeing this all along even with the ellipsoid we\u0027ve had the sphere,"},{"Start":"00:52.450 ","End":"00:57.180","Text":"certainly with the cone, we\u0027ve had elliptical and circular."},{"Start":"00:58.000 ","End":"01:02.485","Text":"Let\u0027s first of all start with a picture."},{"Start":"01:02.485 ","End":"01:05.365","Text":"I brought all 3 pictures at once."},{"Start":"01:05.365 ","End":"01:12.350","Text":"This is the elliptic cylinder"},{"Start":"01:12.350 ","End":"01:15.550","Text":"and this is the circular,"},{"Start":"01:15.550 ","End":"01:18.200","Text":"which is a special case."},{"Start":"01:18.590 ","End":"01:22.080","Text":"Just like a circle is a special case of an ellipse."},{"Start":"01:22.080 ","End":"01:26.395","Text":"This 1 is the hyperbolic cylinder,"},{"Start":"01:26.395 ","End":"01:29.305","Text":"less commonly heard about."},{"Start":"01:29.305 ","End":"01:35.815","Text":"Actually, they have unusual equations in 3D."},{"Start":"01:35.815 ","End":"01:39.310","Text":"Because if we take the 2D equations,"},{"Start":"01:39.310 ","End":"01:42.790","Text":"in fact, there\u0027s some the idea of making a table."},{"Start":"01:42.790 ","End":"01:46.665","Text":"Let me write here ellipse,"},{"Start":"01:46.665 ","End":"01:49.365","Text":"circle, and hyperbola."},{"Start":"01:49.365 ","End":"01:56.215","Text":"Then I write the equation for each of these and the ellipse,"},{"Start":"01:56.215 ","End":"01:57.895","Text":"we already did this before,"},{"Start":"01:57.895 ","End":"02:00.310","Text":"would be, let\u0027s see,"},{"Start":"02:00.310 ","End":"02:09.150","Text":"it was x squared over a squared plus y squared over b squared equals 1."},{"Start":"02:09.150 ","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":"02:20.470","Text":"we get x squared plus y squared equals r-squared."},{"Start":"02:20.470 ","End":"02:26.859","Text":"The hyperbola, you may or may not have studied it in 2D."},{"Start":"02:26.859 ","End":"02:30.245","Text":"The hyperbola has the equation,"},{"Start":"02:30.245 ","End":"02:37.880","Text":"x squared 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":"03:02.940","Text":"that\u0027s this curve here that has 2 branches,"},{"Start":"03:02.940 ","End":"03:06.280","Text":"I think they\u0027re called sheets here."},{"Start":"03:08.570 ","End":"03:11.825","Text":"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":"03:28.975","Text":"We bring them in 1 particular orientation like this."},{"Start":"03:28.975 ","End":"03:35.900","Text":"We\u0027ll just interchange the variables to get them in the other orientations."},{"Start":"03:35.900 ","End":"03:42.715","Text":"I just wanted to give you an idea that this is a hyperbola."},{"Start":"03:42.715 ","End":"03:45.045","Text":"Now I\u0027m coming to the point."},{"Start":"03:45.045 ","End":"03:49.065","Text":"These 2D shapes."},{"Start":"03:49.065 ","End":"03:52.255","Text":"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":"04:00.085","Text":"How can that be? Well, there\u0027s a missing z in 3D."},{"Start":"04:00.085 ","End":"04:01.960","Text":"This is an equation in x, y, and z,"},{"Start":"04:01.960 ","End":"04:04.735","Text":"which just happens to be without z."},{"Start":"04:04.735 ","End":"04:09.395","Text":"Then these 3 things become the elliptic,"},{"Start":"04:09.395 ","End":"04:14.069","Text":"circular, and hyperbolic cylinder."},{"Start":"04:14.069 ","End":"04:17.645","Text":"I wrote that. The reason of this no z,"},{"Start":"04:17.645 ","End":"04:21.790","Text":"the implication is that if we take a curve in 2-dimensions,"},{"Start":"04:21.790 ","End":"04:23.440","Text":"I\u0027ll just do a quick freehand."},{"Start":"04:23.440 ","End":"04:25.110","Text":"I mean, if whether we have the ellipse,"},{"Start":"04:25.110 ","End":"04:27.215","Text":"whether we have the circle,"},{"Start":"04:27.215 ","End":"04:32.905","Text":"or whether we have the hyperbola in the plane."},{"Start":"04:32.905 ","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":"04:48.349","Text":"It\u0027s just that we raise vertical lines through, z is unbounded,"},{"Start":"04:48.349 ","End":"04:51.160","Text":"which means like taking a shape in"},{"Start":"04:51.160 ","End":"04:56.240","Text":"2-dimensions are 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":"05:04.850","Text":"any curve in the plane."},{"Start":"05:04.850 ","End":"05:09.980","Text":"Then we just give it a third dimension by raising the z."},{"Start":"05:10.340 ","End":"05:13.340","Text":"The same equations in 2D,"},{"Start":"05:13.340 ","End":"05:21.290","Text":"they mean 1 thing and in 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":"05:34.545","Text":"That would be z is the axis which this shape is centered around so to speak."},{"Start":"05:34.545 ","End":"05:37.330","Text":"Of course, 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":"05:48.660","Text":"One like this and the other like this."},{"Start":"05:48.950 ","End":"05:55.410","Text":"Done with cylinder, we\u0027re going to move on next to the hyperboloid. No, wait."},{"Start":"05:55.410 ","End":"05:56.780","Text":"There\u0027s 1 kind I forgot."},{"Start":"05:56.780 ","End":"05:58.790","Text":"I suddenly remember it\u0027s not very common."},{"Start":"05:58.790 ","End":"06:02.855","Text":"But there is actually a thing called a parabolic cylinder."},{"Start":"06:02.855 ","End":"06:04.445","Text":"If I take the equation,"},{"Start":"06:04.445 ","End":"06:10.130","Text":"any equation of a parabola in the plane,"},{"Start":"06:10.130 ","End":"06:11.600","Text":"y equals x squared,"},{"Start":"06:11.600 ","End":"06:15.150","Text":"I prefer to take x equals y squared,"},{"Start":"06:15.590 ","End":"06:20.900","Text":"have it on the side and without any z."},{"Start":"06:20.900 ","End":"06:23.719","Text":"Then this is a parabola in the plane,"},{"Start":"06:23.719 ","End":"06:33.390","Text":"but here it becomes a parabolic cylinder and I\u0027ll throw in the picture."},{"Start":"06:33.390 ","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":"06:40.190","Text":"but just for completeness sake."},{"Start":"06:40.190 ","End":"06:43.540","Text":"Now we can go on to the hyperboloid."},{"Start":"06:43.540 ","End":"06:47.180","Text":"Here we are. Next one is hyperboloid."},{"Start":"06:47.180 ","End":"06:51.690","Text":"There\u0027s 2 kinds, 1 sheet, and 2 sheet."},{"Start":"06:51.690 ","End":"06:56.435","Text":"Actually, it can also be elliptic or circular,"},{"Start":"06:56.435 ","End":"07:00.005","Text":"just assuming the elliptic and the circular will be a special case."},{"Start":"07:00.005 ","End":"07:02.375","Text":"If I write that here, have room,"},{"Start":"07:02.375 ","End":"07:05.105","Text":"so also can be elliptic or circular."},{"Start":"07:05.105 ","End":"07:07.520","Text":"Now before I give you the pictures,"},{"Start":"07:07.520 ","End":"07:09.350","Text":"diagrams for each of these,"},{"Start":"07:09.350 ","End":"07:12.900","Text":"I want to just look again in 2-dimensions."},{"Start":"07:12.900 ","End":"07:14.770","Text":"We\u0027ve had this picture before,"},{"Start":"07:14.770 ","End":"07:16.660","Text":"there\u0027s way too much information on it."},{"Start":"07:16.660 ","End":"07:18.250","Text":"Just wants you to look at the general shape,"},{"Start":"07:18.250 ","End":"07:20.980","Text":"but there were 2 variations and the plane."},{"Start":"07:20.980 ","End":"07:25.510","Text":"We had this and this depending on,"},{"Start":"07:25.510 ","End":"07:28.330","Text":"well, we discussed this."},{"Start":"07:28.330 ","End":"07:33.490","Text":"Now, I\u0027d like to imagine that this,"},{"Start":"07:33.490 ","End":"07:36.760","Text":"It\u0027s not like the x-axis and this is the y-axis."},{"Start":"07:36.760 ","End":"07:42.820","Text":"I want you to imagine this as being the whole x, y plane."},{"Start":"07:42.820 ","End":"07:46.525","Text":"This will be the z-axis in 3D,"},{"Start":"07:46.525 ","End":"07:47.815","Text":"like looking at the side,"},{"Start":"07:47.815 ","End":"07:51.950","Text":"this whole thing is a plane and the same here,"},{"Start":"07:51.950 ","End":"07:56.955","Text":"this will be the x, y plane and this will be the z-axis."},{"Start":"07:56.955 ","End":"08:03.070","Text":"Now, imagine that we rotate this shape around this axis."},{"Start":"08:03.070 ","End":"08:05.229","Text":"If you rotate this shape,"},{"Start":"08:05.229 ","End":"08:11.290","Text":"we will get the 1 sheeted hyperboloid that you will see in a moment,"},{"Start":"08:11.290 ","End":"08:14.215","Text":"but it only has 1 bit still connected."},{"Start":"08:14.215 ","End":"08:19.570","Text":"But if I rotate this about the z-axis,"},{"Start":"08:19.570 ","End":"08:22.660","Text":"I\u0027ll have 1 bit here and 1 bit here."},{"Start":"08:22.660 ","End":"08:25.885","Text":"Let\u0027s see how this looks in 3D."},{"Start":"08:25.885 ","End":"08:28.495","Text":"Push this to the side,"},{"Start":"08:28.495 ","End":"08:34.015","Text":"and here they are, the 2 hyperboloids."},{"Start":"08:34.015 ","End":"08:37.405","Text":"Obviously, this is the 1 sheeted and this is the 2 sheeted."},{"Start":"08:37.405 ","End":"08:39.985","Text":"We\u0027ll assume that this is elliptical."},{"Start":"08:39.985 ","End":"08:48.010","Text":"Not necessarily circular like I implied if we just did a rotation and both of them,"},{"Start":"08:48.010 ","End":"08:49.840","Text":"the center is the z-axis."},{"Start":"08:49.840 ","End":"08:52.540","Text":"The z-axis is the exceptional one."},{"Start":"08:52.540 ","End":"08:59.590","Text":"The equation is that x squared over"},{"Start":"08:59.590 ","End":"09:03.490","Text":"a squared plus y squared over b squared"},{"Start":"09:03.490 ","End":"09:10.315","Text":"minus z squared over c squared equals 1."},{"Start":"09:10.315 ","End":"09:12.670","Text":"See the odd one out is the z,"},{"Start":"09:12.670 ","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":"09:21.035","Text":"Now, for the 2 sheeted one,"},{"Start":"09:21.035 ","End":"09:27.630","Text":"actually the equation is not hard to remember wherever you see on the left,"},{"Start":"09:27.630 ","End":"09:29.625","Text":"a plus, you put a minus, and vice versa."},{"Start":"09:29.625 ","End":"09:36.089","Text":"We have minus x squared over a squared minus y squared"},{"Start":"09:36.089 ","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":"09:59.660","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":"10:19.780","Text":"I think I want to go straight on to the paraboloid."},{"Start":"10:19.780 ","End":"10:21.565","Text":"Okay, paraboloid."},{"Start":"10:21.565 ","End":"10:23.769","Text":"Let\u0027s start with the elliptic, circular."},{"Start":"10:23.769 ","End":"10:26.500","Text":"I already put the picture here,"},{"Start":"10:26.500 ","End":"10:31.165","Text":"and let me give you the equation."},{"Start":"10:31.165 ","End":"10:34.045","Text":"The equation for this will be,"},{"Start":"10:34.045 ","End":"10:35.845","Text":"I\u0027ll write it over here,"},{"Start":"10:35.845 ","End":"10:43.375","Text":"x squared over a squared plus y squared"},{"Start":"10:43.375 ","End":"10:52.435","Text":"over b squared equals z over c. Now,"},{"Start":"10:52.435 ","End":"10:58.570","Text":"z is the exceptional one and it certainly is centered around the z-axis,"},{"Start":"10:58.570 ","End":"11:01.765","Text":"and as usual, we can rearrange the letters if you want it differently."},{"Start":"11:01.765 ","End":"11:05.950","Text":"The thing is that it actually makes a difference if c is positive or negative."},{"Start":"11:05.950 ","End":"11:11.335","Text":"I have just mentioned that if c is bigger than 0, it\u0027s like this,"},{"Start":"11:11.335 ","End":"11:13.420","Text":"and if c is less than 0,"},{"Start":"11:13.420 ","End":"11:15.010","Text":"it just faces down,"},{"Start":"11:15.010 ","End":"11:20.270","Text":"just like we have a parabola that concaves up or concaves down."},{"Start":"11:22.290 ","End":"11:26.770","Text":"This is just a mnemonic to indicate that 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":"11:34.420","Text":"If a equals b,"},{"Start":"11:34.420 ","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":"11:45.865","Text":"If a equals b is equal to r,"},{"Start":"11:45.865 ","End":"11:51.175","Text":"then it\u0027s a circular rather than"},{"Start":"11:51.175 ","End":"11:59.060","Text":"an elliptic paraboloid, circular paraboloid."},{"Start":"12:00.300 ","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":"12:18.685","Text":"Notice that if I take sections with vertical planes,"},{"Start":"12:18.685 ","End":"12:25.300","Text":"if I take sections parallel to the x,"},{"Start":"12:25.300 ","End":"12:30.100","Text":"z plane, I\u0027m going to get parabolas."},{"Start":"12:30.100 ","End":"12:33.040","Text":"All these lines are parabolas that are facing up."},{"Start":"12:33.040 ","End":"12:36.760","Text":"On the other hand, if I take planes parallel to the z,"},{"Start":"12:36.760 ","End":"12:39.310","Text":"y plane, 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":"12:48.580","Text":"I mean parallel to the x, y plane,"},{"Start":"12:48.580 ","End":"12:53.980","Text":"I\u0027ll get hyperbolas on the top half of the space above."},{"Start":"12:53.980 ","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":"13:00.685","Text":"the branches and on the other way,"},{"Start":"13:00.685 ","End":"13:02.050","Text":"if I take it below the x,"},{"Start":"13:02.050 ","End":"13:04.850","Text":"y plane, they\u0027ll go the other way."},{"Start":"13:04.890 ","End":"13:08.470","Text":"Well, anyway, you just have to imagine it."},{"Start":"13:08.470 ","End":"13:12.850","Text":"I\u0027ll give you the equation of this one."},{"Start":"13:12.850 ","End":"13:15.835","Text":"It\u0027s actually very similar to this one."},{"Start":"13:15.835 ","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 difference 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":"13:50.980","Text":"the way reverses instead of going a parabola up in the x,"},{"Start":"13:50.980 ","End":"13:54.580","Text":"z plane and a downward parabola in the y,"},{"Start":"13:54.580 ","End":"13:56.875","Text":"z plane, it will be the other way round,"},{"Start":"13:56.875 ","End":"14:02.005","Text":"but the general shape is the same and the z is the odd one out because it"},{"Start":"14:02.005 ","End":"14:09.950","Text":"is the center of symmetry, the z-axis."},{"Start":"14:11.310 ","End":"14:13.720","Text":"I think we\u0027re done."},{"Start":"14:13.720 ","End":"14:19.435","Text":"The only intention was a preliminary acquaintance with all these quadric surfaces,"},{"Start":"14:19.435 ","End":"14:22.555","Text":"and they\u0027re actually even others that are variations on this."},{"Start":"14:22.555 ","End":"14:26.540","Text":"I\u0027m declaring this subject to be done."}],"ID":10682},{"Watched":false,"Name":"Exercise 1","Duration":"3m 41s","ChapterTopicVideoID":10339,"CourseChapterTopicPlaylistID":12295,"HasSubtitles":true,"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":"00:12.030","Text":"For that you need software,"},{"Start":"00:12.030 ","End":"00:14.054","Text":"and 3D is complicated."},{"Start":"00:14.054 ","End":"00:16.950","Text":"There are exercises that really more for me to"},{"Start":"00:16.950 ","End":"00:19.710","Text":"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 1 for example."},{"Start":"00:28.435 ","End":"00:30.810","Text":"Normally, they\u0027d be x, y, and z."},{"Start":"00:30.810 ","End":"00:32.700","Text":"But here there\u0027s only y and z,"},{"Start":"00:32.700 ","End":"00:34.925","Text":"so there\u0027s only 2 variables."},{"Start":"00:34.925 ","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":"00:50.270","Text":"This is a sketch of a 2D ellipse in x and"},{"Start":"00:50.270 ","End":"00:56.750","Text":"y. I want to use this to give you an idea of how to draw something like this."},{"Start":"00:56.750 ","End":"00:58.040","Text":"Now, for 1 thing,"},{"Start":"00:58.040 ","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":"01:10.145","Text":"and here to y and z."},{"Start":"01:10.145 ","End":"01:12.770","Text":"This is the equation of an ellipse."},{"Start":"01:12.770 ","End":"01:16.115","Text":"In our case, we have, well,"},{"Start":"01:16.115 ","End":"01:19.370","Text":"I could write this as z squared over 1 squared,"},{"Start":"01:19.370 ","End":"01:21.845","Text":"so it looks even more like this."},{"Start":"01:21.845 ","End":"01:26.939","Text":"A is like the big radius,"},{"Start":"01:26.939 ","End":"01:29.340","Text":"and b is the small radius."},{"Start":"01:29.340 ","End":"01:35.080","Text":"An ellipse has 2 radii."},{"Start":"01:36.770 ","End":"01:41.860","Text":"This a is 3 and this b is 1."},{"Start":"01:42.260 ","End":"01:45.680","Text":"This is what it looks like in 2D."},{"Start":"01:45.680 ","End":"01:47.690","Text":"But remember we\u0027re now in 3D,"},{"Start":"01:47.690 ","End":"01:50.240","Text":"and just x happens to be missing."},{"Start":"01:50.240 ","End":"01:57.140","Text":"All you have to do is take this ellipse and extend it upwards and"},{"Start":"01:57.140 ","End":"01:59.570","Text":"downwards infinitely like in"},{"Start":"01:59.570 ","End":"02:05.640","Text":"vertical straight lines vertical to the plane of the drawing."},{"Start":"02:06.470 ","End":"02:15.215","Text":"Best thing I can do is just give you a 3-dimensional sketch that was made by computer."},{"Start":"02:15.215 ","End":"02:17.180","Text":"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":"02:21.480","Text":"these 2 points,"},{"Start":"02:21.480 ","End":"02:24.435","Text":"I marked these 2 points here."},{"Start":"02:24.435 ","End":"02:29.120","Text":"This is where y is 3 factor,"},{"Start":"02:29.120 ","End":"02:34.675","Text":"this point here would be the point x is 0,"},{"Start":"02:34.675 ","End":"02:38.475","Text":"y is 3, z is 0."},{"Start":"02:38.475 ","End":"02:41.235","Text":"This point would be the point."},{"Start":"02:41.235 ","End":"02:45.825","Text":"Well, x is again 0 because we\u0027re in the z,y plane."},{"Start":"02:45.825 ","End":"02:50.835","Text":"This time y is 0 and z is 1."},{"Start":"02:50.835 ","End":"02:58.360","Text":"Is there any place we can make a cross-section parallel to the z,"},{"Start":"02:58.360 ","End":"03:01.235","Text":"y plane and get this ellipse."},{"Start":"03:01.235 ","End":"03:06.000","Text":"The ellipse extends infinitely in the x"},{"Start":"03:06.000 ","End":"03:11.400","Text":"direction like the x-axis is perpendicular to this."},{"Start":"03:11.400 ","End":"03:16.030","Text":"Anyway, that\u0027s just to give you an idea."},{"Start":"03:16.580 ","End":"03:22.970","Text":"The most important thing here was that because we had a variable missing x,"},{"Start":"03:22.970 ","End":"03:26.810","Text":"we could sketch it in the plane of the other 2 variables, y,"},{"Start":"03:26.810 ","End":"03:30.845","Text":"z, and then extend it infinitely"},{"Start":"03:30.845 ","End":"03:37.280","Text":"in the direction perpendicular to the plane of the sketch, like so."},{"Start":"03:37.280 ","End":"03:40.260","Text":"That\u0027s all for this 1."}],"ID":10683},{"Watched":false,"Name":"Exercise 2","Duration":"3m 2s","ChapterTopicVideoID":10340,"CourseChapterTopicPlaylistID":12295,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.870","Text":"This exercise is not really an exercise."},{"Start":"00:03.870 ","End":"00:08.280","Text":"I don\u0027t expect you to graph surfaces in 3D,"},{"Start":"00:08.280 ","End":"00:14.830","Text":"I\u0027m just getting you more familiar with these quadric surfaces."},{"Start":"00:14.900 ","End":"00:21.840","Text":"I look at this formula and it reminds me of something more general."},{"Start":"00:21.840 ","End":"00:25.240","Text":"This is what I\u0027m talking about."},{"Start":"00:25.430 ","End":"00:34.545","Text":"It definitely fits this where a is equal to square root of 4 is 2,"},{"Start":"00:34.545 ","End":"00:38.010","Text":"b is 3,"},{"Start":"00:38.010 ","End":"00:43.755","Text":"and c is the square root of 6."},{"Start":"00:43.755 ","End":"00:51.640","Text":"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":"01:00.580","Text":"then it would be a sphere,"},{"Start":"01:00.580 ","End":"01:04.580","Text":"even though a sphere is also a special kind of ellipsoid."},{"Start":"01:04.580 ","End":"01:08.300","Text":"But in this case, these numbers are not all the same,"},{"Start":"01:08.300 ","End":"01:11.790","Text":"so it\u0027s a real ellipsoid."},{"Start":"01:12.650 ","End":"01:16.440","Text":"I\u0027ll just show you what it looks like."},{"Start":"01:16.440 ","End":"01:19.500","Text":"These numbers, 2, 3,"},{"Start":"01:19.500 ","End":"01:21.135","Text":"square root of 6,"},{"Start":"01:21.135 ","End":"01:25.410","Text":"can be seen in the sketch."},{"Start":"01:25.410 ","End":"01:34.485","Text":"The 2 would be where the ellipsoid cuts the x-axis."},{"Start":"01:34.485 ","End":"01:37.390","Text":"This point here,"},{"Start":"01:38.900 ","End":"01:45.780","Text":"this would be 2, 0, 0,"},{"Start":"01:45.780 ","End":"01:48.630","Text":"that\u0027s where the x cuts the a."},{"Start":"01:48.630 ","End":"01:54.195","Text":"This point here would be 0,"},{"Start":"01:54.195 ","End":"02:04.820","Text":"3, 0, and this point here would be 0, 0, root 6."},{"Start":"02:04.820 ","End":"02:07.655","Text":"That\u0027s the meaning of these a, b, and c,"},{"Start":"02:07.655 ","End":"02:13.260","Text":"it\u0027s where the ellipsoid cuts the axis,"},{"Start":"02:13.260 ","End":"02:16.730","Text":"each one at its own value,"},{"Start":"02:16.730 ","End":"02:20.160","Text":"a, b, or c respectively."},{"Start":"02:20.780 ","End":"02:23.905","Text":"Nothing much more to say,"},{"Start":"02:23.905 ","End":"02:28.340","Text":"except perhaps I\u0027ll note that these lines that are drawn on"},{"Start":"02:28.340 ","End":"02:34.035","Text":"the ellipsoid are actually contours or level curves."},{"Start":"02:34.035 ","End":"02:40.065","Text":"The ones going this way are where the value of z is constant"},{"Start":"02:40.065 ","End":"02:47.145","Text":"and the ones going this way are where y is constant,"},{"Start":"02:47.145 ","End":"02:50.955","Text":"and the ones, like these,"},{"Start":"02:50.955 ","End":"02:55.925","Text":"are where the values of x are constant."},{"Start":"02:55.925 ","End":"03:02.280","Text":"Anyway, that\u0027s neither here nor there. That\u0027s it."}],"ID":10684},{"Watched":false,"Name":"Exercise 3","Duration":"4m 27s","ChapterTopicVideoID":10341,"CourseChapterTopicPlaylistID":12295,"HasSubtitles":true,"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":"00:49.325","Text":"I find this 1 which really fits this except for the bit with the minus 6."},{"Start":"00:49.325 ","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":"01:09.785","Text":"well, this is like z over 1,"},{"Start":"01:09.785 ","End":"01:13.590","Text":"so c equals 1."},{"Start":"01:14.210 ","End":"01:19.725","Text":"In general, the shape of this graph,"},{"Start":"01:19.725 ","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":"01:46.670","Text":"a squared plus x squared over b squared equals y over c,"},{"Start":"01:46.670 ","End":"01:49.205","Text":"then y would be the odd 1 out."},{"Start":"01:49.205 ","End":"01:52.190","Text":"The odd 1 out is the axis that it\u0027s centered"},{"Start":"01:52.190 ","End":"01:55.190","Text":"along and in this case z is the odd 1 out so,"},{"Start":"01:55.190 ","End":"01:58.115","Text":"z is the axis it\u0027s centered on."},{"Start":"01:58.115 ","End":"02:00.065","Text":"The sine of c,"},{"Start":"02:00.065 ","End":"02:01.970","Text":"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":"02:14.330","Text":"this is what it\u0027s going to look like with z and a positive c."},{"Start":"02:14.330 ","End":"02:22.715","Text":"This shape is called an elliptic paraboloid."},{"Start":"02:22.715 ","End":"02:31.575","Text":"I\u0027ll write that, elliptic"},{"Start":"02:31.575 ","End":"02:34.185","Text":"and paraboloid."},{"Start":"02:34.185 ","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":"02:49.970","Text":"you can see, in ellipses."},{"Start":"02:49.970 ","End":"02:54.235","Text":"Now, in our particular case,"},{"Start":"02:54.235 ","End":"02:57.900","Text":"we have that a equals b,"},{"Start":"02:57.900 ","End":"03:00.970","Text":"so 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":"03:06.635","Text":"Well, a circle is a special case of an ellipse,"},{"Start":"03:06.635 ","End":"03:10.970","Text":"but then it\u0027s circular cross-sections."},{"Start":"03:10.970 ","End":"03:14.165","Text":"But I don\u0027t know if there\u0027s a word circular paraboloid."},{"Start":"03:14.165 ","End":"03:16.300","Text":"I suppose it could be."},{"Start":"03:16.300 ","End":"03:18.480","Text":"That\u0027s what we have in our case,"},{"Start":"03:18.480 ","End":"03:19.730","Text":"the cross-sections are circles."},{"Start":"03:19.730 ","End":"03:22.820","Text":"Now, 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 lower 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":"03:58.480","Text":"normally the apex,"},{"Start":"03:58.480 ","End":"04:00.155","Text":"is at the origin,"},{"Start":"04:00.155 ","End":"04:04.880","Text":"but here the apex is moved 6 units down,"},{"Start":"04:04.880 ","End":"04:09.485","Text":"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, don\u0027t want to get into any more detail than that."},{"Start":"04:18.425 ","End":"04:23.400","Text":"Here we have a elliptical or circular paraboloid"},{"Start":"04:23.400 ","End":"04:28.660","Text":"that\u0027s being shifted down 6 units. That\u0027s it."}],"ID":10685},{"Watched":false,"Name":"Exercise 4","Duration":"2m 42s","ChapterTopicVideoID":10342,"CourseChapterTopicPlaylistID":12295,"HasSubtitles":true,"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 quadratic surfaces."},{"Start":"00:13.125 ","End":"00:18.540","Text":"Now, I look among the list of surfaces that we\u0027ve studied,"},{"Start":"00:18.540 ","End":"00:22.950","Text":"and the closest 1 to this is the following,"},{"Start":"00:22.950 ","End":"00:24.855","Text":"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":"00:34.430","Text":"it\u0027s the y variable that\u0027s the odd 1 out and it\u0027s on the other side,"},{"Start":"00:34.430 ","End":"00:37.095","Text":"whereas here it\u0027s the z variable."},{"Start":"00:37.095 ","End":"00:45.855","Text":"This would be actually a cone that is centered along the z-axis."},{"Start":"00:45.855 ","End":"00:48.360","Text":"We\u0027ll have to rotate the variables,"},{"Start":"00:48.360 ","End":"00:51.060","Text":"because we have y here,"},{"Start":"00:51.060 ","End":"00:54.345","Text":"and ours will be centered on the y-axis."},{"Start":"00:54.345 ","End":"00:57.090","Text":"This 1 looks in general,"},{"Start":"00:57.090 ","End":"00:59.505","Text":"something like this,"},{"Start":"00:59.505 ","End":"01:01.460","Text":"and as I said,"},{"Start":"01:01.460 ","End":"01:05.840","Text":"it\u0027s centered along the z-axis because that\u0027s the odd 1 out variable,"},{"Start":"01:05.840 ","End":"01:10.790","Text":"and the other thing is that if A and B happen to be equal,"},{"Start":"01:10.790 ","End":"01:13.400","Text":"then these cross sections are circles,"},{"Start":"01:13.400 ","End":"01:15.280","Text":"otherwise, they\u0027re ellipses."},{"Start":"01:15.280 ","End":"01:19.940","Text":"You could say that this in general is an elliptic cone,"},{"Start":"01:19.940 ","End":"01:21.485","Text":"or a circular cone."},{"Start":"01:21.485 ","End":"01:25.655","Text":"In our case, what we have is,"},{"Start":"01:25.655 ","End":"01:30.480","Text":"we have x squared over a squared, in our case,"},{"Start":"01:30.480 ","End":"01:31.890","Text":"a would be a 1/2,"},{"Start":"01:31.890 ","End":"01:34.940","Text":"we could write this x squared over a 1/2 squared,"},{"Start":"01:34.940 ","End":"01:37.190","Text":"putting the 4 and the denominator,"},{"Start":"01:37.190 ","End":"01:43.050","Text":"plus z squared over 1 1/4"},{"Start":"01:43.050 ","End":"01:48.660","Text":"squared equals y squared over 1 squared."},{"Start":"01:48.660 ","End":"01:50.265","Text":"We have our a, b, and c,"},{"Start":"01:50.265 ","End":"01:52.995","Text":"except that y is the odd 1 out."},{"Start":"01:52.995 ","End":"01:54.840","Text":"These 2 are different,"},{"Start":"01:54.840 ","End":"01:55.960","Text":"like the a and the b,"},{"Start":"01:55.960 ","End":"01:59.075","Text":"so the cross sections are going to be elliptic,"},{"Start":"01:59.075 ","End":"02:01.175","Text":"and as I mentioned,"},{"Start":"02:01.175 ","End":"02:03.920","Text":"this is centered on the z-axis,"},{"Start":"02:03.920 ","End":"02:06.290","Text":"so this 1 is centered along the y-axis,"},{"Start":"02:06.290 ","End":"02:09.350","Text":"and I\u0027ll just 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":"02:14.750","Text":"that\u0027s what it\u0027s centered on,"},{"Start":"02:14.750 ","End":"02:16.130","Text":"and these are not circles,"},{"Start":"02:16.130 ","End":"02:18.980","Text":"they are ellipses because 1/2 is not equal to a 1/4,"},{"Start":"02:18.980 ","End":"02:21.035","Text":"a is not equal to b."},{"Start":"02:21.035 ","End":"02:24.230","Text":"Not expected to go any deeper into this,"},{"Start":"02:24.230 ","End":"02:29.540","Text":"just to have a general idea of changing the variables round,"},{"Start":"02:29.540 ","End":"02:32.850","Text":"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":10686},{"Watched":false,"Name":"Exercise 5","Duration":"4m 58s","ChapterTopicVideoID":10343,"CourseChapterTopicPlaylistID":12295,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.470","Text":"In this exercise, we have to sketch the following quadric surface,"},{"Start":"00:07.470 ","End":"00:12.810","Text":"but as in the other exercises of this kind you are not really expected to sketch,"},{"Start":"00:12.810 ","End":"00:16.500","Text":"these are more exercises to familiarize you with"},{"Start":"00:16.500 ","End":"00:21.060","Text":"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":"00:44.390","Text":"but if you make a few changes than it would."},{"Start":"00:44.390 ","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":"00:57.565","Text":"In our case, x is the odd one out,"},{"Start":"00:57.565 ","End":"01:00.740","Text":"the other thing is that there are signs here,"},{"Start":"01:00.740 ","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":"01:10.650","Text":"Denominator is another problem I could easily rewrite that,"},{"Start":"01:10.650 ","End":"01:15.020","Text":"in fact, why don\u0027t I just rewrite this a little bit with x on this side."},{"Start":"01:15.020 ","End":"01:21.470","Text":"I could write that x over negative 1,"},{"Start":"01:21.470 ","End":"01:22.640","Text":"if I do that,"},{"Start":"01:22.640 ","End":"01:29.280","Text":"then I can make these plus is equal to,"},{"Start":"01:29.650 ","End":"01:33.715","Text":"it will be minus 4,"},{"Start":"01:33.715 ","End":"01:36.795","Text":"which is not really part of this,"},{"Start":"01:36.795 ","End":"01:44.260","Text":"but plus y squared over root 5,"},{"Start":"01:45.020 ","End":"01:50.495","Text":"sorry, 1 over root 5 squared."},{"Start":"01:50.495 ","End":"01:54.820","Text":"I\u0027m just forcing this to be y squared over something squared and"},{"Start":"01:54.820 ","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 is 1/9,"},{"Start":"02:03.865 ","End":"02:06.345","Text":"which is 1/3 squared."},{"Start":"02:06.345 ","End":"02:07.860","Text":"I have my c,"},{"Start":"02:07.860 ","End":"02:09.885","Text":"my a, and my b,"},{"Start":"02:09.885 ","End":"02:17.665","Text":"and I also see that x is the odd one out and they\u0027re still the matter of the minus 4."},{"Start":"02:17.665 ","End":"02:20.725","Text":"Now, the shape of this,"},{"Start":"02:20.725 ","End":"02:23.885","Text":"if c is positive,"},{"Start":"02:23.885 ","End":"02:31.575","Text":"then it looks like this and 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":"02:41.030","Text":"Here our a and b are different,"},{"Start":"02:41.030 ","End":"02:43.295","Text":"it\u0027s going be elliptical."},{"Start":"02:43.295 ","End":"02:46.530","Text":"The other thing is that this is the c is positive,"},{"Start":"02:46.530 ","End":"02:48.290","Text":"when c is negative,"},{"Start":"02:48.290 ","End":"02:54.280","Text":"it goes the other way around, it goes downwards."},{"Start":"02:54.280 ","End":"02:56.255","Text":"Now, in our case,"},{"Start":"02:56.255 ","End":"02:58.540","Text":"we also have to remember several changes,"},{"Start":"02:58.540 ","End":"03:01.055","Text":"we have to make x the center."},{"Start":"03:01.055 ","End":"03:06.410","Text":"It\u0027s going to face in the negative x direction because of this minus,"},{"Start":"03:06.410 ","End":"03:11.675","Text":"but the other thing is the minus 4 will just mean that we have to shift it"},{"Start":"03:11.675 ","End":"03:17.300","Text":"4 units along the x-axis in the negative x-direction,"},{"Start":"03:17.300 ","End":"03:22.680","Text":"and what we end up with is something like this."},{"Start":"03:22.680 ","End":"03:27.750","Text":"Note, the center is the x-axis"},{"Start":"03:27.750 ","End":"03:36.150","Text":"and 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":"03:56.990","Text":"I think I may have confused you earlier with the minus 4,"},{"Start":"03:56.990 ","End":"04:00.860","Text":"really what I should have been doing is ignore this 4,"},{"Start":"04:00.860 ","End":"04:02.855","Text":"draw the shape of this,"},{"Start":"04:02.855 ","End":"04:07.750","Text":"and then when I look at x not minus x,"},{"Start":"04:11.690 ","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":"04:23.595","Text":"it\u0027s plus 4 in the x-direction because of this,"},{"Start":"04:23.595 ","End":"04:28.985","Text":"but it opens up in the negative x-direction."},{"Start":"04:28.985 ","End":"04:33.560","Text":"Anyway, just think about it and you\u0027ll see that this is roughly what we get and"},{"Start":"04:33.560 ","End":"04:39.775","Text":"the cross-sections are elliptic because these 2 numbers are different."},{"Start":"04:39.775 ","End":"04:41.625","Text":"That\u0027s all you\u0027re expected,"},{"Start":"04:41.625 ","End":"04:46.575","Text":"just follow the general idea when you encounter these quadric surfaces."},{"Start":"04:46.575 ","End":"04:48.780","Text":"You will not be asked to sketch these,"},{"Start":"04:48.780 ","End":"04:51.175","Text":"these were computer aided,"},{"Start":"04:51.175 ","End":"04:54.940","Text":"not something you would do on your own with a table or something."},{"Start":"04:54.940 ","End":"04:59.070","Text":"That\u0027s all I want to say on this. We\u0027re done."}],"ID":10687}],"Thumbnail":null,"ID":12295}]

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