Introduction to 2D and 3D Vectors
0/2 completed

Vector Arithmetic
0/5 completed

Vectors Dot Product
0/8 completed

Vectors Cross Product
0/6 completed

{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction to 2D and 3D Vectors","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors (mostly 2D and 3D)","Duration":"27m 41s","ChapterTopicVideoID":10096,"CourseChapterTopicPlaylistID":57470,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10096.jpeg","UploadDate":"2020-01-16T07:08:50.5370000","DurationForVideoObject":"PT27M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"We were starting a new topic,"},{"Start":"00:01.845 ","End":"00:05.265","Text":"the 1 of the concept vectors,"},{"Start":"00:05.265 ","End":"00:08.745","Text":"which mostly applied in physics,"},{"Start":"00:08.745 ","End":"00:12.300","Text":"but they\u0027re used also abstractly in mathematics."},{"Start":"00:12.300 ","End":"00:18.270","Text":"They\u0027re used to represent quantities that have both a magnitude,"},{"Start":"00:18.270 ","End":"00:26.085","Text":"magnitude is size, and direction,"},{"Start":"00:26.085 ","End":"00:28.455","Text":"both of these things."},{"Start":"00:28.455 ","End":"00:33.450","Text":"Examples which I will bring from physics,"},{"Start":"00:33.450 ","End":"00:36.225","Text":"1 example is force,"},{"Start":"00:36.225 ","End":"00:40.350","Text":"which doesn\u0027t just have a magnitude measured in newtons,"},{"Start":"00:40.350 ","End":"00:42.180","Text":"but it also has a direction."},{"Start":"00:42.180 ","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":"01:02.030","Text":"Speed is just how many miles per hour or kilometers per hour you\u0027re going,"},{"Start":"01:02.030 ","End":"01:05.925","Text":"say 60 kilometers an hour,"},{"Start":"01:05.925 ","End":"01:10.640","Text":"the velocity would be maybe 60 kilometers an hour going north,"},{"Start":"01:10.640 ","End":"01:12.125","Text":"and the same thing with force."},{"Start":"01:12.125 ","End":"01:15.005","Text":"Let me show you a diagram."},{"Start":"01:15.005 ","End":"01:20.510","Text":"Typically, a vector is represented with an arrow of a certain length,"},{"Start":"01:20.510 ","End":"01:23.245","Text":"and going in a certain direction."},{"Start":"01:23.245 ","End":"01:28.174","Text":"All these vectors would be considered to be the same vector,"},{"Start":"01:28.174 ","End":"01:30.200","Text":"each of them if you look,"},{"Start":"01:30.200 ","End":"01:33.600","Text":"would go, let\u0027s say,"},{"Start":"01:33.600 ","End":"01:36.269","Text":"2 units to the left,"},{"Start":"01:36.269 ","End":"01:40.325","Text":"and 5 units up the same thing."},{"Start":"01:40.325 ","End":"01:44.185","Text":"They\u0027re all parallel, and they all have the same size."},{"Start":"01:44.185 ","End":"01:47.045","Text":"These are all considered to be the same vector,"},{"Start":"01:47.045 ","End":"01:51.440","Text":"and the position at which they are applied, like here,"},{"Start":"01:51.440 ","End":"01:53.735","Text":"or here, or here,"},{"Start":"01:53.735 ","End":"01:56.390","Text":"or here is not significant."},{"Start":"01:56.390 ","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":"02:09.030","Text":"it could be 80 kilometers an hour in this direction."},{"Start":"02:09.030 ","End":"02:10.890","Text":"It doesn\u0027t matter where the car is,"},{"Start":"02:10.890 ","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":"02:22.040","Text":"This is an example of vectors,"},{"Start":"02:22.040 ","End":"02:25.130","Text":"and this is in 2-dimensions."},{"Start":"02:25.130 ","End":"02:29.220","Text":"There\u0027s also such concept in 3-dimensions,"},{"Start":"02:29.220 ","End":"02:30.940","Text":"it\u0027s exactly the same,"},{"Start":"02:30.940 ","End":"02:33.680","Text":"a magnitude and a direction, but in 3D,"},{"Start":"02:33.680 ","End":"02:36.770","Text":"but we often use 2D especially with drawings."},{"Start":"02:36.770 ","End":"02:40.910","Text":"It\u0027s easier to draw diagrams in 2-dimensions."},{"Start":"02:40.910 ","End":"02:48.235","Text":"Graphically, vectors are presented as what we call a directed line segment."},{"Start":"02:48.235 ","End":"02:50.360","Text":"When we have a picture like this,"},{"Start":"02:50.360 ","End":"02:55.025","Text":"each of these is called a representation of the vector."},{"Start":"02:55.025 ","End":"02:56.405","Text":"They\u0027re all the same,"},{"Start":"02:56.405 ","End":"03:01.310","Text":"each 1 of them could be a representation of the same vector."},{"Start":"03:01.310 ","End":"03:06.250","Text":"The way we write the vector numerically not graphically,"},{"Start":"03:06.250 ","End":"03:11.220","Text":"the notation is something like you take a left up,"},{"Start":"03:11.220 ","End":"03:14.500","Text":"often V, and put a little arrow above it."},{"Start":"03:14.500 ","End":"03:16.160","Text":"When you see the arrow above it,"},{"Start":"03:16.160 ","End":"03:18.995","Text":"it means it\u0027s a vector, not a number."},{"Start":"03:18.995 ","End":"03:23.655","Text":"Numbers are also called scalars, we\u0027ll get to that."},{"Start":"03:23.655 ","End":"03:27.159","Text":"This might be a vector V,"},{"Start":"03:27.159 ","End":"03:29.780","Text":"and this would be the same V,"},{"Start":"03:29.780 ","End":"03:33.780","Text":"they\u0027re all the same vector."},{"Start":"03:35.840 ","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 an x,"},{"Start":"03:45.920 ","End":"03:51.379","Text":"y coordinate system, but the number of units"},{"Start":"03:51.379 ","End":"03:57.290","Text":"we go in the x-direction positively goes here,"},{"Start":"03:57.290 ","End":"04:01.025","Text":"and the y-direction positively goes here,"},{"Start":"04:01.025 ","End":"04:04.860","Text":"so the x and the y."},{"Start":"04:04.860 ","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":"04:12.720","Text":"which means negative 2,"},{"Start":"04:12.720 ","End":"04:16.305","Text":"and they go up 5 which is a positive 5."},{"Start":"04:16.305 ","End":"04:27.105","Text":"We use angular brackets to distinguish from the point minus 2, 5."},{"Start":"04:27.105 ","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":"04:35.950 ","End":"04:44.915","Text":"then the other end is going to be the point minus 2, 5."},{"Start":"04:44.915 ","End":"04:48.260","Text":"But there\u0027s a difference between the point."},{"Start":"04:48.260 ","End":"04:50.450","Text":"This is a vector,"},{"Start":"04:50.450 ","End":"04:52.970","Text":"and if I just say minus 2,"},{"Start":"04:52.970 ","End":"04:55.805","Text":"5, then it\u0027s a point."},{"Start":"04:55.805 ","End":"05:00.830","Text":"But this is called the position vector for this point."},{"Start":"05:00.830 ","End":"05:04.775","Text":"When you join the origin to a point,"},{"Start":"05:04.775 ","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":"05:32.540","Text":"We\u0027ll probably put that under examples of a vector is something called a position vector."},{"Start":"05:32.540 ","End":"05:37.530","Text":"I just want to have it written so the concept has been introduced."},{"Start":"05:37.840 ","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 end points,"},{"Start":"05:44.270 ","End":"05:48.385","Text":"let\u0027s say this is the point A and this is the point B,"},{"Start":"05:48.385 ","End":"05:53.130","Text":"A would be the point in this case, let\u0027s see,"},{"Start":"05:53.130 ","End":"05:55.830","Text":"the x of it would be minus 3,"},{"Start":"05:55.830 ","End":"05:58.545","Text":"and the y of it would be minus 4."},{"Start":"05:58.545 ","End":"06:06.345","Text":"B would be the point minus 5, 1."},{"Start":"06:06.345 ","End":"06:12.650","Text":"I\u0027d like to point out the relationship between the tail of the vector,"},{"Start":"06:12.650 ","End":"06:14.705","Text":"the head of the vector, and the vector itself."},{"Start":"06:14.705 ","End":"06:21.550","Text":"The vector we said was minus 2, 5."},{"Start":"06:21.550 ","End":"06:32.070","Text":"Note that if I take my minus 3 minus 4 and I add the minus 2,"},{"Start":"06:32.070 ","End":"06:34.440","Text":"5 to it, that minus 2,"},{"Start":"06:34.440 ","End":"06:38.380","Text":"5, I\u0027m just going to write that here, minus 2, 5."},{"Start":"06:38.480 ","End":"06:43.619","Text":"Coordinate-wise, what I\u0027ll get is minus 3,"},{"Start":"06:43.619 ","End":"06:46.395","Text":"minus 2, I\u0027m adding negative 2,"},{"Start":"06:46.395 ","End":"06:50.250","Text":"and then minus 4,"},{"Start":"06:50.250 ","End":"06:57.150","Text":"plus 5 which is just equal to minus 3,"},{"Start":"06:57.150 ","End":"07:02.010","Text":"minus 2 is minus 5, 1."},{"Start":"07:02.010 ","End":"07:04.940","Text":"In other words, if I have a point A at the tail of"},{"Start":"07:04.940 ","End":"07:08.600","Text":"the vector and the point B at the head of the vector,"},{"Start":"07:08.600 ","End":"07:13.665","Text":"and this is the vector that goes from here to here,"},{"Start":"07:13.665 ","End":"07:18.995","Text":"we can get the coordinates of the tail from the head."},{"Start":"07:18.995 ","End":"07:24.860","Text":"We could get any 1 of these 3 from the other 2 if we\u0027re given the tail and the head,"},{"Start":"07:24.860 ","End":"07:28.715","Text":"we could subtract minus 5 takeaway minus 3 is minus 2."},{"Start":"07:28.715 ","End":"07:34.595","Text":"In some ways, the tail plus the vector is the head as far as numbers go,"},{"Start":"07:34.595 ","End":"07:37.190","Text":"the coordinates, but they are different entities,"},{"Start":"07:37.190 ","End":"07:39.680","Text":"these 2 points, and this is a vector."},{"Start":"07:39.680 ","End":"07:41.720","Text":"But the math works that way."},{"Start":"07:41.720 ","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":"07:55.205","Text":"let\u0027s say this is my A,"},{"Start":"07:55.205 ","End":"07:59.365","Text":"and I want to go to a point B,"},{"Start":"07:59.365 ","End":"08:02.120","Text":"and the vector that takes me there is,"},{"Start":"08:02.120 ","End":"08:05.555","Text":"let\u0027s say, a_1, a_2,"},{"Start":"08:05.555 ","End":"08:14.145","Text":"then the B will be x plus a_1,"},{"Start":"08:14.145 ","End":"08:17.265","Text":"and y plus a_2."},{"Start":"08:17.265 ","End":"08:20.150","Text":"I should really put a bar at least toward the whole vector,"},{"Start":"08:20.150 ","End":"08:24.335","Text":"sometimes just the bar, sometimes you forget the thing all together."},{"Start":"08:24.335 ","End":"08:27.965","Text":"Now, so far we\u0027ve just talked about 2D,"},{"Start":"08:27.965 ","End":"08:32.180","Text":"2-dimensional, but the same thing works in 3D."},{"Start":"08:32.180 ","End":"08:39.480","Text":"In 3D, a point would be 3 coordinates,"},{"Start":"08:39.480 ","End":"08:42.390","Text":"x, y, and z."},{"Start":"08:42.390 ","End":"08:43.910","Text":"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":"08:48.830","Text":"We also have a height,"},{"Start":"08:48.830 ","End":"08:54.500","Text":"and we have a vector V,"},{"Start":"08:54.500 ","End":"08:57.530","Text":"and this time the vector V we will have"},{"Start":"08:57.530 ","End":"09:05.405","Text":"3 components; a_1, a_2, a_3."},{"Start":"09:05.405 ","End":"09:10.040","Text":"This will take us from the point A to the point B,"},{"Start":"09:10.040 ","End":"09:13.715","Text":"which will be x plus the first coordinate,"},{"Start":"09:13.715 ","End":"09:17.785","Text":"y plus the second coordinate,"},{"Start":"09:17.785 ","End":"09:27.250","Text":"and running out of space here, z plus a_3."},{"Start":"09:28.050 ","End":"09:31.810","Text":"This concept works the other way around also."},{"Start":"09:31.810 ","End":"09:37.285","Text":"Given 2 points, I can find the vector that takes me from 1 to the other."},{"Start":"09:37.285 ","End":"09:41.185","Text":"So suppose I had a point A,"},{"Start":"09:41.185 ","End":"09:44.620","Text":"which was, let\u0027s say,"},{"Start":"09:44.620 ","End":"09:47.035","Text":"I\u0027ll call it a1, a2,"},{"Start":"09:47.035 ","End":"09:51.490","Text":"a3, and I have a point B, which is,"},{"Start":"09:51.490 ","End":"09:54.250","Text":"call it say b1, b2,"},{"Start":"09:54.250 ","End":"09:58.840","Text":"b3, and I want to know the vector that takes me from here to here."},{"Start":"09:58.840 ","End":"10:01.600","Text":"First of all, the name for that vector,"},{"Start":"10:01.600 ","End":"10:05.395","Text":"it\u0027s called AB with an arrow on top,"},{"Start":"10:05.395 ","End":"10:06.880","Text":"and the order is important,"},{"Start":"10:06.880 ","End":"10:08.230","Text":"as we\u0027ll soon see."},{"Start":"10:08.230 ","End":"10:14.425","Text":"This will equal the second coordinate minus the first coordinate,"},{"Start":"10:14.425 ","End":"10:21.955","Text":"will be this minus this and then this minus this,"},{"Start":"10:21.955 ","End":"10:27.745","Text":"b2 minus a2, 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":"10:36.820","Text":"let\u0027s suppose I want to know what is BA."},{"Start":"10:36.820 ","End":"10:38.830","Text":"Give it another name,"},{"Start":"10:38.830 ","End":"10:43.450","Text":"say w, this would be the other way."},{"Start":"10:43.450 ","End":"10:48.340","Text":"This minus this, this is actually going to be a1 minus b1,"},{"Start":"10:48.340 ","End":"10:55.915","Text":"a2 minus b2, and a3 minus b3,"},{"Start":"10:55.915 ","End":"10:58.345","Text":"and this is actually the negatives of this."},{"Start":"10:58.345 ","End":"11:02.515","Text":"For example, in here if I took in 2-dimensions,"},{"Start":"11:02.515 ","End":"11:06.850","Text":"let\u0027s say w was the opposite of this,"},{"Start":"11:06.850 ","End":"11:11.200","Text":"plus 2 and minus 5."},{"Start":"11:11.200 ","End":"11:14.305","Text":"Then it would take me the other way"},{"Start":"11:14.305 ","End":"11:21.580","Text":"from B to A and you can check minus 3 takeaway minus 5 is 2,"},{"Start":"11:21.580 ","End":"11:24.850","Text":"minus 4 minus 1, minus 5."},{"Start":"11:24.850 ","End":"11:29.665","Text":"Or you could say that minus 5 plus 2 is minus 3 and so on."},{"Start":"11:29.665 ","End":"11:32.350","Text":"So you get the opposite signs."},{"Start":"11:32.350 ","End":"11:36.220","Text":"It\u0027s important the order, the vector from A to B,"},{"Start":"11:36.220 ","End":"11:38.170","Text":"opposite from the vector from B to A,"},{"Start":"11:38.170 ","End":"11:41.890","Text":"and it\u0027s the second minus the first when you do the coordinates,"},{"Start":"11:41.890 ","End":"11:44.240","Text":"so don\u0027t get it backwards."},{"Start":"11:44.280 ","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":"11:59.040","Text":"If I take 0,0 and add minus 2,5,"},{"Start":"11:59.040 ","End":"12:01.335","Text":"I just get minus 2,5."},{"Start":"12:01.335 ","End":"12:03.690","Text":"So for position vectors,"},{"Start":"12:03.690 ","End":"12:09.330","Text":"the vector and the point at the head of the vector actually have the same coordinates,"},{"Start":"12:09.330 ","End":"12:10.965","Text":"so ones with a round brackets,"},{"Start":"12:10.965 ","End":"12:13.830","Text":"ones with angular brackets and their different entities."},{"Start":"12:13.830 ","End":"12:16.555","Text":"Let\u0027s do just some exercises."},{"Start":"12:16.555 ","End":"12:21.834","Text":"I will show you the exercises you might get at this very basic level."},{"Start":"12:21.834 ","End":"12:24.130","Text":"In this example exercise,"},{"Start":"12:24.130 ","End":"12:31.310","Text":"I\u0027m going to ask you to find the following vectors."},{"Start":"12:31.770 ","End":"12:34.180","Text":"Let\u0027s do 3 examples,"},{"Start":"12:34.180 ","End":"12:35.845","Text":"will 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":"12:57.140","Text":"3, to the point minus 7,4,5."},{"Start":"12:57.140 ","End":"13:05.365","Text":"In part b, I\u0027ll give a 2-dimensional example."},{"Start":"13:05.365 ","End":"13:15.980","Text":"So the vector from minus 2,7 to 1, minus 5."},{"Start":"13:16.080 ","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":"13:34.650 ","End":"13:36.910","Text":"So solve them."},{"Start":"13:36.910 ","End":"13:39.790","Text":"When we take a vector from here to here,"},{"Start":"13:39.790 ","End":"13:42.760","Text":"remember we do the second minus the first."},{"Start":"13:42.760 ","End":"13:47.850","Text":"So what we do is minus 7 takeaway 1,"},{"Start":"13:47.850 ","End":"13:51.855","Text":"and then second coordinate 4 takeaway 2,"},{"Start":"13:51.855 ","End":"13:58.120","Text":"and 5 takeaway 3, in angular brackets."},{"Start":"13:58.120 ","End":"13:59.710","Text":"Of course you do the calculation,"},{"Start":"13:59.710 ","End":"14:09.160","Text":"so it comes out to be minus 8,2,2 and that would be the answer."},{"Start":"14:09.160 ","End":"14:11.860","Text":"The same thing in 2-dimensions."},{"Start":"14:11.860 ","End":"14:15.730","Text":"We\u0027ll take the second minus the first,"},{"Start":"14:15.730 ","End":"14:24.595","Text":"so 1 takeaway minus 2 and minus 5 takeaway 7."},{"Start":"14:24.595 ","End":"14:27.970","Text":"The answer is a 2-dimensional vector,"},{"Start":"14:27.970 ","End":"14:36.695","Text":"3, minus 5 minus 7 is minus 12."},{"Start":"14:36.695 ","End":"14:38.910","Text":"The last one is the easiest,"},{"Start":"14:38.910 ","End":"14:42.210","Text":"the position vector for a point is from the origin to the point,"},{"Start":"14:42.210 ","End":"14:44.640","Text":"so 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":"15:04.075","Text":"Let\u0027s move on a bit."},{"Start":"15:04.075 ","End":"15:07.045","Text":"I\u0027m going to talk about the magnitude of a vector."},{"Start":"15:07.045 ","End":"15:09.985","Text":"We\u0027ve seen various examples of vectors,"},{"Start":"15:09.985 ","End":"15:11.560","Text":"and remember at the beginning,"},{"Start":"15:11.560 ","End":"15:12.880","Text":"I\u0027m going to go back up,"},{"Start":"15:12.880 ","End":"15:16.600","Text":"we said that the vector has magnitude and direction."},{"Start":"15:16.600 ","End":"15:19.210","Text":"Well, I want to focus on"},{"Start":"15:19.210 ","End":"15:24.085","Text":"the magnitude of a vector and I\u0027m going to erase what I don\u0027t need."},{"Start":"15:24.085 ","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":"15:39.745","Text":"I might have a vector minus 3,4 and I want to know what its magnitude is."},{"Start":"15:39.745 ","End":"15:45.010","Text":"Or in 3-dimensions, I might have a vector,"},{"Start":"15:45.010 ","End":"15:52.870","Text":"let\u0027s say 3,4,12 and"},{"Start":"15:52.870 ","End":"15:56.270","Text":"I want to know what its 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":"16:07.495","Text":"which will help us with all this."},{"Start":"16:07.495 ","End":"16:10.120","Text":"Let\u0027s just leave the vectors for a moment."},{"Start":"16:10.120 ","End":"16:14.110","Text":"Now, Pythagoras\u0027s theorem in 2-dimensions,"},{"Start":"16:14.110 ","End":"16:17.515","Text":"if we just look at the base of this box."},{"Start":"16:17.515 ","End":"16:20.185","Text":"This is 90 degrees;"},{"Start":"16:20.185 ","End":"16:24.430","Text":"then what Pythagoras\u0027s theorem says is that the hypotenuse,"},{"Start":"16:24.430 ","End":"16:30.940","Text":"in this case c, 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":"16:37.870","Text":"Now suppose I have something in 3-dimensions,"},{"Start":"16:37.870 ","End":"16:42.340","Text":"I have this line here inside a box."},{"Start":"16:42.340 ","End":"16:45.159","Text":"It\u0027s got an x, a y,"},{"Start":"16:45.159 ","End":"16:47.170","Text":"and it\u0027s z up."},{"Start":"16:47.170 ","End":"16:53.380","Text":"So turns out that you can just extend this formula and say that the square of this is"},{"Start":"16:53.380 ","End":"17:00.010","Text":"the sums of the squares of all the 3 equal at 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":"17:07.480","Text":"is that you can use Pythagoras on this triangle"},{"Start":"17:07.480 ","End":"17:12.025","Text":"here and get that z squared plus c squared is s squared,"},{"Start":"17:12.025 ","End":"17:15.010","Text":"but c squared from here is x squared plus y squared."},{"Start":"17:15.010 ","End":"17:19.330","Text":"So that proves it. That\u0027s a generalization of Pythagoras\u0027s theorem."},{"Start":"17:19.330 ","End":"17:24.040","Text":"So instead of in 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":"17:38.410","Text":"In general, in 2-dimensions,"},{"Start":"17:38.410 ","End":"17:39.835","Text":"vector V would be,"},{"Start":"17:39.835 ","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":"17:56.335","Text":"we could draw a vector that goes 3 units this way and 4 units this way."},{"Start":"17:56.335 ","End":"17:59.710","Text":"This would be 3 in this direction,"},{"Start":"17:59.710 ","End":"18:03.865","Text":"and I go up 4, and then I get the vector."},{"Start":"18:03.865 ","End":"18:06.940","Text":"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":"18:10.525","Text":"and it goes from here to here."},{"Start":"18:10.525 ","End":"18:13.930","Text":"We called it V. The question is,"},{"Start":"18:13.930 ","End":"18:15.100","Text":"what is the magnitude?"},{"Start":"18:15.100 ","End":"18:16.735","Text":"What is the size of V?"},{"Start":"18:16.735 ","End":"18:18.655","Text":"We just use Pythagoras."},{"Start":"18:18.655 ","End":"18:26.935","Text":"We would say the square root of 3 squared plus 4 squared."},{"Start":"18:26.935 ","End":"18:30.820","Text":"Notice that if I\u0027d actually put the minus 3 squared here,"},{"Start":"18:30.820 ","End":"18:33.235","Text":"it wouldn\u0027t have made any difference."},{"Start":"18:33.235 ","End":"18:36.520","Text":"In other words, the magnitude of"},{"Start":"18:36.520 ","End":"18:39.580","Text":"a vector in 2-dimensions is the square root of the sum of the squares,"},{"Start":"18:39.580 ","End":"18:41.425","Text":"and let\u0027s just write this in general."},{"Start":"18:41.425 ","End":"18:45.430","Text":"What I would say was we used a notation,"},{"Start":"18:45.430 ","End":"18:48.250","Text":"a double bar around the vector,"},{"Start":"18:48.250 ","End":"18:51.430","Text":"and a double vertical bar."},{"Start":"18:51.430 ","End":"18:52.720","Text":"Like the absolute value,"},{"Start":"18:52.720 ","End":"18:54.850","Text":"but just doubled up."},{"Start":"18:54.850 ","End":"18:58.225","Text":"This is called the magnitude of the vector V,"},{"Start":"18:58.225 ","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":"19:08.335","Text":"and this is using Pythagoras."},{"Start":"19:08.335 ","End":"19:13.120","Text":"Now, let\u0027s just extend the same thing to 3-dimensions."},{"Start":"19:13.120 ","End":"19:15.190","Text":"If I have a vector,"},{"Start":"19:15.190 ","End":"19:17.170","Text":"well, use the same letter again."},{"Start":"19:17.170 ","End":"19:21.175","Text":"It doesn\u0027t matter, but I use a different context,"},{"Start":"19:21.175 ","End":"19:22.675","Text":"maybe a different color."},{"Start":"19:22.675 ","End":"19:26.020","Text":"Yeah, that\u0027ll do it. I have in 3-dimensions."},{"Start":"19:26.020 ","End":"19:28.150","Text":"Let\u0027s take our example here."},{"Start":"19:28.150 ","End":"19:33.835","Text":"Suppose I took 3, 4, 12."},{"Start":"19:33.835 ","End":"19:36.100","Text":"I didn\u0027t actually finish the exercise here."},{"Start":"19:36.100 ","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":"19:53.110","Text":"which is 5, and so the magnitude of V is 5."},{"Start":"19:53.110 ","End":"19:55.209","Text":"Let\u0027s take the other example."},{"Start":"19:55.209 ","End":"20:01.825","Text":"In general, we will have V as being a_1, a_2,"},{"Start":"20:01.825 ","End":"20:07.165","Text":"a_3, and the magnitude of a vector,"},{"Start":"20:07.165 ","End":"20:11.470","Text":"even if it\u0027s 3D, is just the analogy of this."},{"Start":"20:11.470 ","End":"20:14.815","Text":"I won\u0027t draw the sketch because we have a sketch here,"},{"Start":"20:14.815 ","End":"20:17.455","Text":"but this might be a_1,"},{"Start":"20:17.455 ","End":"20:19.285","Text":"this would be a_2,"},{"Start":"20:19.285 ","End":"20:21.220","Text":"this would be a_3,"},{"Start":"20:21.220 ","End":"20:30.790","Text":"and this line here would be the vector V. So what we would get that the length of this,"},{"Start":"20:30.790 ","End":"20:34.900","Text":"the magnitude as the length is the square root of a_1"},{"Start":"20:34.900 ","End":"20:39.490","Text":"squared plus a_2 squared plus a_3 squared."},{"Start":"20:39.490 ","End":"20:41.320","Text":"Again, by Pythagoras."},{"Start":"20:41.320 ","End":"20:46.284","Text":"Notice that minuses don\u0027t make any difference because everything\u0027s being squared anyway,"},{"Start":"20:46.284 ","End":"20:48.850","Text":"so we don\u0027t have to worry about that."},{"Start":"20:48.850 ","End":"20:58.330","Text":"In our case, let\u0027s just see what we get for the magnitude."},{"Start":"20:58.330 ","End":"21:02.305","Text":"In our case, we will get the square root"},{"Start":"21:02.305 ","End":"21:09.400","Text":"of 3 squared plus 4 squared plus 12 squared."},{"Start":"21:09.400 ","End":"21:17.275","Text":"9 plus 16 is 25 plus 144, that\u0027s 169."},{"Start":"21:17.275 ","End":"21:22.850","Text":"The square root of 169 is 13 in this case."},{"Start":"21:24.120 ","End":"21:29.320","Text":"Now suppose V was the vector 0,"},{"Start":"21:29.320 ","End":"21:33.700","Text":"0 in 2-dimensions and the absolute value,"},{"Start":"21:33.700 ","End":"21:35.110","Text":"not absolute value, sorry,"},{"Start":"21:35.110 ","End":"21:38.109","Text":"the magnitude of V,"},{"Start":"21:38.109 ","End":"21:40.225","Text":"I keep forgetting the arrows,"},{"Start":"21:40.225 ","End":"21:42.415","Text":"1 tends to do that,"},{"Start":"21:42.415 ","End":"21:49.854","Text":"is equal by the formula the square root of 0 squared plus 0 squared, which is just 0."},{"Start":"21:49.854 ","End":"21:52.119","Text":"This is a very special vector."},{"Start":"21:52.119 ","End":"21:58.315","Text":"It\u0027s the 0 vector and sometimes we just write it as 0 with an arrow above it,"},{"Start":"21:58.315 ","End":"22:01.225","Text":"not the number 0, but the vector 0."},{"Start":"22:01.225 ","End":"22:10.824","Text":"Similarly, in 3D, we could take a vector V in 3D to be 0, 0, 0,"},{"Start":"22:10.824 ","End":"22:18.010","Text":"and then the magnitude of V would be the square root of 0 squared plus,"},{"Start":"22:18.010 ","End":"22:21.835","Text":"0 squared plus 0 squared."},{"Start":"22:21.835 ","End":"22:27.460","Text":"This would also be denoted as 0 with an arrow,"},{"Start":"22:27.460 ","End":"22:29.200","Text":"but 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":"22:36.835","Text":"depends on the context but that\u0027s the special name."},{"Start":"22:36.835 ","End":"22:40.285","Text":"Besides the 0 vector, which is special,"},{"Start":"22:40.285 ","End":"22:43.840","Text":"another concept that\u0027s important is called the unit vector."},{"Start":"22:43.840 ","End":"22:53.590","Text":"Let\u0027s write what we\u0027ve learned something called the 0 vector."},{"Start":"22:53.590 ","End":"22:56.050","Text":"Now I want to write down,"},{"Start":"22:56.050 ","End":"23:00.200","Text":"the next thing I\u0027m going to do is something called the unit vector."},{"Start":"23:00.900 ","End":"23:08.830","Text":"Well, a 0 vector is a vector whose magnitude is 0 equals 0."},{"Start":"23:08.830 ","End":"23:10.420","Text":"That\u0027s what makes it the 0 vector,"},{"Start":"23:10.420 ","End":"23:12.505","Text":"the magnitude 0, same as here."},{"Start":"23:12.505 ","End":"23:18.190","Text":"The unit vector would be a vector with magnitude 1."},{"Start":"23:18.190 ","End":"23:21.295","Text":"I\u0027ll give an example first of all, in 2D."},{"Start":"23:21.295 ","End":"23:24.280","Text":"First, 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":"23:47.185","Text":"This squared is 1 over 5,"},{"Start":"23:47.185 ","End":"23:51.040","Text":"this squared is 4 over 5, 2 squared is 4,"},{"Start":"23:51.040 ","End":"23:53.065","Text":"and root 5 squared is 5,"},{"Start":"23:53.065 ","End":"23:58.240","Text":"which is the square root of 1/5 plus 4/5 is 1,"},{"Start":"23:58.240 ","End":"24:00.774","Text":"so this is equal to 1."},{"Start":"24:00.774 ","End":"24:04.360","Text":"This vector here is a unit vector,"},{"Start":"24:04.360 ","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":"24:24.114","Text":"This is equal to the square root of 1 squared 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":"24:28.240","Text":"square root of 1 is 1,"},{"Start":"24:28.240 ","End":"24:31.070","Text":"so this is also equal to 1."},{"Start":"24:31.740 ","End":"24:40.120","Text":"This is an example of a unit vector in 2D 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":"24:52.060","Text":"0 is called i,"},{"Start":"24:52.060 ","End":"24:54.460","Text":"I\u0027m not sure if it has a dot or not,"},{"Start":"24:54.460 ","End":"24:58.255","Text":"and that\u0027s the 1, 0, 0."},{"Start":"24:58.255 ","End":"25:02.695","Text":"There is also a special unit vector called j,"},{"Start":"25:02.695 ","End":"25:07.270","Text":"and that is 0, 1, 0."},{"Start":"25:07.270 ","End":"25:09.220","Text":"It\u0027s also a unit vector, of course,"},{"Start":"25:09.220 ","End":"25:12.295","Text":"because 0 squared plus 1 squared plus 0 squared is 1."},{"Start":"25:12.295 ","End":"25:16.179","Text":"We also have a 3rd unit vector in 3D,"},{"Start":"25:16.179 ","End":"25:19.450","Text":"it\u0027s called k, and this is equal to,"},{"Start":"25:19.450 ","End":"25:23.335","Text":"you might guess, 0, 0, 1."},{"Start":"25:23.335 ","End":"25:27.250","Text":"In 2D, there are also unit vectors."},{"Start":"25:27.250 ","End":"25:29.050","Text":"We only have 2 of them."},{"Start":"25:29.050 ","End":"25:33.655","Text":"We have i, which is 1, 0,"},{"Start":"25:33.655 ","End":"25:40.210","Text":"and we have a j which is 0, 1."},{"Start":"25:40.210 ","End":"25:47.290","Text":"There\u0027s actually a special name for these vectors with single 1 and the rest 0s,"},{"Start":"25:47.290 ","End":"25:54.370","Text":"they\u0027re actually called standard basis vectors."},{"Start":"25:54.370 ","End":"25:57.640","Text":"It will always be clear in which context we\u0027re talking about,"},{"Start":"25:57.640 ","End":"26:00.310","Text":"the 2D or the 3D case."},{"Start":"26:00.310 ","End":"26:08.230","Text":"I just want to mention that although we\u0027ve talked about 2D and 3D vectors, in general,"},{"Start":"26:08.230 ","End":"26:15.580","Text":"there are n-dimensional vectors and it\u0027s not restricted to 2 or 3,"},{"Start":"26:15.580 ","End":"26:16.825","Text":"there\u0027s a 4th, 5th, 6th,"},{"Start":"26:16.825 ","End":"26:18.355","Text":"any number of dimensions."},{"Start":"26:18.355 ","End":"26:19.990","Text":"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":"26:30.790","Text":"But an n-dimensional vector V would be something like a_1,"},{"Start":"26:30.790 ","End":"26:34.105","Text":"a_2, and depending on anyway, and so on,"},{"Start":"26:34.105 ","End":"26:38.740","Text":"up to a_n, and could be 4 or 5, 6 whatever."},{"Start":"26:38.740 ","End":"26:43.540","Text":"Most of the formulas are just straightforward extensions."},{"Start":"26:43.540 ","End":"26:48.550","Text":"The magnitude of a vector instead of a_1 squared plus a_2 squared plus"},{"Start":"26:48.550 ","End":"26:53.815","Text":"a_3 squared would be the sum of the squares of all of them and you take the square root."},{"Start":"26:53.815 ","End":"26:58.150","Text":"Each of these would be called the ai\u0027s,"},{"Start":"26:58.150 ","End":"27:01.940","Text":"a_1, a_2, also called components of the vector."},{"Start":"27:02.520 ","End":"27:05.440","Text":"Sometimes we say the x component,"},{"Start":"27:05.440 ","End":"27:07.480","Text":"the y component, the z component of the 1st,"},{"Start":"27:07.480 ","End":"27:08.980","Text":"2nd, and 3rd component,"},{"Start":"27:08.980 ","End":"27:11.485","Text":"and so on. Just mentioning it."},{"Start":"27:11.485 ","End":"27:13.150","Text":"We\u0027re mostly going to deal with"},{"Start":"27:13.150 ","End":"27:18.175","Text":"just 2- and 3-dimensions and the formulas will be almost the same,"},{"Start":"27:18.175 ","End":"27:24.235","Text":"just like we had that in 2-dimensions,"},{"Start":"27:24.235 ","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":"27:30.685","Text":"a_1 square plus a_2, a_3 squared."},{"Start":"27:30.685 ","End":"27:34.045","Text":"All the formulas are analogous pretty much in 2- or 3-dimensions."},{"Start":"27:34.045 ","End":"27:41.300","Text":"That\u0027s what we\u0027ll be sticking with mostly. Done for now."}],"ID":10282},{"Watched":false,"Name":"Exercise 1","Duration":"6m 13s","ChapterTopicVideoID":10097,"CourseChapterTopicPlaylistID":57470,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10097.jpeg","UploadDate":"2017-08-20T10:18:17.8530000","DurationForVideoObject":"PT6M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.720","Text":"This exercise has several parts 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":"00:23.370","Text":"In these 2, we have what is called a displacement vector."},{"Start":"00:23.370 ","End":"00:25.469","Text":"I\u0027m not sure if I mentioned the term displacement,"},{"Start":"00:25.469 ","End":"00:27.555","Text":"but 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":"00:39.890","Text":"goes from 1 to the other displacement vector."},{"Start":"00:39.890 ","End":"00:43.010","Text":"What we do is we take the coordinates of"},{"Start":"00:43.010 ","End":"00:49.050","Text":"the first point from the coordinates of the second point."},{"Start":"00:49.250 ","End":"00:53.580","Text":"In Part a, we just take,"},{"Start":"00:53.580 ","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":"01:01.820 ","End":"01:09.020","Text":"and the second 1 would be minus 2 minus 3,"},{"Start":"01:09.020 ","End":"01:13.530","Text":"which gives us 13,"},{"Start":"01:14.230 ","End":"01:18.920","Text":"minus 2 minus 3 is minus 5."},{"Start":"01:18.920 ","End":"01:23.000","Text":"I\u0027ll be using the 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":"01:31.190","Text":"either 1 of these."},{"Start":"01:31.190 ","End":"01:34.325","Text":"In Part b, we have a 3D vector."},{"Start":"01:34.325 ","End":"01:37.025","Text":"Once again we do the same thing."},{"Start":"01:37.025 ","End":"01:43.860","Text":"We have 2 minus 2 is,"},{"Start":"01:43.860 ","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":"02:06.690","Text":"The magnitude, I\u0027ll just stick to the angular brackets,"},{"Start":"02:06.690 ","End":"02:09.065","Text":"and give me some space."},{"Start":"02:09.065 ","End":"02:16.010","Text":"Magnitude of 13, minus 5 is"},{"Start":"02:16.010 ","End":"02:24.775","Text":"just the square root of 13 squared plus negative 5 squared,"},{"Start":"02:24.775 ","End":"02:31.855","Text":"which comes out, 169 and 25 is 194."},{"Start":"02:31.855 ","End":"02:34.580","Text":"That\u0027s the magnitude. In any event,"},{"Start":"02:34.580 ","End":"02:41.690","Text":"the magnitude is not equal to 1 so the answer is that it\u0027s not a unit vector,"},{"Start":"02:41.690 ","End":"02:45.085","Text":"I\u0027ll just write the word not a unit vector."},{"Start":"02:45.085 ","End":"02:47.020","Text":"In the next 1,"},{"Start":"02:47.020 ","End":"02:52.490","Text":"we have the vector and now its magnitude,"},{"Start":"02:53.360 ","End":"02:56.010","Text":"put it in bars."},{"Start":"02:56.010 ","End":"03:00.145","Text":"What it means is the square root of this squared,"},{"Start":"03:00.145 ","End":"03:04.760","Text":"0 squared plus 1 squared plus 0 squared,"},{"Start":"03:04.760 ","End":"03:06.200","Text":"which is the square root of 1,"},{"Start":"03:06.200 ","End":"03:08.105","Text":"which is 1 so yes,"},{"Start":"03:08.105 ","End":"03:13.175","Text":"this is a unit vector."},{"Start":"03:13.175 ","End":"03:16.840","Text":"Now this one\u0027s in 3D, this one\u0027s in 2D."},{"Start":"03:16.840 ","End":"03:19.500","Text":"Now in Part c and d,"},{"Start":"03:19.500 ","End":"03:21.800","Text":"we\u0027re talking about the position vector,"},{"Start":"03:21.800 ","End":"03:25.055","Text":"when we have the position vector of a point,"},{"Start":"03:25.055 ","End":"03:31.710","Text":"is actually a displacement vector from the origin to the point."},{"Start":"03:31.710 ","End":"03:35.990","Text":"What it is is just the origin is 0,0."},{"Start":"03:35.990 ","End":"03:37.925","Text":"We just subtract zeros 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":"03:49.200","Text":"In Part c, I can straight away write the answer as 1/2,"},{"Start":"03:49.200 ","End":"03:51.435","Text":"root 3/2 and in Part d,"},{"Start":"03:51.435 ","End":"03:58.560","Text":"the answer will be minus 8,3,5."},{"Start":"03:58.560 ","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":"04:10.260 ","End":"04:13.660","Text":"By the way, this happens to be a unit vector."},{"Start":"04:13.660 ","End":"04:17.125","Text":"I\u0027m familiar with it but that wasn\u0027t what was asked for here."},{"Start":"04:17.125 ","End":"04:18.790","Text":"If you take this square it\u0027s 1/4,"},{"Start":"04:18.790 ","End":"04:20.440","Text":"this square 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":"04:26.769","Text":"This 1 in the other notation,"},{"Start":"04:26.769 ","End":"04:36.820","Text":"minus 8i plus 3j plus 5k wasn\u0027t asked for 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":"04:41.835","Text":"That\u0027s Part 1."},{"Start":"04:41.835 ","End":"04:50.595","Text":"In Part 2, 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":"05:02.730","Text":"We\u0027re given the point and we\u0027re given a vector,"},{"Start":"05:02.730 ","End":"05:05.880","Text":"the tail and the head of the vector."},{"Start":"05:05.880 ","End":"05:11.115","Text":"This is given, the vector,"},{"Start":"05:11.115 ","End":"05:17.675","Text":"and this is what we have to find is what\u0027s the head of the vector, where it ends."},{"Start":"05:17.675 ","End":"05:25.490","Text":"What we do is we take for this the coordinates of P and we"},{"Start":"05:25.490 ","End":"05:33.330","Text":"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":"05:48.030","Text":"and then minus 1 plus 0, that\u0027s the endpoint."},{"Start":"05:48.030 ","End":"05:53.000","Text":"Because if this vector is the endpoint minus the start point,"},{"Start":"05:53.000 ","End":"05:59.160","Text":"the start point plus the vector is the endpoint when I say plus I mean coordinate-wise."},{"Start":"05:59.420 ","End":"06:02.535","Text":"This will be, let\u0027s see,"},{"Start":"06:02.535 ","End":"06:06.540","Text":"4 this comes out to be 1."},{"Start":"06:06.540 ","End":"06:11.780","Text":"This comes out to be minus 1 and that\u0027s it,"},{"Start":"06:11.780 ","End":"06:14.039","Text":"we\u0027re done for this exercise."}],"ID":10283}],"Thumbnail":null,"ID":57470},{"Name":"Vector Arithmetic","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vector Arithmetic","Duration":"23m 37s","ChapterTopicVideoID":10098,"CourseChapterTopicPlaylistID":57471,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10098.jpeg","UploadDate":"2020-01-16T07:08:22.6200000","DurationForVideoObject":"PT23M37S","Description":null,"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":"00:08.280","Text":"This will apply to all dimensions,"},{"Start":"00:08.280 ","End":"00:12.165","Text":"the 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":"00:22.320","Text":"Let\u0027s start with the simplest,"},{"Start":"00:22.320 ","End":"00:29.470","Text":"which is addition, and then we\u0027ll go on to subtraction and other operations."},{"Start":"00:29.660 ","End":"00:34.350","Text":"Let\u0027s say in 3-dimensions that I have 2 vectors, 1 of them,"},{"Start":"00:34.350 ","End":"00:41.210","Text":"I\u0027ll call a, and this will be a_1, a_2, a_3."},{"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 b_1, b_2, b_3."},{"Start":"00:53.220 ","End":"00:56.575","Text":"What I\u0027m going to do is define the addition,"},{"Start":"00:56.575 ","End":"01:01.000","Text":"a plus b, to be,"},{"Start":"01:01.000 ","End":"01:04.100","Text":"the obvious thing to do is component-wise addition."},{"Start":"01:04.100 ","End":"01:06.350","Text":"In other words, we add the first components,"},{"Start":"01:06.350 ","End":"01:09.140","Text":"so that would be a_1 plus b_1,"},{"Start":"01:09.140 ","End":"01:12.980","Text":"and the next 1 would be a_2 plus b_2,"},{"Start":"01:12.980 ","End":"01:17.390","Text":"and then a_3 plus b_3."},{"Start":"01:17.390 ","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 a_1 plus a_2 and b_1 plus b_2 in 2-dimensions."},{"Start":"01:29.990 ","End":"01:33.840","Text":"I\u0027ll just give you a simple example."},{"Start":"01:33.840 ","End":"01:36.209","Text":"Just make up some numbers."},{"Start":"01:36.209 ","End":"01:41.340","Text":"2, minus 4, 7 plus,"},{"Start":"01:41.340 ","End":"01:47.700","Text":"I don\u0027t know, 3, 8, minus 11."},{"Start":"01:47.700 ","End":"01:49.760","Text":"Then what this equals,"},{"Start":"01:49.760 ","End":"01:51.410","Text":"this will be my a, this will be my b,"},{"Start":"01:51.410 ","End":"01:54.515","Text":"we add 2 plus 3 is 5,"},{"Start":"01:54.515 ","End":"01:57.665","Text":"negative 4 and 8 is 4,"},{"Start":"01:57.665 ","End":"02:00.920","Text":"7 and minus 11 is minus 4."},{"Start":"02:00.920 ","End":"02:02.180","Text":"That\u0027s all that\u0027s to it,"},{"Start":"02:02.180 ","End":"02:05.540","Text":"and similarly for in 2D."},{"Start":"02:05.540 ","End":"02:10.550","Text":"There is an illustration geometrically of what this means."},{"Start":"02:10.550 ","End":"02:13.700","Text":"I\u0027ll illustrate it in 2-dimensions where this is"},{"Start":"02:13.700 ","End":"02:16.780","Text":"the x-direction and this is the y-direction."},{"Start":"02:16.780 ","End":"02:20.155","Text":"Suppose I have 2 vectors a,"},{"Start":"02:20.155 ","End":"02:22.960","Text":"let\u0027s say that a is a_1,"},{"Start":"02:22.960 ","End":"02:29.225","Text":"a_2 and b is b_1, b_2,"},{"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":"02:47.610","Text":"What we would expect this to be is a_1 plus a_2, b_1 plus b_2."},{"Start":"02:47.610 ","End":"02:49.960","Text":"Let me just show you briefly why this is so"},{"Start":"02:49.960 ","End":"02:52.910","Text":"whether you could accept it on faith without proof."},{"Start":"02:52.910 ","End":"02:54.920","Text":"Say that this vector is a_1,"},{"Start":"02:54.920 ","End":"03:03.900","Text":"a_2 means that this part here would be a_1,"},{"Start":"03:03.900 ","End":"03:06.315","Text":"and this part here would be a_2."},{"Start":"03:06.315 ","End":"03:16.220","Text":"Similarly, I could make a little vertical line here and another line here,"},{"Start":"03:16.220 ","End":"03:20.790","Text":"and this would be b_1, just this bit,"},{"Start":"03:20.790 ","End":"03:25.440","Text":"and this part would be b_2,"},{"Start":"03:25.440 ","End":"03:26.940","Text":"from here to here."},{"Start":"03:26.940 ","End":"03:29.650","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":"03:36.715","Text":"then from here to here,"},{"Start":"03:36.715 ","End":"03:39.865","Text":"it\u0027s also going to be b_1."},{"Start":"03:39.865 ","End":"03:42.455","Text":"I meant to draw this over here."},{"Start":"03:42.455 ","End":"03:45.800","Text":"Since this 1 is parallel to this 1,"},{"Start":"03:45.800 ","End":"03:51.585","Text":"this bit here is also going to be a_2,"},{"Start":"03:51.585 ","End":"03:53.519","Text":"the same as this,"},{"Start":"03:53.519 ","End":"03:57.340","Text":"and so we can say that the coordinates of this up to here,"},{"Start":"03:57.340 ","End":"03:58.570","Text":"it\u0027s a_1 plus b_1,"},{"Start":"03:58.570 ","End":"04:00.490","Text":"up to here it\u0027s a_2 plus b_2,"},{"Start":"04:00.490 ","End":"04:03.640","Text":"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":"04:08.275","Text":"and this is the proof in 2D."},{"Start":"04:08.275 ","End":"04:12.280","Text":"Back to vector arithmetic,"},{"Start":"04:12.280 ","End":"04:18.805","Text":"that explains addition, 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":"04:25.485","Text":"For subtraction, we use the same a and b,"},{"Start":"04:25.485 ","End":"04:28.300","Text":"and I\u0027ll start out by copying this formula."},{"Start":"04:28.300 ","End":"04:34.755","Text":"All we have to do is change the pluses into minuses,"},{"Start":"04:34.755 ","End":"04:39.640","Text":"and there we have the formula for subtraction of vectors."},{"Start":"04:39.640 ","End":"04:42.815","Text":"If I take the same 2 as an example,"},{"Start":"04:42.815 ","End":"04:45.080","Text":"I copied the addition,"},{"Start":"04:45.080 ","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":"04:55.625","Text":"2 minus 3 is minus 1,"},{"Start":"04:55.625 ","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":"05:13.930","Text":"There\u0027s also a diagram pictorially to show what\u0027s going on."},{"Start":"05:13.930 ","End":"05:18.905","Text":"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":"05:31.025","Text":"then the vector a minus b is what you get when you draw a vector from the tip of b,"},{"Start":"05:31.025 ","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":"05:44.315","Text":"By the way, the addition rule sometimes is"},{"Start":"05:44.315 ","End":"05:49.520","Text":"called the parallelogram law for obvious reasons."},{"Start":"05:49.520 ","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":"06:07.260","Text":"we could put the vector b here."},{"Start":"06:07.820 ","End":"06:12.520","Text":"Remember what we said about vectors that they have magnitude and direction,"},{"Start":"06:12.520 ","End":"06:15.745","Text":"but it doesn\u0027t matter where you attach the tail to."},{"Start":"06:15.745 ","End":"06:17.440","Text":"This is vector b,"},{"Start":"06:17.440 ","End":"06:19.495","Text":"this is also vector b."},{"Start":"06:19.495 ","End":"06:23.560","Text":"Then if we place the head of the 1 to the tail of the other,"},{"Start":"06:23.560 ","End":"06:26.800","Text":"then we get a triangle, so 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":"06:50.260","Text":"Instead of 1, 2 3, 1,"},{"Start":"06:50.260 ","End":"06:51.920","Text":"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":"07:04.430","Text":"but I want to emphasize."},{"Start":"07:04.430 ","End":"07:06.335","Text":"That because we now have vectors,"},{"Start":"07:06.335 ","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":"07:27.694","Text":"which actually means multiplication of a scalar by a vector."},{"Start":"07:27.694 ","End":"07:34.860","Text":"I\u0027ll explain. Suppose we have a vector a, same as before,"},{"Start":"07:34.860 ","End":"07:38.130","Text":"you can take it in 3D, a_1, a_2,"},{"Start":"07:38.130 ","End":"07:40.305","Text":"a_3 are its components,"},{"Start":"07:40.305 ","End":"07:42.910","Text":"and that\u0027s a vector."},{"Start":"07:44.930 ","End":"07:47.940","Text":"It\u0027s a different color C,"},{"Start":"07:47.940 ","End":"07:49.980","Text":"which is a scalar,"},{"Start":"07:49.980 ","End":"07:52.335","Text":"which means a number."},{"Start":"07:52.335 ","End":"07:58.440","Text":"We have that C times a,"},{"Start":"07:58.440 ","End":"08:01.320","Text":"we just write it as Ca without anything in between,"},{"Start":"08:01.320 ","End":"08:05.190","Text":"or you could put a dot optionally."},{"Start":"08:05.190 ","End":"08:09.380","Text":"Scalar times a vector is going to give a new vector,"},{"Start":"08:09.380 ","End":"08:11.765","Text":"and this new vector,"},{"Start":"08:11.765 ","End":"08:15.815","Text":"we start off by a_3,"},{"Start":"08:15.815 ","End":"08:23.510","Text":"a_2, a_1, and we multiply each of the components by that same"},{"Start":"08:23.510 ","End":"08:27.770","Text":"C. I think the reason it\u0027s called the scalar is that it"},{"Start":"08:27.770 ","End":"08:32.900","Text":"makes a scaled version larger or smaller of the original."},{"Start":"08:32.900 ","End":"08:36.365","Text":"I\u0027ll give some examples and then we show it on the diagram."},{"Start":"08:36.365 ","End":"08:41.630","Text":"Then I\u0027m going to give a 2D example so that afterwards I can draw it better."},{"Start":"08:41.630 ","End":"08:45.890","Text":"So let\u0027s take a to be the vector 2, 4,"},{"Start":"08:45.890 ","End":"08:49.650","Text":"and let\u0027s take the scalar C,"},{"Start":"08:49.650 ","End":"08:52.070","Text":"well, I want 3 different examples."},{"Start":"08:52.070 ","End":"08:57.920","Text":"I want to take C first as 3, then as 1/2,"},{"Start":"08:57.920 ","End":"09:00.920","Text":"just to show you that we can take fractions or smaller than 1,"},{"Start":"09:00.920 ","End":"09:04.190","Text":"and 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":"09:09.640","Text":"I have to compute C times a for all of these."},{"Start":"09:09.640 ","End":"09:16.425","Text":"If I take the 3, then 3a is going to be just 3 times 2 is 6,"},{"Start":"09:16.425 ","End":"09:18.659","Text":"3 times 4 is 12."},{"Start":"09:18.659 ","End":"09:22.990","Text":"I didn\u0027t even bother to write that 6 is 3 times 2, we can see that."},{"Start":"09:22.990 ","End":"09:26.555","Text":"Then the next example, 1/2,"},{"Start":"09:26.555 ","End":"09:32.885","Text":"1/2 of a is just take the components of a and multiply each 1 by 1/2,"},{"Start":"09:32.885 ","End":"09:36.275","Text":"so it\u0027ll be 1, 2."},{"Start":"09:36.275 ","End":"09:39.665","Text":"For the third example, minus 2."},{"Start":"09:39.665 ","End":"09:42.365","Text":"If I multiply that by a,"},{"Start":"09:42.365 ","End":"09:44.060","Text":"I just get, again,"},{"Start":"09:44.060 ","End":"09:45.950","Text":"just negative makes no difference."},{"Start":"09:45.950 ","End":"09:49.710","Text":"Minus 2 times 2 is minus 4,"},{"Start":"09:49.710 ","End":"09:53.940","Text":"minus 2 times 4 is minus 8,"},{"Start":"09:53.940 ","End":"09:59.260","Text":"and I\u0027ll show you what these look like on a graph."},{"Start":"09:59.260 ","End":"10:05.790","Text":"Here\u0027s our diagram. The original a,"},{"Start":"10:05.790 ","End":"10:07.815","Text":"which is the 2,"},{"Start":"10:07.815 ","End":"10:10.799","Text":"4, is in black,"},{"Start":"10:10.799 ","End":"10:13.140","Text":"we just see the head of it."},{"Start":"10:13.140 ","End":"10:19.945","Text":"The 3a is this vector here, the blue 1."},{"Start":"10:19.945 ","End":"10:24.140","Text":"It goes all the way from the origin,"},{"Start":"10:24.140 ","End":"10:26.675","Text":"but we just didn\u0027t color it,"},{"Start":"10:26.675 ","End":"10:29.030","Text":"becomes 1 color on top of the other."},{"Start":"10:29.030 ","End":"10:36.245","Text":"1/2a is the green 1 and minus 2 a 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":"10:46.910","Text":"it goes in exactly the same direction,"},{"Start":"10:46.910 ","End":"10:49.100","Text":"and 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":"11:06.160","Text":"Let me add a fourth example here, 0."},{"Start":"11:07.010 ","End":"11:10.860","Text":"0a, in our case,"},{"Start":"11:10.860 ","End":"11:17.100","Text":"it\u0027s 0 times 2 is 0,"},{"Start":"11:17.100 ","End":"11:20.955","Text":"0 times 4 is 0,"},{"Start":"11:20.955 ","End":"11:25.085","Text":"and in fact, what we get is the 0 vector."},{"Start":"11:25.085 ","End":"11:29.015","Text":"This is true in general for vectors of any dimension,"},{"Start":"11:29.015 ","End":"11:39.184","Text":"that the scalar 0 times any vector will always give me the 0 vector."},{"Start":"11:39.184 ","End":"11:43.745","Text":"Just make a note, this is always in any dimension."},{"Start":"11:43.745 ","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":"11:54.150 ","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":"12:01.300","Text":"A and B vectors,"},{"Start":"12:01.300 ","End":"12:02.605","Text":"and they are parallel,"},{"Start":"12:02.605 ","End":"12:07.165","Text":"just means that 1 of them is a scalar times the other so that A might"},{"Start":"12:07.165 ","End":"12:12.310","Text":"be equal to a scalar C times the other vector,"},{"Start":"12:12.310 ","End":"12:20.330","Text":"and usually, we exclude the 0 vectors."},{"Start":"12:20.490 ","End":"12:23.630","Text":"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":"12:29.680","Text":"If 1 is a constant,"},{"Start":"12:29.680 ","End":"12:31.915","Text":"a scalar times the other."},{"Start":"12:31.915 ","End":"12:35.140","Text":"Even if they\u0027re in opposite directions, it\u0027s 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":"12:41.590","Text":"it would still be parallel."},{"Start":"12:41.590 ","End":"12:44.935","Text":"Take another example with parallel."},{"Start":"12:44.935 ","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":"12:53.275","Text":"The first 1 will be, I don\u0027t know,"},{"Start":"12:53.275 ","End":"12:57.265","Text":"3 minus 5, 2,"},{"Start":"12:57.265 ","End":"12:59.745","Text":"and that will be,"},{"Start":"12:59.745 ","End":"13:05.115","Text":"we\u0027ll call that a, and the other 1 will be,"},{"Start":"13:05.115 ","End":"13:13.000","Text":"let\u0027s say, minus 9, 15, minus 6."},{"Start":"13:13.000 ","End":"13:14.785","Text":"This 1, we\u0027ll call it B."},{"Start":"13:14.785 ","End":"13:18.070","Text":"My question is, are A and B parallel?"},{"Start":"13:18.070 ","End":"13:22.720","Text":"Well, we have to find a constant that multiplies 1 to give the other,"},{"Start":"13:22.720 ","End":"13:27.145","Text":"it means we have to find the same constant that multiplies each of these 3 to give these."},{"Start":"13:27.145 ","End":"13:33.235","Text":"We look at the first 1, 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 comp1nts 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":"13:50.260","Text":"Good. I can say,"},{"Start":"13:50.260 ","End":"13:53.140","Text":"if I take C equals minus 3,"},{"Start":"13:53.140 ","End":"13:58.735","Text":"then C times A is equal to B."},{"Start":"13:58.735 ","End":"14:01.855","Text":"It doesn\u0027t really matter on which side you put the C 1."},{"Start":"14:01.855 ","End":"14:05.005","Text":"Let\u0027s take another example in 2 dimensions."},{"Start":"14:05.005 ","End":"14:11.350","Text":"Let\u0027s take 2, 6,"},{"Start":"14:11.350 ","End":"14:19.105","Text":"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, and 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 constant times the other."},{"Start":"14:29.230 ","End":"14:34.080","Text":"Let\u0027s see. I could multiply 2 by 1/2 to get 1."},{"Start":"14:34.080 ","End":"14:36.795","Text":"But if C is going to be 1/2,"},{"Start":"14:36.795 ","End":"14:39.569","Text":"and I have to multiply it by 6,"},{"Start":"14:39.569 ","End":"14:41.310","Text":"1/2 times 6 is 3,"},{"Start":"14:41.310 ","End":"14:42.720","Text":"which is not 5."},{"Start":"14:42.720 ","End":"14:45.975","Text":"I can\u0027t find 1 C that\u0027s good for all the comp1nts."},{"Start":"14:45.975 ","End":"14:48.425","Text":"These are not parallel,"},{"Start":"14:48.425 ","End":"14:55.960","Text":"and there is no such C. That\u0027s just how we can tell by looking at the numbers."},{"Start":"14:55.960 ","End":"14:59.680","Text":"The next minor topic I want to discuss,"},{"Start":"14:59.680 ","End":"15:02.245","Text":"I\u0027ll just give it a name."},{"Start":"15:02.245 ","End":"15:05.650","Text":"A common task that we need to do with vectors,"},{"Start":"15:05.650 ","End":"15:09.100","Text":"and we\u0027ll see this later on is that when we\u0027re given a vector,"},{"Start":"15:09.100 ","End":"15:13.165","Text":"we want to find a unit vector parallel to it."},{"Start":"15:13.165 ","End":"15:14.950","Text":"Not just parallel, but actually,"},{"Start":"15:14.950 ","End":"15:16.600","Text":"in the same direction."},{"Start":"15:16.600 ","End":"15:18.040","Text":"Because if we have 2 unit vectors,"},{"Start":"15:18.040 ","End":"15:20.245","Text":"1 could be in the opposite direction."},{"Start":"15:20.245 ","End":"15:28.315","Text":"I brought an example that we previously did about finding the magnitude of a vector."},{"Start":"15:28.315 ","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":"15:35.380","Text":"we took its magnitude using the formula,"},{"Start":"15:35.380 ","End":"15:37.165","Text":"and we got 13."},{"Start":"15:37.165 ","End":"15:47.515","Text":"Now, it turns out that if I take 1 over 13 times our vector, in other words,"},{"Start":"15:47.515 ","End":"15:57.955","Text":"we get, I\u0027ll call it W. W is in fact, let\u0027s see,"},{"Start":"15:57.955 ","End":"16:03.895","Text":"3 over 13, 4 over 13,"},{"Start":"16:03.895 ","End":"16:10.165","Text":"12 over 13, and a little arrow on top."},{"Start":"16:10.165 ","End":"16:15.135","Text":"Then, it turns out that W meets our criterion,"},{"Start":"16:15.135 ","End":"16:18.660","Text":"but W is parallel to V. Well,"},{"Start":"16:18.660 ","End":"16:19.980","Text":"the parallel is obvious."},{"Start":"16:19.980 ","End":"16:22.484","Text":"When you take a vector and multiply it by a constant,"},{"Start":"16:22.484 ","End":"16:24.315","Text":"that\u0027s a definition of parallel."},{"Start":"16:24.315 ","End":"16:26.720","Text":"But why is it a unit vector?"},{"Start":"16:26.720 ","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":"16:36.880","Text":"The magnitude of W is equal"},{"Start":"16:36.880 ","End":"16:44.470","Text":"to the square root of 3 over 13 squared,"},{"Start":"16:44.470 ","End":"16:47.485","Text":"plus 4 over 13 squared,"},{"Start":"16:47.485 ","End":"16:51.145","Text":"plus 12 over 13 squared,"},{"Start":"16:51.145 ","End":"16:58.915","Text":"and this will just equal the square root."},{"Start":"16:58.915 ","End":"17:03.190","Text":"Now, 13 squared goes in the denominator."},{"Start":"17:03.190 ","End":"17:07.390","Text":"I can write over 13 squared."},{"Start":"17:07.390 ","End":"17:10.885","Text":"Here, I have 3 squared, plus 4 squared,"},{"Start":"17:10.885 ","End":"17:16.810","Text":"plus 12 squared, and basically,"},{"Start":"17:16.810 ","End":"17:20.980","Text":"what we get is just like here because the square root of 3 squared, plus 4 squared,"},{"Start":"17:20.980 ","End":"17:23.515","Text":"plus 12 squared is 13,"},{"Start":"17:23.515 ","End":"17:25.570","Text":"we get 13 over 13,"},{"Start":"17:25.570 ","End":"17:30.650","Text":"which is exactly equal to 1."},{"Start":"17:33.150 ","End":"17:36.384","Text":"This means that this is a unit vector,"},{"Start":"17:36.384 ","End":"17:40.915","Text":"and it\u0027s also in the same direction."},{"Start":"17:40.915 ","End":"17:43.540","Text":"Perhaps I should have menti1d that with parallel,"},{"Start":"17:43.540 ","End":"17:45.970","Text":"if the C is positive,"},{"Start":"17:45.970 ","End":"17:49.015","Text":"then they\u0027re parallel and in the same direction,"},{"Start":"17:49.015 ","End":"17:52.765","Text":"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":"18:04.660","Text":"In general, if I\u0027m given a vector V,"},{"Start":"18:04.660 ","End":"18:10.040","Text":"and then I define a new vector, I\u0027ll call it U."},{"Start":"18:10.560 ","End":"18:16.690","Text":"In fact, I think I should have called this 1 U because U for unit."},{"Start":"18:16.690 ","End":"18:19.285","Text":"There, just give it a name change."},{"Start":"18:19.285 ","End":"18:21.490","Text":"If I let U, in general,"},{"Start":"18:21.490 ","End":"18:27.130","Text":"be 1 over the magnitude of V,"},{"Start":"18:27.130 ","End":"18:31.750","Text":"a little arrow here, and multiply it."},{"Start":"18:31.750 ","End":"18:36.865","Text":"This is a scalar times the original vector V, then U,"},{"Start":"18:36.865 ","End":"18:42.010","Text":"this 1 will always be a unit vector,"},{"Start":"18:42.010 ","End":"18:46.825","Text":"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 its magnitude,"},{"Start":"18:51.265 ","End":"18:53.155","Text":"take 1 over the magnitude,"},{"Start":"18:53.155 ","End":"18:54.550","Text":"multiply by the vector,"},{"Start":"18:54.550 ","End":"18:57.759","Text":"we get a unit vector in the same direction."},{"Start":"18:57.759 ","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":"19:05.830","Text":"I menti1d the concept standard basis vectors,"},{"Start":"19:05.830 ","End":"19:07.240","Text":"and just to remind you,"},{"Start":"19:07.240 ","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":"19:19.570","Text":"There\u0027s a 1, and the rest are 0s,"},{"Start":"19:19.570 ","End":"19:22.090","Text":"and in 3D they\u0027re called I, j,"},{"Start":"19:22.090 ","End":"19:28.820","Text":"and k. In 2D they are I and j. I\u0027m not sure what letters we use in higher dimensions."},{"Start":"19:29.250 ","End":"19:32.140","Text":"I\u0027m going to do something backward here."},{"Start":"19:32.140 ","End":"19:33.880","Text":"I\u0027m going to show you what I\u0027m heading towards,"},{"Start":"19:33.880 ","End":"19:36.370","Text":"and then I\u0027ll give you the theory."},{"Start":"19:36.370 ","End":"19:39.550","Text":"There\u0027s going to be another way to write vectors."},{"Start":"19:39.550 ","End":"19:42.295","Text":"If I have a vector, like 3, 4,"},{"Start":"19:42.295 ","End":"19:47.680","Text":"5, either, I could say that this is equal to 3i,"},{"Start":"19:47.680 ","End":"19:52.370","Text":"plus 4j, plus 5k,"},{"Start":"19:54.180 ","End":"19:57.280","Text":"and 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":"20:03.580","Text":"Let\u0027s take A_1, A_2,"},{"Start":"20:03.580 ","End":"20:07.525","Text":"A_3 to be a vector in 3D."},{"Start":"20:07.525 ","End":"20:10.165","Text":"Now, because of addition of vectors,"},{"Start":"20:10.165 ","End":"20:14.800","Text":"I can write this as A_1, 0, 0,"},{"Start":"20:14.800 ","End":"20:20.290","Text":"plus 0, A_2, 0,"},{"Start":"20:20.290 ","End":"20:25.180","Text":"plus 0, 0, A_3."},{"Start":"20:25.180 ","End":"20:28.525","Text":"We discussed addition of vectors."},{"Start":"20:28.525 ","End":"20:31.045","Text":"Basically, you just add the first comp1nt,"},{"Start":"20:31.045 ","End":"20:32.830","Text":"A_1 plus 0, plus 0,"},{"Start":"20:32.830 ","End":"20:35.230","Text":"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":"20:56.110","Text":"The second 1 I can say is A_2 times 0, 1, 0."},{"Start":"20:56.110 ","End":"20:57.670","Text":"Just multiply each 1,"},{"Start":"20:57.670 ","End":"21:00.205","Text":"you get 0, A_2, 0,"},{"Start":"21:00.205 ","End":"21:01.450","Text":"and the third 1, of course,"},{"Start":"21:01.450 ","End":"21:05.065","Text":"is A_3 times 0, 0, 1."},{"Start":"21:05.065 ","End":"21:07.720","Text":"But look, definition of this is i."},{"Start":"21:07.720 ","End":"21:09.950","Text":"This is just A_1i,"},{"Start":"21:10.350 ","End":"21:15.930","Text":"plus A_2j, this 1 is j,"},{"Start":"21:15.930 ","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":"21:24.415","Text":"A_2, A_3, in general,"},{"Start":"21:24.415 ","End":"21:25.480","Text":"is equal to this,"},{"Start":"21:25.480 ","End":"21:27.130","Text":"and if I choose 3, 4,"},{"Start":"21:27.130 ","End":"21:30.395","Text":"5, and 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":"21:43.420","Text":"Another way is to write them with the standard basis vectors I,"},{"Start":"21:43.420 ","End":"21:47.695","Text":"j, and k. I just wanted to mention this for reference."},{"Start":"21:47.695 ","End":"21:57.835","Text":"I want to just end with some final arithmetic formulas that we haven\u0027t covered up to now."},{"Start":"21:57.835 ","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":"22:10.120","Text":"When we add 3 different vectors,"},{"Start":"22:10.120 ","End":"22:14.125","Text":"it doesn\u0027t matter which 2 you take first,"},{"Start":"22:14.125 ","End":"22:15.490","Text":"we learned addition of 2."},{"Start":"22:15.490 ","End":"22:18.685","Text":"It doesn\u0027t matter if you add these 2 and then add this to the third."},{"Start":"22:18.685 ","End":"22:21.640","Text":"Or you add these 2 and you take this plus this."},{"Start":"22:21.640 ","End":"22:23.320","Text":"That\u0027s like in arithmetic,"},{"Start":"22:23.320 ","End":"22:25.495","Text":"what we call the associative law,"},{"Start":"22:25.495 ","End":"22:27.730","Text":"and then there\u0027s another rule is that it doesn\u0027t"},{"Start":"22:27.730 ","End":"22:30.250","Text":"matter in which order you take 2 of them."},{"Start":"22:30.250 ","End":"22:35.780","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":"22:54.400","Text":"it doesn\u0027t change the vector."},{"Start":"22:54.400 ","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 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":"23:12.025","Text":"we could take 3 times each of them and then add,"},{"Start":"23:12.025 ","End":"23:15.145","Text":"and the other thing is another distributive."},{"Start":"23:15.145 ","End":"23:18.549","Text":"If I have 2 plus 3 times a vector,"},{"Start":"23:18.549 ","End":"23:20.695","Text":"in other words, 5 times a vector,"},{"Start":"23:20.695 ","End":"23:22.360","Text":"it\u0027s the same as twice the vector,"},{"Start":"23:22.360 ","End":"23:25.225","Text":"plus 3 times the vector, for example,"},{"Start":"23:25.225 ","End":"23:27.610","Text":"and I\u0027m not going to prove these,"},{"Start":"23:27.610 ","End":"23:29.740","Text":"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":"23:37.430","Text":"We\u0027re d1 with this clip,"}],"ID":10277},{"Watched":false,"Name":"Exercise 1","Duration":"3m 54s","ChapterTopicVideoID":10099,"CourseChapterTopicPlaylistID":57471,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10099.jpeg","UploadDate":"2017-08-20T10:28:12.8230000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.555","Text":"In this exercise, there\u0027s really 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":"00:40.575","Text":"5 times 4, 20."},{"Start":"00:40.575 ","End":"00:43.680","Text":"That\u0027s it. In B,"},{"Start":"00:43.680 ","End":"00:48.905","Text":"we have to do a bit of addition and subtraction as well as scalar multiplication."},{"Start":"00:48.905 ","End":"00:50.720","Text":"I\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":"01:16.470","Text":"let\u0027s do the scalar by vector multiplication."},{"Start":"01:16.470 ","End":"01:20.550","Text":"6 times minus 2 is minus 12,"},{"Start":"01:20.550 ","End":"01:23.145","Text":"6 times 5 is 30."},{"Start":"01:23.145 ","End":"01:24.660","Text":"That\u0027s the first 1."},{"Start":"01:24.660 ","End":"01:29.070","Text":"The second 1, 3 times 7 is 21,"},{"Start":"01:29.070 ","End":"01:31.559","Text":"3 times 4 is 12."},{"Start":"01:31.559 ","End":"01:33.210","Text":"2 scalar multiplications."},{"Start":"01:33.210 ","End":"01:34.650","Text":"Now, subtraction."},{"Start":"01:34.650 ","End":"01:37.790","Text":"Subtraction, just subtract component-wise."},{"Start":"01:37.790 ","End":"01:41.435","Text":"Minus 12 minus 21."},{"Start":"01:41.435 ","End":"01:50.030","Text":"I make that minus 33 and 30 minus"},{"Start":"01:50.030 ","End":"01:52.735","Text":"12 is 18."},{"Start":"01:52.735 ","End":"02:00.680","Text":"Part C. Let\u0027s, first of all,"},{"Start":"02:00.680 ","End":"02:07.960","Text":"do the inside and then we\u0027ll take the norm 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":"02:56.630 ","End":"02:58.850","Text":"Then the second component,"},{"Start":"02:58.850 ","End":"03:01.115","Text":"9 times 4 is 36,"},{"Start":"03:01.115 ","End":"03:07.735","Text":"4 times 5 is 20."},{"Start":"03:07.735 ","End":"03:10.560","Text":"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 6161."},{"Start":"03:35.795 ","End":"03:37.475","Text":"I would leave it like that,"},{"Start":"03:37.475 ","End":"03:41.610","Text":"but if you want a numerical result,"},{"Start":"03:41.610 ","End":"03:43.655","Text":"a calculator will help."},{"Start":"03:43.655 ","End":"03:53.800","Text":"78.492 something approximately. That\u0027s it."}],"ID":10278},{"Watched":false,"Name":"Exercise 2","Duration":"4m 57s","ChapterTopicVideoID":10100,"CourseChapterTopicPlaylistID":57471,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10100.jpeg","UploadDate":"2017-08-20T10:29:42.4870000","DurationForVideoObject":"PT4M57S","Description":null,"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":"00:18.750","Text":"Let\u0027s start with the Part A."},{"Start":"00:18.750 ","End":"00:21.450","Text":"What we have is that v is this,"},{"Start":"00:21.450 ","End":"00:23.085","Text":"and we want minus 4v,"},{"Start":"00:23.085 ","End":"00:31.395","Text":"so we want minus 4 times 6j minus 2k,"},{"Start":"00:31.395 ","End":"00:33.900","Text":"so scalar by a vector,"},{"Start":"00:33.900 ","End":"00:37.680","Text":"we just multiply it as you would expect."},{"Start":"00:37.680 ","End":"00:43.305","Text":"Multiply the scalar by the j component and by the k component."},{"Start":"00:43.305 ","End":"00:44.910","Text":"There is no i component,"},{"Start":"00:44.910 ","End":"00:48.040","Text":"so 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":"01:07.040","Text":"What we have here is 10 times the vector u, which is 7i."},{"Start":"01:07.040 ","End":"01:11.720","Text":"I\u0027m tired of writing the arrows all the time."},{"Start":"01:11.720 ","End":"01:20.390","Text":"Let\u0027s write minus 2j plus 4k and plus v,"},{"Start":"01:20.390 ","End":"01:25.530","Text":"which is 6j minus 2k,"},{"Start":"01:27.050 ","End":"01:30.405","Text":"and this is equal to."},{"Start":"01:30.405 ","End":"01:35.750","Text":"First of all, I\u0027ll do the multiplication of the scalar by the vector,"},{"Start":"01:35.750 ","End":"01:39.470","Text":"so this gives me 70i minus"},{"Start":"01:39.470 ","End":"01:46.185","Text":"20j plus 40k, that\u0027s 1."},{"Start":"01:46.185 ","End":"01:49.470","Text":"Then the other 1, just as is,"},{"Start":"01:49.470 ","End":"01:53.900","Text":"6j minus 2k, there is no i component."},{"Start":"01:53.900 ","End":"01:58.760","Text":"Some people like to write 0i just to have it complete, possible,"},{"Start":"01:58.760 ","End":"02:01.980","Text":"and this equals component-wise,"},{"Start":"02:01.980 ","End":"02:04.155","Text":"70i, there is no i here,"},{"Start":"02:04.155 ","End":"02:06.045","Text":"so it\u0027s just 70i."},{"Start":"02:06.045 ","End":"02:11.625","Text":"Then minus 20 plus 6 with the j."},{"Start":"02:11.625 ","End":"02:17.565","Text":"That\u0027s minus 14j, and 40"},{"Start":"02:17.565 ","End":"02:24.090","Text":"minus 2 is 38k."},{"Start":"02:24.090 ","End":"02:26.265","Text":"Now I\u0027ll go fix those arrows,"},{"Start":"02:26.265 ","End":"02:28.590","Text":"and there we are."},{"Start":"02:28.590 ","End":"02:31.245","Text":"Finally, in Part C,"},{"Start":"02:31.245 ","End":"02:34.725","Text":"we also have a magnitude,"},{"Start":"02:34.725 ","End":"02:37.545","Text":"so what we have is minus 8,"},{"Start":"02:37.545 ","End":"02:45.465","Text":"and then u is 7i minus 2j plus 4k,"},{"Start":"02:45.465 ","End":"02:53.860","Text":"and minus 3 vector v is 6j minus 2k."},{"Start":"02:56.240 ","End":"03:00.035","Text":"This is equal to, first of all,"},{"Start":"03:00.035 ","End":"03:03.965","Text":"I\u0027ll do the scalar product, scalar with vector."},{"Start":"03:03.965 ","End":"03:08.100","Text":"Minus 8 with this just component-wise,"},{"Start":"03:10.930 ","End":"03:20.870","Text":"minus 56i and then plus 16j minus 32k."},{"Start":"03:20.870 ","End":"03:28.895","Text":"Then I\u0027ll take it as a minus and just multiply the 3,"},{"Start":"03:28.895 ","End":"03:34.870","Text":"so I have 18j minus 6k."},{"Start":"03:35.480 ","End":"03:40.930","Text":"Now I have to do the subtraction."},{"Start":"03:41.660 ","End":"03:45.555","Text":"What I have is minus 56i,"},{"Start":"03:45.555 ","End":"03:47.490","Text":"there\u0027s nothing here with i,"},{"Start":"03:47.490 ","End":"03:57.030","Text":"so it stays minus 56i,16j minus 18j is minus 2j."},{"Start":"03:57.030 ","End":"04:04.480","Text":"Minus 32 plus 6 is minus 26k."},{"Start":"04:05.330 ","End":"04:11.900","Text":"That\u0027s just up to the magnitude."},{"Start":"04:11.900 ","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":"04:22.180","Text":"the square root of each 1 squared."},{"Start":"04:22.180 ","End":"04:25.705","Text":"Obviously, the minuses don\u0027t make any difference,"},{"Start":"04:25.705 ","End":"04:32.350","Text":"56 squared plus 2 squared plus 26 squared,"},{"Start":"04:32.350 ","End":"04:36.465","Text":"and this is equal to,"},{"Start":"04:36.465 ","End":"04:41.895","Text":"I make this 3816."},{"Start":"04:41.895 ","End":"04:44.664","Text":"If you have to have a numerical answer,"},{"Start":"04:44.664 ","End":"04:51.890","Text":"then 61.77, something roughly, not so important."},{"Start":"04:51.890 ","End":"04:54.650","Text":"The idea is just to know how to do it."},{"Start":"04:54.650 ","End":"04:57.750","Text":"That\u0027s it for this exercise."}],"ID":10279},{"Watched":false,"Name":"Exercise 3","Duration":"3m 27s","ChapterTopicVideoID":10101,"CourseChapterTopicPlaylistID":57471,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10101.jpeg","UploadDate":"2017-08-20T10:30:55.7530000","DurationForVideoObject":"PT3M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.915","Text":"This exercise has 2 parts to it that just start with."},{"Start":"00:03.915 ","End":"00:07.870","Text":"Part 1, we\u0027re given this vector and we want a unit vector"},{"Start":"00:07.870 ","End":"00:10.360","Text":"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":"00:18.370","Text":"In any event, we\u0027ll need the magnitude of V,"},{"Start":"00:18.370 ","End":"00:22.060","Text":"and this is equal to just using the formula,"},{"Start":"00:22.060 ","End":"00:29.784","Text":"the square root of 1 squared and minus 4 squared and 8 squared,"},{"Start":"00:29.784 ","End":"00:38.515","Text":"which is just 1 plus 16 plus 64."},{"Start":"00:38.515 ","End":"00:44.050","Text":"This comes out as 81."},{"Start":"00:44.050 ","End":"00:48.670","Text":"It\u0027s the square root of 81, which is 9."},{"Start":"00:48.670 ","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":"01:09.385","Text":"This is equal to 1/9"},{"Start":"01:09.385 ","End":"01:19.025","Text":"i minus 4/9 j plus 8/9 k,"},{"Start":"01:19.025 ","End":"01:21.595","Text":"and 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":"01:42.760","Text":"In part 2, we want 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":"01:53.194","Text":"The other difference is that this is in 3D and this is in 2D,"},{"Start":"01:53.194 ","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 are the same idea that 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, that 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":"02:35.060","Text":"but I don\u0027t, I want a vector with magnitude 10."},{"Start":"02:35.060 ","End":"02:40.340","Text":"I\u0027m going to multiply this by 10 and multiply it by 10."},{"Start":"02:40.340 ","End":"02:46.320","Text":"Then I\u0027ll get 10 over root 29 times W,"},{"Start":"02:46.320 ","End":"02:49.750","Text":"and this will come out to be"},{"Start":"02:51.440 ","End":"02:59.690","Text":"angular brackets minus 2 times 10 over 29 is minus 20 over root 29."},{"Start":"02:59.690 ","End":"03:09.650","Text":"Did I just say 29, and then 5 times the 10 over root 29,"},{"Start":"03:09.650 ","End":"03:15.875","Text":"which is 50 over root 29, and that\u0027ll do it."},{"Start":"03:15.875 ","End":"03:20.390","Text":"The idea again is divided by its magnitude,"},{"Start":"03:20.390 ","End":"03:27.570","Text":"but then multiply by 10. That\u0027s it."}],"ID":10280},{"Watched":false,"Name":"Exercise 4","Duration":"3m 54s","ChapterTopicVideoID":10102,"CourseChapterTopicPlaylistID":57471,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10102.jpeg","UploadDate":"2017-08-20T10:32:09.3130000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.524","Text":"In this exercise, we\u0027re given,"},{"Start":"00:03.524 ","End":"00:06.900","Text":"in each of the parts a and b,"},{"Start":"00:06.900 ","End":"00:08.180","Text":"a pair of vectors,"},{"Start":"00:08.180 ","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 to be parallel,"},{"Start":"00:15.300 ","End":"00:18.255","Text":"1 has to be some scalar times the other."},{"Start":"00:18.255 ","End":"00:27.140","Text":"Let\u0027s say that w is going to equal some k times v. Usually,"},{"Start":"00:27.140 ","End":"00:30.740","Text":"we require k not equal to 0."},{"Start":"00:30.740 ","End":"00:34.010","Text":"If k was 0, then w would be 0 and"},{"Start":"00:34.010 ","End":"00:38.330","Text":"it\u0027s debatable whether the 0 vector is parallel to another vector,"},{"Start":"00:38.330 ","End":"00:39.985","Text":"some people might say yes."},{"Start":"00:39.985 ","End":"00:43.180","Text":"Usually, we take k to be non-zero."},{"Start":"00:43.180 ","End":"00:45.570","Text":"Suppose there is such a k,"},{"Start":"00:45.570 ","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":"00:54.890","Text":"Let\u0027s take the first component, the i component,"},{"Start":"00:54.890 ","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":"01:16.610","Text":"which is, divide top and bottom by 3,"},{"Start":"01:16.610 ","End":"01:22.020","Text":"5 over 2, 15 over 6."},{"Start":"01:22.250 ","End":"01:33.315","Text":"That means that w would have to be 5 over 2 times v. If this is the case,"},{"Start":"01:33.315 ","End":"01:35.915","Text":"let\u0027s check, 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":"01:48.650","Text":"We just multiply out 5 over 2 v is equal to 5 over 2 times 6 is 15,"},{"Start":"01:48.650 ","End":"01:50.780","Text":"15_i, 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":"01:58.935","Text":"5 over 2 times 4 is,"},{"Start":"01:58.935 ","End":"02:02.100","Text":"in fact, 10, and there\u0027s a minus,"},{"Start":"02:02.100 ","End":"02:04.575","Text":"so we get minus 10_j."},{"Start":"02:04.575 ","End":"02:09.650","Text":"If I take 5 over 2 times 16,"},{"Start":"02:09.650 ","End":"02:12.560","Text":"16 over 2 is 8 times 5 is 40,"},{"Start":"02:12.560 ","End":"02:15.750","Text":"so we\u0027ve got minus 40_k."},{"Start":"02:16.960 ","End":"02:19.490","Text":"Is this equal to w?"},{"Start":"02:19.490 ","End":"02:21.770","Text":"Well, the answer is yes."},{"Start":"02:21.770 ","End":"02:24.830","Text":"There\u0027s no question mark about it,"},{"Start":"02:24.830 ","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":"02:42.445","Text":"If they\u0027re parallel, then let\u0027s say that the second is some constant times the first."},{"Start":"02:42.445 ","End":"02:45.620","Text":"Constant I mean Scalar."},{"Start":"02:46.760 ","End":"02:55.335","Text":"Let\u0027s take it that b is some k times a."},{"Start":"02:55.335 ","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":"03:22.215","Text":"My question is, does b really equal 2 times a where I put the k equals 2 here?"},{"Start":"03:22.215 ","End":"03:25.980","Text":"Well, let\u0027s see, 2 a is equal to,"},{"Start":"03:25.980 ","End":"03:28.425","Text":"it\u0027s an angular bracket form,"},{"Start":"03:28.425 ","End":"03:32.310","Text":"so we got 6 minus 4,"},{"Start":"03:32.310 ","End":"03:34.290","Text":"so far, so good."},{"Start":"03:34.290 ","End":"03:39.525","Text":"Then the third component is 10, not so good."},{"Start":"03:39.525 ","End":"03:46.379","Text":"This is not equal to B."},{"Start":"03:46.379 ","End":"03:48.315","Text":"The answer is no,"},{"Start":"03:48.315 ","End":"03:53.860","Text":"not parallel and that\u0027s it."}],"ID":10281}],"Thumbnail":null,"ID":57471},{"Name":"Vectors Dot Product","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors - Dot Product","Duration":"20m 22s","ChapterTopicVideoID":10103,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10103.jpeg","UploadDate":"2020-01-16T07:08:36.7370000","DurationForVideoObject":"PT20M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.080","Text":"Continuing with vectors, I\u0027m going to talk about something called the dot product."},{"Start":"00:05.080 ","End":"00:08.200","Text":"Let me get straight to the definition,"},{"Start":"00:08.200 ","End":"00:12.160","Text":"but I\u0027ll define it in 3-dimensions and it will be similar in other dimensions."},{"Start":"00:12.160 ","End":"00:14.860","Text":"If I have 1 vector, a,"},{"Start":"00:14.860 ","End":"00:19.285","Text":"and let\u0027s say a has components a_1, a_2,"},{"Start":"00:19.285 ","End":"00:22.970","Text":"and a_3 and I have another vector b,"},{"Start":"00:22.970 ","End":"00:28.265","Text":"which is naturally b_1, b_2, b_3."},{"Start":"00:28.265 ","End":"00:32.650","Text":"Then I\u0027m going to define the dot product,"},{"Start":"00:32.650 ","End":"00:38.580","Text":"a and the dot with b to be actually a number,"},{"Start":"00:38.580 ","End":"00:47.385","Text":"a scalar, a_1b_1 plus a_2b_2 plus a_3b_3."},{"Start":"00:47.385 ","End":"00:48.735","Text":"The first with the first,"},{"Start":"00:48.735 ","End":"00:49.800","Text":"second with the second,"},{"Start":"00:49.800 ","End":"00:53.325","Text":"the third with the third multiply and then add."},{"Start":"00:53.325 ","End":"00:57.290","Text":"I\u0027ll get some examples in a moment."},{"Start":"00:57.290 ","End":"01:05.720","Text":"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":"01:12.030","Text":"Later there\u0027ll be another product called the cross-product,"},{"Start":"01:12.030 ","End":"01:14.930","Text":"where a vector times a vector will be a vector, but not yet."},{"Start":"01:14.930 ","End":"01:20.945","Text":"Examples, let\u0027s say we have 0, 4,"},{"Start":"01:20.945 ","End":"01:29.520","Text":"minus 2 times 2, minus 1, 7,"},{"Start":"01:29.520 ","End":"01:36.575","Text":"then we get, 0 times 2 is 0,"},{"Start":"01:36.575 ","End":"01:39.895","Text":"4 times 1 is minus 4,"},{"Start":"01:39.895 ","End":"01:42.910","Text":"adding minus 4, so I just put minus 4."},{"Start":"01:42.910 ","End":"01:51.300","Text":"Minus 2 times 7 is here, minus 14."},{"Start":"01:51.300 ","End":"01:55.875","Text":"All together the answer is minus 18."},{"Start":"01:55.875 ","End":"02:01.540","Text":"Let\u0027s take an example in 2-dimensions."},{"Start":"02:01.700 ","End":"02:09.630","Text":"5 minus 8.1, 2."},{"Start":"02:09.630 ","End":"02:11.590","Text":"There\u0027s just 2 things to add."},{"Start":"02:11.590 ","End":"02:17.920","Text":"5 times 1 is 5 minus 8 times 2 is minus 16 minus 11."},{"Start":"02:17.920 ","End":"02:21.070","Text":"Now let\u0027s take an example with the other notation."},{"Start":"02:21.070 ","End":"02:31.705","Text":"Let\u0027s take 3j minus 7k dot-product with"},{"Start":"02:31.705 ","End":"02:37.900","Text":"2i plus 3j plus"},{"Start":"02:37.900 ","End":"02:44.530","Text":"k. I\u0027m supposed to put arrows on these."},{"Start":"02:45.380 ","End":"02:51.825","Text":"Now in this case, notice that there is no item."},{"Start":"02:51.825 ","End":"02:57.075","Text":"I can think of a 0i here,"},{"Start":"02:57.075 ","End":"02:58.770","Text":"even though it\u0027s not there."},{"Start":"02:58.770 ","End":"03:01.340","Text":"Also notice that if it\u0027s just k on its own,"},{"Start":"03:01.340 ","End":"03:04.110","Text":"that\u0027s like as a 1 here."},{"Start":"03:04.110 ","End":"03:08.519","Text":"What I get is 0 times 2 is 0."},{"Start":"03:08.519 ","End":"03:11.250","Text":"The i with the i and the j with the j."},{"Start":"03:11.250 ","End":"03:14.610","Text":"3 times 3 is 9,"},{"Start":"03:14.610 ","End":"03:19.890","Text":"and the k with the k minus 7 times 1 is minus 7,"},{"Start":"03:19.890 ","End":"03:22.110","Text":"so the answer is 2."},{"Start":"03:22.110 ","End":"03:25.460","Text":"Also be careful if you\u0027re not for some reason 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":"03:30.650","Text":"here there\u0027s a 1."},{"Start":"03:30.650 ","End":"03:33.030","Text":"That\u0027s things to look out for."},{"Start":"03:34.670 ","End":"03:42.395","Text":"I\u0027ll give 1 more example just to show you that it works in 4 dimensions."},{"Start":"03:42.395 ","End":"03:47.820","Text":"We\u0027ll take 9, 5, minus 4,"},{"Start":"03:47.820 ","End":"03:52.650","Text":"2 dot product with minus 3,"},{"Start":"03:52.650 ","End":"03:57.120","Text":"minus 2, 7, minus 1."},{"Start":"03:57.120 ","End":"03:58.950","Text":"Same thing in 4-dimensions."},{"Start":"03:58.950 ","End":"04:01.830","Text":"This with this, minus 27,"},{"Start":"04:01.830 ","End":"04:04.800","Text":"5 times minus 2, minus 10,"},{"Start":"04:04.800 ","End":"04:07.950","Text":"minus 4 times 7, minus 28,"},{"Start":"04:07.950 ","End":"04:12.250","Text":"2 times minus 1, minus 2."},{"Start":"04:12.500 ","End":"04:16.900","Text":"I make it minus 67."},{"Start":"04:17.870 ","End":"04:20.810","Text":"Here are some formulas."},{"Start":"04:20.810 ","End":"04:22.580","Text":"Write the word formulas."},{"Start":"04:22.580 ","End":"04:26.090","Text":"Actually the correct word is formulae, Latin."},{"Start":"04:26.090 ","End":"04:31.250","Text":"The 6 of them, 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":"04:38.335","Text":"There\u0027s a distributive law that looks a little bit like,"},{"Start":"04:38.335 ","End":"04:41.495","Text":"if it\u0027s multiplication and addition"},{"Start":"04:41.495 ","End":"04:45.215","Text":"with the regular algebra then it looks just like this."},{"Start":"04:45.215 ","End":"04:53.120","Text":"The product of u with the sum of v plus w is u.v plus w is u.v plus u.w."},{"Start":"04:53.120 ","End":"04:54.785","Text":"It looks intuitive."},{"Start":"04:54.785 ","End":"05:00.395","Text":"If I take a constant to scalar and multiply it by the first 1,"},{"Start":"05:00.395 ","End":"05:02.090","Text":"or by the second 1,"},{"Start":"05:02.090 ","End":"05:04.835","Text":"or by the product, it\u0027s all the same."},{"Start":"05:04.835 ","End":"05:08.410","Text":"It doesn\u0027t matter where you put the scalar."},{"Start":"05:08.410 ","End":"05:12.805","Text":"The order doesn\u0027t matter with a dot product."},{"Start":"05:12.805 ","End":"05:16.070","Text":"Obviously, if I interchange the order,"},{"Start":"05:16.070 ","End":"05:21.125","Text":"I just get b_1a_1, which is the same as a_1b_1 and so on."},{"Start":"05:21.125 ","End":"05:26.030","Text":"Dot product with the 0 vector means that if 1 of them is 0,"},{"Start":"05:26.030 ","End":"05:30.630","Text":"like 0, 0, 0, the dot product is going to be 0."},{"Start":"05:31.790 ","End":"05:34.010","Text":"Now what does this say?"},{"Start":"05:34.010 ","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":"05:48.000","Text":"Suppose my vector v is a_1,"},{"Start":"05:48.000 ","End":"05:51.760","Text":"a_2, a_3, these are the 3 components."},{"Start":"05:51.760 ","End":"05:57.095","Text":"We mentioned something called the magnitude of a vector where we put it in bars."},{"Start":"05:57.095 ","End":"06:07.265","Text":"We defined it to be the square root of a_1 squared plus a_2 squared plus a_3 squared."},{"Start":"06:07.265 ","End":"06:11.070","Text":"Let\u0027s see that we really get an equality."},{"Start":"06:12.680 ","End":"06:16.935","Text":"V.v vectors is going to"},{"Start":"06:16.935 ","End":"06:24.940","Text":"equal a_1, a_2, a_3."},{"Start":"06:28.190 ","End":"06:34.215","Text":"again, with a_1, a_2, a_3."},{"Start":"06:34.215 ","End":"06:38.625","Text":"This equals, this with this is a_1 squared,"},{"Start":"06:38.625 ","End":"06:41.910","Text":"a_2 with a_2, a_2 squared,"},{"Start":"06:41.910 ","End":"06:45.885","Text":"a_3 with a_3 is a_ 3 squared."},{"Start":"06:45.885 ","End":"06:48.035","Text":"That\u0027s the left-hand side."},{"Start":"06:48.035 ","End":"06:52.055","Text":"On the other hand, the right-hand side,"},{"Start":"06:52.055 ","End":"07:01.470","Text":"magnitude of v squared is this thing squared."},{"Start":"07:01.470 ","End":"07:04.535","Text":"If I do that, the square root of something squared,"},{"Start":"07:04.535 ","End":"07:07.695","Text":"this is just equal to the thing without the square roots."},{"Start":"07:07.695 ","End":"07:12.930","Text":"It\u0027s a_1 squared plus a_2 squared plus a_3 squared."},{"Start":"07:12.930 ","End":"07:15.120","Text":"We get the same result."},{"Start":"07:15.120 ","End":"07:17.210","Text":"These really are equal."},{"Start":"07:17.210 ","End":"07:19.490","Text":"I mean, this equals this."},{"Start":"07:19.490 ","End":"07:22.549","Text":"Left-hand side equals right-hand side,"},{"Start":"07:22.549 ","End":"07:25.045","Text":"so we\u0027ve verified."},{"Start":"07:25.045 ","End":"07:30.105","Text":"Now look, this says that if we get 0,"},{"Start":"07:30.105 ","End":"07:32.865","Text":"the vector must be 0. Why is that?"},{"Start":"07:32.865 ","End":"07:36.210","Text":"Because if v.v is 0, v.v,"},{"Start":"07:36.210 ","End":"07:41.670","Text":"is just a_1 squared plus a_2 squared plus a_3 squared."},{"Start":"07:41.670 ","End":"07:48.935","Text":"Suppose a_1 squared plus a_2 squared plus a_3 squared equals 0."},{"Start":"07:48.935 ","End":"07:50.840","Text":"Now each of these is non-negative,"},{"Start":"07:50.840 ","End":"07:52.700","Text":"bigger, or equal to 0."},{"Start":"07:52.700 ","End":"07:56.420","Text":"The only way that 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":"08:03.219","Text":"This means that a_1 is 0,"},{"Start":"08:03.219 ","End":"08:06.875","Text":"a_2 is 0, and a_3 is 0."},{"Start":"08:06.875 ","End":"08:10.624","Text":"In short, that the vector v is 0,"},{"Start":"08:10.624 ","End":"08:12.440","Text":"because all its components are 0."},{"Start":"08:12.440 ","End":"08:14.330","Text":"That explains that 1."},{"Start":"08:14.330 ","End":"08:17.615","Text":"It turns out that there is actually"},{"Start":"08:17.615 ","End":"08:26.445","Text":"a geometric or trigonometric interpretation of the dot product."},{"Start":"08:26.445 ","End":"08:30.515","Text":"I need to bring in a diagram for that."},{"Start":"08:30.515 ","End":"08:32.605","Text":"Here\u0027s the picture."},{"Start":"08:32.605 ","End":"08:36.205","Text":"This is the x-axis, the y-axis."},{"Start":"08:36.205 ","End":"08:40.100","Text":"I take 2 vectors, a and b."},{"Start":"08:40.100 ","End":"08:42.475","Text":"We\u0027ll do it in 2D. It\u0027s easier to sketch."},{"Start":"08:42.475 ","End":"08:44.860","Text":"Let\u0027s say they have an angle,"},{"Start":"08:44.860 ","End":"08:47.620","Text":"call it Theta between them."},{"Start":"08:47.620 ","End":"08:51.970","Text":"The magnitude is just a between 2 bars."},{"Start":"08:51.970 ","End":"08:54.040","Text":"The magnitude is the length of the vector."},{"Start":"08:54.040 ","End":"08:56.725","Text":"If I didn\u0027t mention it before then I\u0027m mentioning it now."},{"Start":"08:56.725 ","End":"09:00.230","Text":"The magnitude of b is the length of b."},{"Start":"09:00.230 ","End":"09:03.655","Text":"This is just magnitude of b."},{"Start":"09:03.655 ","End":"09:07.945","Text":"Now it turns out that there\u0027s an important formula"},{"Start":"09:07.945 ","End":"09:16.790","Text":"that a.b is also equal to this length,"},{"Start":"09:16.790 ","End":"09:20.860","Text":"which is this times just a regular,"},{"Start":"09:20.860 ","End":"09:24.955","Text":"not a dot product, just a times multiplication of numbers,"},{"Start":"09:24.955 ","End":"09:34.270","Text":"b, times the cosine of the angle in between them."},{"Start":"09:34.850 ","End":"09:42.840","Text":"It\u0027s not just the product of the lengths times the cosine of the angle in between them."},{"Start":"09:42.840 ","End":"09:45.490","Text":"I\u0027m going to assume that the angle is"},{"Start":"09:45.490 ","End":"09:52.070","Text":"between 0 and 180 degrees or in radians between 0 and Pi."},{"Start":"09:52.070 ","End":"09:55.910","Text":"Because if it\u0027s bigger than 180 degrees,"},{"Start":"09:55.910 ","End":"09:57.890","Text":"I can just look at it from the other side,"},{"Start":"09:57.890 ","End":"10:01.700","Text":"so that\u0027s the result."},{"Start":"10:01.700 ","End":"10:04.115","Text":"I\u0027m not going to prove it."},{"Start":"10:04.115 ","End":"10:09.230","Text":"1 of the uses of this formulae or in geometric interpretation,"},{"Start":"10:09.230 ","End":"10:13.025","Text":"is to find the angle between 2 vectors,"},{"Start":"10:13.025 ","End":"10:20.150","Text":"because I can write it as cosine of Theta is equal to,"},{"Start":"10:20.150 ","End":"10:22.250","Text":"and if I take this over to the other side,"},{"Start":"10:22.250 ","End":"10:29.645","Text":"I get a.b over magnitude of a,"},{"Start":"10:29.645 ","End":"10:35.765","Text":"magnitude of b. I\u0027ll just put the vector sign over each of these."},{"Start":"10:35.765 ","End":"10:40.490","Text":"I\u0027ll show you an example of how we use it to get to the angle."},{"Start":"10:40.490 ","End":"10:41.840","Text":"We get to the cosine of the angle,"},{"Start":"10:41.840 ","End":"10:46.930","Text":"and then on the calculator we do the arc cosine."},{"Start":"10:46.930 ","End":"10:51.700","Text":"In the example, let\u0027s take a and b, both 3-dimensional."},{"Start":"10:51.700 ","End":"10:57.120","Text":"Let\u0027s take a to be 3, minus 4,"},{"Start":"10:57.120 ","End":"11:04.925","Text":"minus 1, and we\u0027ll take b as 0, 5."},{"Start":"11:04.925 ","End":"11:07.225","Text":"If we use this formula,"},{"Start":"11:07.225 ","End":"11:09.670","Text":"we get that the cosine of Theta,"},{"Start":"11:09.670 ","End":"11:14.830","Text":"which is the angle between these 2 cosine of Theta is, first of all,"},{"Start":"11:14.830 ","End":"11:16.345","Text":"a dot b,"},{"Start":"11:16.345 ","End":"11:25.945","Text":"which is 3 times 0 is 0 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 ignore the minus plus 4 squared plus 1 squared times the square root,"},{"Start":"11:48.460 ","End":"11:54.490","Text":"0 squared plus 5 squared plus 2 squared."},{"Start":"11:54.490 ","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":"12:13.915","Text":"We\u0027ve got the square root of 26 times the square root of 25 plus 4 is 29."},{"Start":"12:13.915 ","End":"12:16.555","Text":"If we do this on the calculator,"},{"Start":"12:16.555 ","End":"12:21.770","Text":"this comes out to approximately 0.8011927,"},{"Start":"12:23.580 ","End":"12:26.350","Text":"blah, blah, blah, sorry,"},{"Start":"12:26.350 ","End":"12:31.450","Text":"I forgot the minus then leave this on the calculator to ever many places."},{"Start":"12:31.450 ","End":"12:41.200","Text":"Let\u0027s do the arc cosine We need the arc cosine of the above or rather theta."},{"Start":"12:41.200 ","End":"12:46.300","Text":"Theta is equal to the arc cosine of the I don\u0027t want to copy it."},{"Start":"12:46.300 ","End":"12:51.130","Text":"It\u0027s just whatever this is and depending on what your calculator is set on,"},{"Start":"12:51.130 ","End":"12:53.005","Text":"if it\u0027s on degrees,"},{"Start":"12:53.005 ","End":"12:58.405","Text":"the answer you\u0027ll get is 143.24 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":"13:06.415","Text":"We write with a little c here."},{"Start":"13:06.415 ","End":"13:09.670","Text":"Depending on how you want your result."},{"Start":"13:09.670 ","End":"13:12.910","Text":"Another thing we can deduce from this formula,"},{"Start":"13:12.910 ","End":"13:17.245","Text":"is we can tell when 2 vectors are perpendicular."},{"Start":"13:17.245 ","End":"13:20.050","Text":"Actually instead of the word perpendicular."},{"Start":"13:20.050 ","End":"13:23.470","Text":"In this context, we are using another word."},{"Start":"13:23.470 ","End":"13:25.255","Text":"We use the word orthogonal."},{"Start":"13:25.255 ","End":"13:27.100","Text":"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":"13:35.155","Text":"which means that 90 degrees to each other."},{"Start":"13:35.155 ","End":"13:38.349","Text":"Let\u0027s assume that these vectors are not 0."},{"Start":"13:38.349 ","End":"13:40.270","Text":"If one 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":"13:45.205","Text":"Assume we\u0027re talking about nonzero vectors."},{"Start":"13:45.205 ","End":"13:49.000","Text":"Then if the lines are orthogonal,"},{"Start":"13:49.000 ","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":"14:04.660","Text":"which means that we get that this thing is 0 and 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":"14:17.200","Text":"then the cosine is 0 so the angle is"},{"Start":"14:17.200 ","End":"14:23.965","Text":"90 degrees or Pi over 2 if we\u0027re talking about radians."},{"Start":"14:23.965 ","End":"14:26.920","Text":"The condition for orthogonal,"},{"Start":"14:26.920 ","End":"14:35.155","Text":"and we\u0027re talking about vectors a and B is that a dot b equals 0."},{"Start":"14:35.155 ","End":"14:38.725","Text":"That\u0027s the condition for perpendicular or orthogonal."},{"Start":"14:38.725 ","End":"14:41.230","Text":"While we\u0027re at it, we might as well talk about"},{"Start":"14:41.230 ","End":"14:44.680","Text":"the condition for 2 vectors to be parallel."},{"Start":"14:44.680 ","End":"14:48.415","Text":"There\u0027s 2 ways for vectors to be parallel;"},{"Start":"14:48.415 ","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":"15:07.540","Text":"I\u0027ll talk in degrees or 180 degrees"},{"Start":"15:07.540 ","End":"15:13.585","Text":"and each of these has an interpretation in terms of the dot product."},{"Start":"15:13.585 ","End":"15:15.505","Text":"Just move this aside."},{"Start":"15:15.505 ","End":"15:22.315","Text":"What it means is that if Theta is 0 degrees 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":"15:33.340","Text":"We get the formula that dot-product is equal to"},{"Start":"15:33.340 ","End":"15:43.014","Text":"just the magnitude of the first times the magnitude of the second vector signs."},{"Start":"15:43.014 ","End":"15:46.285","Text":"That\u0027s one way of being parallel."},{"Start":"15:46.285 ","End":"15:52.855","Text":"The other way of being parallel is for the 180 degrees cosine of 180 degrees is minus 1."},{"Start":"15:52.855 ","End":"15:58.150","Text":"We would get that a dot b vector,"},{"Start":"15:58.150 ","End":"16:05.980","Text":"vector 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 numbers. Let\u0027s check."},{"Start":"16:13.620 ","End":"16:15.765","Text":"In my first example,"},{"Start":"16:15.765 ","End":"16:23.130","Text":"I\u0027ll take a to equal 6 minus 2 minus"},{"Start":"16:23.130 ","End":"16:32.380","Text":"1 and 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":"16:38.245","Text":"Let\u0027s see what is a dot b."},{"Start":"16:38.245 ","End":"16:43.900","Text":"Let\u0027s see if they are orthogonal or parallel or maybe neither."},{"Start":"16:43.900 ","End":"16:49.195","Text":"A dot b is 6 times 2 is 12,"},{"Start":"16:49.195 ","End":"16:52.405","Text":"minus 2 times 5 is minus 10,"},{"Start":"16:52.405 ","End":"16:56.335","Text":"minus 1 times 2 is minus 2."},{"Start":"16:56.335 ","End":"17:04.010","Text":"This is equal to 0. I know that these 2 vectors are orthogonal."},{"Start":"17:05.850 ","End":"17:09.550","Text":"Now let\u0027s take another example."},{"Start":"17:09.550 ","End":"17:13.585","Text":"The other example, I\u0027ll use the other notation,"},{"Start":"17:13.585 ","End":"17:17.785","Text":"u is 2i minus j."},{"Start":"17:17.785 ","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":"17:36.290","Text":"be minus a half Pi plus a quarter j."},{"Start":"17:36.750 ","End":"17:39.730","Text":"It\u0027s easier to work with this notation."},{"Start":"17:39.730 ","End":"17:42.220","Text":"Let me just convert it right away."},{"Start":"17:42.220 ","End":"17:45.580","Text":"2 minus 1 is u,"},{"Start":"17:45.580 ","End":"17:54.850","Text":"and here I have minus 1/2, 1/4."},{"Start":"17:54.850 ","End":"17:59.170","Text":"If I want to know what is u dot v,"},{"Start":"17:59.170 ","End":"18:01.870","Text":"I\u0027ll take this representation."},{"Start":"18:01.870 ","End":"18:08.590","Text":"It\u0027s easier and do it in our heads 2 times minus 1/2, I\u0027ll write it,"},{"Start":"18:08.590 ","End":"18:16.375","Text":"2 times minus 1/2 is minus 1 and minus 1 times 1/4"},{"Start":"18:16.375 ","End":"18:25.540","Text":"is minus 1/4 and I\u0027ll write it as an improper fraction minus 5 over 4."},{"Start":"18:25.540 ","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":"19:18.655","Text":"Yes, 2D, there\u0027s only 2 terms times the square root of"},{"Start":"19:18.655 ","End":"19:25.645","Text":"1/2 squared plus 1/4 squared."},{"Start":"19:25.645 ","End":"19:29.920","Text":"Let\u0027s see, this is the square root of 5,"},{"Start":"19:29.920 ","End":"19:35.950","Text":"and this is the square root of 1/4 plus 1/16."},{"Start":"19:35.950 ","End":"19:38.485","Text":"I put it all over 16 is 4 plus 1."},{"Start":"19:38.485 ","End":"19:41.140","Text":"It\u0027s 5/16."},{"Start":"19:41.140 ","End":"19:43.435","Text":"We have 5/16."},{"Start":"19:43.435 ","End":"19:48.370","Text":"What this equals is the square root of 5 times the square root of"},{"Start":"19:48.370 ","End":"19:54.860","Text":"5 over the square root of 16."},{"Start":"19:55.860 ","End":"20:03.535","Text":"This equals 5 over 4 because this times this is 5 and this is 4."},{"Start":"20:03.535 ","End":"20:07.285","Text":"Now, this is not the same as this."},{"Start":"20:07.285 ","End":"20:10.030","Text":"It\u0027s the minus of this."},{"Start":"20:10.030 ","End":"20:17.590","Text":"It turns out that they are parallel and in fact they are in opposite directions."},{"Start":"20:17.590 ","End":"20:20.485","Text":"I think we\u0027re done with that topic."},{"Start":"20:20.485 ","End":"20:23.210","Text":"Let\u0027s take a break now."}],"ID":10284},{"Watched":false,"Name":"Vectors - Dot Product (continued)","Duration":"15m 36s","ChapterTopicVideoID":10104,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10104.jpeg","UploadDate":"2017-08-20T10:49:05.1200000","DurationForVideoObject":"PT15M36S","Description":null,"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":"00:25.320","Text":"Let\u0027s see, projection of b onto a,"},{"Start":"00:25.320 ","End":"00:28.755","Text":"and we\u0027re going to denote it as"},{"Start":"00:28.755 ","End":"00:36.450","Text":"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":"01:00.680","Text":"It says if, I don\u0027t know,"},{"Start":"01:00.680 ","End":"01:06.530","Text":"you had a projector like the Sun or a light source from above,"},{"Start":"01:06.530 ","End":"01:10.250","Text":"and this would be like the shadow that b would form onto a."},{"Start":"01:10.250 ","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":"01:22.050 ","End":"01:26.270","Text":"What could happen though 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":"01:33.185","Text":"but the continuation of a, and in fact,"},{"Start":"01:33.185 ","End":"01:35.675","Text":"the projection in this case,"},{"Start":"01:35.675 ","End":"01:39.440","Text":"which is this vector in blue as it is here."},{"Start":"01:39.440 ","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":"01:48.215","Text":"These are the 2 situations."},{"Start":"01:48.215 ","End":"01:49.760","Text":"If it\u0027s 90 degrees,"},{"Start":"01:49.760 ","End":"01:53.300","Text":"then the projection is 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":"02:08.640","Text":"it\u0027s going to be some scalar times a."},{"Start":"02:08.640 ","End":"02:12.340","Text":"So I know that there\u0027s going to be an a in here,"},{"Start":"02:12.340 ","End":"02:15.830","Text":"and 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":"02:40.220","Text":"I will be drawing a perpendicular onto b,"},{"Start":"02:40.220 ","End":"02:46.250","Text":"and it would be parallel to be that b in the same direction as b."},{"Start":"02:46.250 ","End":"02:47.855","Text":"It\u0027ll 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":"03:13.965","Text":"What does that equal?"},{"Start":"03:13.965 ","End":"03:17.060","Text":"You know what? That\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 one we just use the formula straight as is."},{"Start":"03:33.740 ","End":"03:37.100","Text":"Because copied them out here, these things,"},{"Start":"03:37.100 ","End":"03:42.260","Text":"and now we\u0027ll start projection of b onto a using this formula."},{"Start":"03:42.260 ","End":"03:45.284","Text":"First of all, a dot b,"},{"Start":"03:45.284 ","End":"03:48.635","Text":"so we have a fraction here and on the top,"},{"Start":"03:48.635 ","End":"03:51.200","Text":"1 times 2 is 2,"},{"Start":"03:51.200 ","End":"03:53.690","Text":"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":"04:10.175","Text":"but then I square it again,"},{"Start":"04:10.175 ","End":"04:12.230","Text":"so don\u0027t bother even doing the square root."},{"Start":"04:12.230 ","End":"04:16.910","Text":"All this times the vector a,"},{"Start":"04:16.910 ","End":"04:22.200","Text":"which is 1, 0 minus 2. Now, what do we have here?"},{"Start":"04:22.200 ","End":"04:28.244","Text":"2 plus 2 is 4, 1 plus 4 is 5, so this is 4,"},{"Start":"04:28.244 ","End":"04:34.710","Text":"this part here is 4/5 of this 1,"},{"Start":"04:34.710 ","End":"04:41.820","Text":"0 minus 2, and this equals 4/5."},{"Start":"04:41.820 ","End":"04:46.605","Text":"4/5 times 0 is 0, 4/5 times minus 2 is minus 8/5."},{"Start":"04:46.605 ","End":"04:49.950","Text":"That\u0027s the first one. Now, how about the second one?"},{"Start":"04:49.950 ","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":"04:59.765","Text":"On this numerator, we still get 4,"},{"Start":"04:59.765 ","End":"05:03.250","Text":"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":"05:16.880","Text":"Also here, different vector, 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":"05:23.820","Text":"This time it\u0027s equal to,"},{"Start":"05:24.460 ","End":"05:27.560","Text":"4 plus 1 plus 1 is 6,"},{"Start":"05:27.560 ","End":"05:35.520","Text":"4/6 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":"05:46.650","Text":"1 times 2/3 is 2/3 and minus 1 is minus 2/3."},{"Start":"05:46.650 ","End":"05:49.055","Text":"That\u0027s it for projections."},{"Start":"05:49.055 ","End":"05:51.100","Text":"Now, let\u0027s move on."},{"Start":"05:51.100 ","End":"05:55.390","Text":"The last topic under a dot-product,"},{"Start":"05:55.390 ","End":"06:00.315","Text":"something called direction cosines and direction angles."},{"Start":"06:00.315 ","End":"06:03.560","Text":"This is peculiar to 3D."},{"Start":"06:03.560 ","End":"06:08.585","Text":"Normally, everything we said so far is okay for 2D, 3D, 4D, whatever."},{"Start":"06:08.585 ","End":"06:11.065","Text":"This one is, just for 3D."},{"Start":"06:11.065 ","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":"06:33.500","Text":"What we want to know are the angles that this vector forms with the axis, let\u0027s say,"},{"Start":"06:33.500 ","End":"06:39.140","Text":"it 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":"06:42.999","Text":"and Gamma with the z-axis."},{"Start":"06:42.999 ","End":"06:46.020","Text":"We usually are more concerned with the cosines of the angles,"},{"Start":"06:46.020 ","End":"06:48.460","Text":"well, both of them."},{"Start":"06:48.920 ","End":"06:52.620","Text":"Turns out that there is a formula."},{"Start":"06:52.620 ","End":"06:54.480","Text":"Here\u0027s the formula, well,"},{"Start":"06:54.480 ","End":"06:58.280","Text":"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 will be able to find the angles too."},{"Start":"07:05.180 ","End":"07:09.500","Text":"But let\u0027s look at 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":"07:18.000","Text":"the standard basis vector I,"},{"Start":"07:18.000 ","End":"07:20.850","Text":"and over the magnitude of a."},{"Start":"07:20.850 ","End":"07:23.855","Text":"Everything else is just the same except that instead of I,"},{"Start":"07:23.855 ","End":"07:29.405","Text":"we have j and k. I\u0027m not going to repeat the meaning of I,"},{"Start":"07:29.405 ","End":"07:32.400","Text":"j, and k. You\u0027re supposed to remember."},{"Start":"07:32.400 ","End":"07:36.650","Text":"This is like 1 0 0 this is 0 1 0, 0 0 1."},{"Start":"07:36.650 ","End":"07:39.840","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 a_1,"},{"Start":"07:51.455 ","End":"07:56.575","Text":"a_2, a_3, just to give it some components, coordinates."},{"Start":"07:56.575 ","End":"08:04.080","Text":"In that case, the first one would be, a_1,"},{"Start":"08:04.080 ","End":"08:12.790","Text":"a_2, a_3, 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 zeros,"},{"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 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":"08:41.619","Text":"Similarly, here we\u0027d get a_2,"},{"Start":"08:41.619 ","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":"09:03.940","Text":"but I think you can follow separators here, 3 formulas."},{"Start":"09:03.940 ","End":"09:06.625","Text":"I think I\u0027m going to erase this."},{"Start":"09:06.625 ","End":"09:15.700","Text":"Now, these 3 cosines are called direction cosines of the vector a and the Alpha,"},{"Start":"09:15.700 ","End":"09:19.885","Text":"Beta, Gamma themselves are called direction angles."},{"Start":"09:19.885 ","End":"09:21.804","Text":"Like I said, if we have the cosine,"},{"Start":"09:21.804 ","End":"09:24.535","Text":"then 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 will see that in the example."},{"Start":"09:30.895 ","End":"09:36.490","Text":"Let\u0027s say the vector a was this 2, 1, and 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":"09:53.634","Text":"Magnitude of a is the square root"},{"Start":"09:53.634 ","End":"09:59.950","Text":"of 2 squared plus 1 squared and we ignore the minus, 4 squared."},{"Start":"09:59.950 ","End":"10:07.225","Text":"That comes out to be 4 plus 1 plus 16,"},{"Start":"10:07.225 ","End":"10:10.220","Text":"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":"10:24.160","Text":"which is the first component 2 over the magnitude of a square root of 21."},{"Start":"10:24.160 ","End":"10:30.400","Text":"Similarly, cosine Beta is 1 over square root of 21,"},{"Start":"10:30.400 ","End":"10:37.400","Text":"and cosine Gamma; minus 4 over root of 21."},{"Start":"10:38.040 ","End":"10:48.085","Text":"All we need now is to do this on the calculator and then if we do the arc cosine,"},{"Start":"10:48.085 ","End":"10:51.355","Text":"we can get Alpha, 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":"11:11.095","Text":"Here, 77.396 degrees, I\u0027m rounding to 3 places of course."},{"Start":"11:11.095 ","End":"11:18.415","Text":"The last one is 150.794 degrees."},{"Start":"11:18.415 ","End":"11:24.860","Text":"Notice that the negative ones come out bigger than 90 degrees and 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":"11:31.900","Text":"I\u0027ll just show you what you would get. Well, here they are."},{"Start":"11:31.900 ","End":"11:35.590","Text":"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":"11:48.070","Text":"then it depends on what your calculator is set to. We\u0027re almost done."},{"Start":"11:48.070 ","End":"11:50.290","Text":"I just want to show you some formulas."},{"Start":"11:50.290 ","End":"11:51.640","Text":"There\u0027s 3 of them actually,"},{"Start":"11:51.640 ","End":"11:53.500","Text":"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":"12:35.260","Text":"The question is, is this equal to?"},{"Start":"12:35.260 ","End":"12:39.085","Text":"Let\u0027s see, the right-hand side is magnitude of a."},{"Start":"12:39.085 ","End":"12:47.470","Text":"Now, cosine Alpha is a_1 over magnitude of a,"},{"Start":"12:47.470 ","End":"12:55.795","Text":"and 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":"13:09.280","Text":"you multiply each component."},{"Start":"13:09.280 ","End":"13:11.575","Text":"So this cancels with this, with this, with this."},{"Start":"13:11.575 ","End":"13:14.780","Text":"This one 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":"13:22.555","Text":"let\u0027s call it u, I have a reason for calling it u,"},{"Start":"13:22.555 ","End":"13:26.970","Text":"which is the cosine of Alpha,"},{"Start":"13:26.970 ","End":"13:30.810","Text":"cosine Beta, cosine Gamma."},{"Start":"13:30.810 ","End":"13:33.030","Text":"This is a unit vector."},{"Start":"13:33.030 ","End":"13:35.710","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 your mind about the highlighting."},{"Start":"13:42.550 ","End":"13:48.160","Text":"The reason is, is that u is just a over the magnitude of"},{"Start":"13:48.160 ","End":"13:54.310","Text":"a and this thing is 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":"14:13.435","Text":"you can take it outside."},{"Start":"14:13.435 ","End":"14:21.205","Text":"It\u0027s 1 over the magnitude of a times the magnitude of a."},{"Start":"14:21.205 ","End":"14:24.265","Text":"Well, and this is obviously just equal to 1."},{"Start":"14:24.265 ","End":"14:26.785","Text":"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":"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":"14:59.810","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":"15:14.830","Text":"Maybe this would be 1,"},{"Start":"15:14.830 ","End":"15:21.039","Text":"2, 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":"15:30.115","Text":"Here they are for our future reference and we\u0027re done with the subject"},{"Start":"15:30.115 ","End":"15:36.290","Text":"of dot products on the subtopic of Direction Cosines, we\u0027re done."}],"ID":10285},{"Watched":false,"Name":"Exercise 1","Duration":"2m 46s","ChapterTopicVideoID":10105,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10105.jpeg","UploadDate":"2017-08-20T10:50:29.7330000","DurationForVideoObject":"PT2M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"This exercise is in 3 parts,"},{"Start":"00:02.130 ","End":"00:05.490","Text":"and each of them is an a and a b vectors,"},{"Start":"00:05.490 ","End":"00:07.665","Text":"and we have to find the dot product."},{"Start":"00:07.665 ","End":"00:13.335","Text":"In Number 1, we have both of them are in 2 dimensions,"},{"Start":"00:13.335 ","End":"00:15.570","Text":"and we have angular bracket notation."},{"Start":"00:15.570 ","End":"00:20.790","Text":"We just use the formula a dot b,"},{"Start":"00:20.790 ","End":"00:23.895","Text":"we multiply component-wise and add."},{"Start":"00:23.895 ","End":"00:34.260","Text":"We\u0027ve got 5 times 4 plus negative 4 times 3."},{"Start":"00:34.260 ","End":"00:39.945","Text":"Let\u0027s see, this is 20 minus 12 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 i, j, k notation,"},{"Start":"00:49.639 ","End":"00:51.454","Text":"but it\u0027s the same idea."},{"Start":"00:51.454 ","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":"01:01.880","Text":"and then we have plus 6 times minus 4."},{"Start":"01:01.880 ","End":"01:05.045","Text":"I\u0027ll 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":"01:11.530","Text":"so minus 3 times 7."},{"Start":"01:11.530 ","End":"01:19.230","Text":"That will give us 48 minus 24 minus 21."},{"Start":"01:19.230 ","End":"01:24.690","Text":"I make that 48 minus 45 is 3."},{"Start":"01:24.690 ","End":"01:26.920","Text":"In the third question,"},{"Start":"01:26.920 ","End":"01:31.645","Text":"we\u0027re not given directly what the vectors a and b are,"},{"Start":"01:31.645 ","End":"01:33.610","Text":"we\u0027re just given some hints about them,"},{"Start":"01:33.610 ","End":"01:35.650","Text":"that this has magnitude 4,"},{"Start":"01:35.650 ","End":"01:38.575","Text":"magnitude 3, and we know the angle between them."},{"Start":"01:38.575 ","End":"01:41.905","Text":"But remember, there\u0027s another way for dot product,"},{"Start":"01:41.905 ","End":"01:47.985","Text":"that if you have 2 vectors, 1 of them a,"},{"Start":"01:47.985 ","End":"01:50.520","Text":"and 1 of them b,"},{"Start":"01:50.520 ","End":"01:53.935","Text":"and if we know the angle between them, Theta,"},{"Start":"01:53.935 ","End":"02:00.925","Text":"then the dot product a dot b is equal to the magnitude of"},{"Start":"02:00.925 ","End":"02:09.800","Text":"a times the magnitude of b times the cosine of the angle between them."},{"Start":"02:09.800 ","End":"02:12.290","Text":"That\u0027s what we\u0027re going to use in this."},{"Start":"02:12.290 ","End":"02:16.420","Text":"We get that a dot b is,"},{"Start":"02:16.420 ","End":"02:19.200","Text":"magnitude of a is 4,"},{"Start":"02:19.200 ","End":"02:22.215","Text":"magnitude of b is 3."},{"Start":"02:22.215 ","End":"02:23.865","Text":"We\u0027re not given the cosine,"},{"Start":"02:23.865 ","End":"02:25.230","Text":"we\u0027re given the angle,"},{"Start":"02:25.230 ","End":"02:29.245","Text":"so I need cosine of Pi over 3."},{"Start":"02:29.245 ","End":"02:32.535","Text":"Now, Pi over 3 is 60 degrees,"},{"Start":"02:32.535 ","End":"02:38.165","Text":"and the cosine of 60 degrees is 1.5."},{"Start":"02:38.165 ","End":"02:41.570","Text":"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":10286},{"Watched":false,"Name":"Exercise 2","Duration":"6m 5s","ChapterTopicVideoID":10106,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10106.jpeg","UploadDate":"2017-08-20T10:53:27.8730000","DurationForVideoObject":"PT6M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.150","Text":"Here we have 2 exercises in 1,"},{"Start":"00:03.150 ","End":"00:06.765","Text":"and each of them we have to find the angle between the 2 vectors."},{"Start":"00:06.765 ","End":"00:09.270","Text":"Let\u0027s give the angle a name,"},{"Start":"00:09.270 ","End":"00:11.895","Text":"we use Greek letter Theta."},{"Start":"00:11.895 ","End":"00:14.610","Text":"We just use a formula."},{"Start":"00:14.610 ","End":"00:19.410","Text":"In a, we will get that 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":"00:35.445","Text":"In this case, it would be a dot b."},{"Start":"00:35.445 ","End":"00:43.340","Text":"We have to divide by the magnitude of a times the magnitude of b."},{"Start":"00:43.340 ","End":"00:48.980","Text":"There\u0027s 3 calculations, the dot product and magnitude for each."},{"Start":"00:48.980 ","End":"00:51.070","Text":"Let\u0027s see what we get."},{"Start":"00:51.070 ","End":"00:56.285","Text":"A dot b, these are 2-dimensional vectors."},{"Start":"00:56.285 ","End":"01:00.165","Text":"Just multiply component-wise and add."},{"Start":"01:00.165 ","End":"01:09.440","Text":"This is going to be 3 times 7 plus 5 times 6, that\u0027s the numerator."},{"Start":"01:09.440 ","End":"01:17.870","Text":"Now, magnitude of a is going to be the square root of 3 squared plus 5 squared,"},{"Start":"01:17.870 ","End":"01:26.665","Text":"and the magnitude of b will be the square root of 7 squared plus 6 squared."},{"Start":"01:26.665 ","End":"01:29.450","Text":"Let\u0027s see what we get."},{"Start":"01:29.450 ","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":"01:37.945","Text":"That makes this 51."},{"Start":"01:37.945 ","End":"01:42.240","Text":"3 squared is 9, 5 squared is 25,"},{"Start":"01:42.240 ","End":"01:46.305","Text":"so we get the square root of 34."},{"Start":"01:46.305 ","End":"01:48.350","Text":"7 squared is 49,"},{"Start":"01:48.350 ","End":"01:52.770","Text":"6 squared is 36, that makes it what?"},{"Start":"01:52.770 ","End":"01:58.120","Text":"85, square root of 85."},{"Start":"01:58.120 ","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 happened 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":"02:16.470","Text":"I can take out root 17 and root 17,"},{"Start":"02:16.470 ","End":"02:18.600","Text":"it\u0027ll cancel out with the 17."},{"Start":"02:18.600 ","End":"02:23.875","Text":"I\u0027ll get 3 over square root of 2,"},{"Start":"02:23.875 ","End":"02:29.075","Text":"square root of 5."},{"Start":"02:29.075 ","End":"02:32.135","Text":"That would be 3 over square root of 10."},{"Start":"02:32.135 ","End":"02:33.830","Text":"But 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":"02:45.500","Text":"Then I get that cosine of Theta is,"},{"Start":"02:45.500 ","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":"03:02.165","Text":"we can just take the inverse cosine."},{"Start":"03:02.165 ","End":"03:04.190","Text":"Depending on your calculator,"},{"Start":"03:04.190 ","End":"03:08.880","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":"03:20.730","Text":"it comes out to approximately 18.435 degrees,"},{"Start":"03:24.950 ","End":"03:27.965","Text":"but if you did it in radians,"},{"Start":"03:27.965 ","End":"03:32.670","Text":"then I would get 0.321,"},{"Start":"03:35.120 ","End":"03:37.530","Text":"to 3 decimal places,"},{"Start":"03:37.530 ","End":"03:42.570","Text":"radians, little c for circular measure."},{"Start":"03:42.570 ","End":"03:45.640","Text":"That\u0027s part a."},{"Start":"03:45.950 ","End":"03:48.620","Text":"Similarly in part b,"},{"Start":"03:48.620 ","End":"03:51.410","Text":"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":"03:56.085","Text":"the same formula, well,"},{"Start":"03:56.085 ","End":"03:57.380","Text":"it won\u0027t be a dot b,"},{"Start":"03:57.380 ","End":"04:00.650","Text":"it\u0027ll be v dot w, but the same idea."},{"Start":"04:00.650 ","End":"04:02.750","Text":"We need the dot products in the numerator,"},{"Start":"04:02.750 ","End":"04:06.960","Text":"so we need 1 times 5 from this and this,"},{"Start":"04:06.960 ","End":"04:11.130","Text":"and then minus 2 times 6,"},{"Start":"04:11.130 ","End":"04:15.985","Text":"minus 3 times 7 over,"},{"Start":"04:15.985 ","End":"04:18.170","Text":"here we get the square root,"},{"Start":"04:18.170 ","End":"04:22.345","Text":"we\u0027re in 3D so we got 3 terms here."},{"Start":"04:22.345 ","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 it\u0027s squaring."},{"Start":"04:29.735 ","End":"04:40.150","Text":"Then the square root of 5 squared plus 6 squared plus 7 squared."},{"Start":"04:40.340 ","End":"04:45.765","Text":"Numerator, 5 minus 12 minus 21,"},{"Start":"04:45.765 ","End":"04:51.885","Text":"which is 5 minus 33, minus 28."},{"Start":"04:51.885 ","End":"04:56.700","Text":"On the denominator, 1 plus 4 plus 9"},{"Start":"04:56.700 ","End":"05:04.995","Text":"would be 14 under the square root sign."},{"Start":"05:04.995 ","End":"05:13.710","Text":"Here we\u0027ve got the square root of 25 and 36 and 49,"},{"Start":"05:13.710 ","End":"05:16.575","Text":"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":"05:26.160","Text":"we get approximately 0.7135,"},{"Start":"05:26.160 ","End":"05:31.470","Text":"but then we straight away take the inverse cosine."},{"Start":"05:31.470 ","End":"05:34.495","Text":"We\u0027ve got that Theta equals,"},{"Start":"05:34.495 ","End":"05:39.755","Text":"and I\u0027ll do it in both degrees and radians."},{"Start":"05:39.755 ","End":"05:47.970","Text":"44.479 degrees"},{"Start":"05:47.970 ","End":"05:56.145","Text":"and 0.7945 radians approximately."},{"Start":"05:56.145 ","End":"06:00.125","Text":"Either way, if you have to choose,"},{"Start":"06:00.125 ","End":"06:05.670","Text":"go for radians. That\u0027s it."}],"ID":10287},{"Watched":false,"Name":"Exercise 3","Duration":"7m 34s","ChapterTopicVideoID":10107,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10107.jpeg","UploadDate":"2017-08-20T10:56:14.9170000","DurationForVideoObject":"PT7M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"This exercise is 3 in 1."},{"Start":"00:02.640 ","End":"00:05.190","Text":"In each case we have a pair of vectors and we have to"},{"Start":"00:05.190 ","End":"00:08.895","Text":"decide if they\u0027re parallel or orthogonal."},{"Start":"00:08.895 ","End":"00:13.515","Text":"Orthogonal meaning perpendicular or neither."},{"Start":"00:13.515 ","End":"00:17.115","Text":"There\u0027s more than 1 way to do this."},{"Start":"00:17.115 ","End":"00:18.930","Text":"I just want to mention 1 way,"},{"Start":"00:18.930 ","End":"00:22.695","Text":"which I\u0027m not going to use 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":"00:34.440","Text":"There is a standard formula for cosine of Theta."},{"Start":"00:34.440 ","End":"00:39.420","Text":"It\u0027s the dot product divided by the magnitude of 1 times the magnitude of the other."},{"Start":"00:39.420 ","End":"00:45.495","Text":"In any event, you could compute the cosine and then divide it into cases."},{"Start":"00:45.495 ","End":"00:52.730","Text":"Say, well, 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 a 180 degrees."},{"Start":"01:03.740 ","End":"01:06.090","Text":"If you like it in radians,"},{"Start":"01:06.090 ","End":"01:08.410","Text":"that\u0027s a 0 or Pi."},{"Start":"01:08.410 ","End":"01:12.390","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":"01:26.870","Text":"the angle whose cosine is 0 is 90 degrees or Pi over 2 in radians."},{"Start":"01:26.870 ","End":"01:31.080","Text":"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 nor 0,"},{"Start":"01:39.920 ","End":"01:44.075","Text":"then it\u0027s neither parallel nor orthogonal."},{"Start":"01:44.075 ","End":"01:45.755","Text":"That\u0027s just 1 way."},{"Start":"01:45.755 ","End":"01:47.600","Text":"I\u0027m not going to use this,"},{"Start":"01:47.600 ","End":"01:52.805","Text":"it\u0027s 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":"Lets try for parallel first."},{"Start":"01:59.534 ","End":"02:04.055","Text":"If they\u0027re parallel, that means that 1 of them, say q,"},{"Start":"02:04.055 ","End":"02:10.010","Text":"is some non-zero number scalar k times the other."},{"Start":"02:10.010 ","End":"02:16.955","Text":"Let\u0027s see if we can find such a k. Now,"},{"Start":"02:16.955 ","End":"02:20.300","Text":"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":"02:25.900","Text":"k has to be 5."},{"Start":"02:25.900 ","End":"02:34.190","Text":"Then because 5 times 1 will give me 5 and I can see that 5 times minus 2 is not minus 8."},{"Start":"02:34.190 ","End":"02:36.870","Text":"But sometimes you can see it more clearly."},{"Start":"02:36.870 ","End":"02:43.880","Text":"Because if I take plus, minus,"},{"Start":"02:43.880 ","End":"02:46.880","Text":"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 could 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":"03:05.810","Text":"then when I multiply by plus or minus the middle"},{"Start":"03:05.810 ","End":"03:08.570","Text":"1 is still going to be the odd 1 out as far as sine."},{"Start":"03:08.570 ","End":"03:11.795","Text":"That\u0027s easier than actually doing computations."},{"Start":"03:11.795 ","End":"03:14.010","Text":"Anyway, so 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":"03:35.705","Text":"Let\u0027s see what is p dot product with q."},{"Start":"03:35.705 ","End":"03:38.840","Text":"Remember we multiply component-wise and add."},{"Start":"03:38.840 ","End":"03:44.460","Text":"This is equal to 1 times 5 minus 2 minus 8,"},{"Start":"03:44.460 ","End":"03:47.705","Text":"so it\u0027s going to be plus 2 times 8."},{"Start":"03:47.705 ","End":"03:51.530","Text":"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":"04:00.050","Text":"It is equal to 0 and if it\u0027s equal to 0,"},{"Start":"04:00.050 ","End":"04:06.050","Text":"then it means that they are orthogonal. That\u0027s a."},{"Start":"04:06.050 ","End":"04:08.695","Text":"Now part b."},{"Start":"04:08.695 ","End":"04:11.190","Text":"Let\u0027s try for parallel first."},{"Start":"04:11.190 ","End":"04:16.040","Text":"They\u0027re parallel, then b is going to be some scalar times"},{"Start":"04:16.040 ","End":"04:25.990","Text":"a. 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":"04:30.965","Text":"then they\u0027re equal component-wise."},{"Start":"04:30.965 ","End":"04:36.630","Text":"I got that 7 is equal to"},{"Start":"04:36.630 ","End":"04:45.260","Text":"k times 3 and 6 is equal to k times 5."},{"Start":"04:45.260 ","End":"04:48.905","Text":"Now, there is no such k. For example,"},{"Start":"04:48.905 ","End":"04:55.565","Text":"we can extract it from the first 1 and say that k must equal 7 over 3."},{"Start":"04:55.565 ","End":"04:57.890","Text":"But then if I substitute it in the second,"},{"Start":"04:57.890 ","End":"05:06.860","Text":"I\u0027ll get that 6 equals 7 over 3 times 5."},{"Start":"05:06.860 ","End":"05:11.290","Text":"That\u0027s certainly not true; false."},{"Start":"05:11.290 ","End":"05:14.840","Text":"They not parallel."},{"Start":"05:14.840 ","End":"05:17.240","Text":"Let\u0027s write that down."},{"Start":"05:17.240 ","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":"05:32.570","Text":"It\u0027s component-wise, I don\u0027t even have to multiply because I\u0027ve got plus,"},{"Start":"05:32.570 ","End":"05:37.925","Text":"plus and everything is a plus 3 times 7 plus 5 times 6."},{"Start":"05:37.925 ","End":"05:41.600","Text":"In any event it\u0027s not equal to 0, no way."},{"Start":"05:41.600 ","End":"05:50.075","Text":"These 2 vectors are not orthogonal and so we write,"},{"Start":"05:50.075 ","End":"05:53.660","Text":"the answer is neither."},{"Start":"05:53.660 ","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":"06:11.325","Text":"If parallel, then w is going to equal a scalar times v. 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":"06:34.245","Text":"Let\u0027s compute minus 5 times v. Let\u0027s see if we get w. Minus 5 times v"},{"Start":"06:34.245 ","End":"06:43.965","Text":"is minus 5 times i minus 2j plus"},{"Start":"06:43.965 ","End":"06:49.785","Text":"3k and this is going to equal minus 5."},{"Start":"06:49.785 ","End":"06:55.845","Text":"i minus 5 times minus 2 is plus 10j."},{"Start":"06:55.845 ","End":"07:03.125","Text":"The third component, minus 5 times 3 is minus 15k."},{"Start":"07:03.125 ","End":"07:13.125","Text":"This is equal to w and so we are good for parallel."},{"Start":"07:13.125 ","End":"07:17.800","Text":"Parallel is the answer."},{"Start":"07:18.170 ","End":"07:25.325","Text":"In part a, I\u0027m 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":"07:34.360","Text":"and in part c we got parallel. We\u0027re done."}],"ID":10288},{"Watched":false,"Name":"Exercise 4","Duration":"5m 58s","ChapterTopicVideoID":10108,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10108.jpeg","UploadDate":"2019-04-08T20:01:42.7100000","DurationForVideoObject":"PT5M58S","Description":null,"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":"00:16.500","Text":"I\u0027m going to copy the formula and a diagram from the tutorial. Here we are."},{"Start":"00:16.500 ","End":"00:18.870","Text":"We did it there with a and b,"},{"Start":"00:18.870 ","End":"00:24.360","Text":"I\u0027m sure we\u0027ll manage to adapt that to u and v. The 1st example is 2D,"},{"Start":"00:24.360 ","End":"00:25.920","Text":"the 2nd 1 is 3D."},{"Start":"00:25.920 ","End":"00:29.550","Text":"Let\u0027s start with the 2D example."},{"Start":"00:29.550 ","End":"00:35.790","Text":"U 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":"00:52.970","Text":"That\u0027s how we say it. This is projected onto this."},{"Start":"00:52.970 ","End":"00:56.370","Text":"It\u0027s equal by the formula."},{"Start":"00:57.530 ","End":"01:04.759","Text":"In this exercise, we\u0027re going to practice projecting 1 vector onto another vector."},{"Start":"01:04.759 ","End":"01:11.465","Text":"Here we\u0027re going to have u and v. We have a 2-dimensional case and a 3-dimensional case."},{"Start":"01:11.465 ","End":"01:14.660","Text":"In case you\u0027ve forgotten what this is all about,"},{"Start":"01:14.660 ","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. No same principles apply."},{"Start":"01:28.580 ","End":"01:33.460","Text":"Let\u0027s start with the 1st 1,"},{"Start":"01:33.460 ","End":"01:38.955","Text":"where u is a and v is b."},{"Start":"01:38.955 ","End":"01:48.539","Text":"What we want is the projection of v onto u,"},{"Start":"01:48.539 ","End":"01:51.860","Text":"and this will equal by the formula,"},{"Start":"01:51.860 ","End":"01:56.390","Text":"the dot product, sorry, not a."},{"Start":"01:56.390 ","End":"01:59.465","Text":"Well, a is just u and I\u0027ll write it already in here."},{"Start":"01:59.465 ","End":"02:02.340","Text":"We need 4 minus 1,"},{"Start":"02:02.340 ","End":"02:04.230","Text":"which is u, dot,"},{"Start":"02:04.230 ","End":"02:05.969","Text":"with the other vector,"},{"Start":"02:05.969 ","End":"02:15.470","Text":"1 comma 7 divided by the magnitude of u squared."},{"Start":"02:15.470 ","End":"02:22.605","Text":"We want the magnitude of 4, minus 1 squared."},{"Start":"02:22.605 ","End":"02:28.400","Text":"This is a scalar, all these times vector u in this case,"},{"Start":"02:28.400 ","End":"02:33.305","Text":"which is 4, minus 1. Let\u0027s see."},{"Start":"02:33.305 ","End":"02:36.170","Text":"The dot product component-wise,"},{"Start":"02:36.170 ","End":"02:38.150","Text":"4 times 1 is 4,"},{"Start":"02:38.150 ","End":"02:43.490","Text":"minus 7 altogether minus 3."},{"Start":"02:43.490 ","End":"02:48.040","Text":"The magnitude squared, we just take this squared plus this squared,"},{"Start":"02:48.040 ","End":"02:52.200","Text":"so 4 squared plus 1 squared,"},{"Start":"02:52.200 ","End":"03:01.620","Text":"in our heads, that will be 17 times vector 4, 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":"03:10.735","Text":"minus 12 over 17,"},{"Start":"03:10.735 ","End":"03:14.605","Text":"and then plus 3 over 17."},{"Start":"03:14.605 ","End":"03:17.820","Text":"That\u0027s the answer for a."},{"Start":"03:17.820 ","End":"03:21.465","Text":"Part b, a 3D case,"},{"Start":"03:21.465 ","End":"03:24.000","Text":"and here we have the i, j, k notation."},{"Start":"03:24.000 ","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":"03:39.455","Text":"Again, we want a dot product."},{"Start":"03:39.455 ","End":"03:43.560","Text":"It\u0027ll be easier for me without the i, j,"},{"Start":"03:43.560 ","End":"03:46.185","Text":"k. I\u0027ll write it as 7,"},{"Start":"03:46.185 ","End":"03:49.590","Text":"minus 1, 1, dot,"},{"Start":"03:49.590 ","End":"03:59.690","Text":"minus 2, 5, minus 6 over this 1 squared, the u squared,"},{"Start":"03:59.690 ","End":"04:03.965","Text":"which will be 7, 1,"},{"Start":"04:03.965 ","End":"04:08.550","Text":"minus 1 magnitude squared,"},{"Start":"04:08.550 ","End":"04:13.140","Text":"and all of this times the 1 we\u0027re projecting onto, which is this."},{"Start":"04:13.140 ","End":"04:17.370","Text":"I\u0027ll write it also, 7, minus 1, 1."},{"Start":"04:17.370 ","End":"04:22.140","Text":"What do we get? The dot product."},{"Start":"04:22.140 ","End":"04:26.385","Text":"Let\u0027s see. I\u0027ll just write it,"},{"Start":"04:26.385 ","End":"04:31.335","Text":"minus 14, minus 5,"},{"Start":"04:31.335 ","End":"04:32.969","Text":"they\u0027re all coming out negative,"},{"Start":"04:32.969 ","End":"04:40.920","Text":"minus 6 over 7 squared plus 1 squared plus 1 squared."},{"Start":"04:40.920 ","End":"04:43.745","Text":"I\u0027m ignoring the minuses of course because I\u0027m squaring,"},{"Start":"04:43.745 ","End":"04:46.290","Text":"and all this times 7,"},{"Start":"04:46.290 ","End":"04:50.790","Text":"minus 1, 1. Let\u0027s see."},{"Start":"04:50.790 ","End":"04:54.040","Text":"In the numerator, it\u0027s all minus,"},{"Start":"04:54.040 ","End":"04:56.940","Text":"so I\u0027ll add them up, 12 and 5 and 6."},{"Start":"04:56.940 ","End":"05:00.710","Text":"12, 17, 25 at the top,"},{"Start":"05:00.710 ","End":"05:03.020","Text":"minus 23 on the bottom."},{"Start":"05:03.020 ","End":"05:07.440","Text":"49 and 1 and 1 is 51."},{"Start":"05:07.910 ","End":"05:12.935","Text":"Just minus 25 over 51 times each of these,"},{"Start":"05:12.935 ","End":"05:16.920","Text":"minus 23 times 7."},{"Start":"05:17.380 ","End":"05:27.910","Text":"Let\u0027s see, minus 175 over 51."},{"Start":"05:27.910 ","End":"05:30.210","Text":"Then here\u0027s just a minus,"},{"Start":"05:30.210 ","End":"05:34.230","Text":"so that makes it plus 25 over 51."},{"Start":"05:34.230 ","End":"05:39.190","Text":"Here, minus 25 over 51."},{"Start":"05:39.650 ","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":10289},{"Watched":false,"Name":"Exercise 5","Duration":"5m 1s","ChapterTopicVideoID":10109,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10109.jpeg","UploadDate":"2017-08-20T10:59:53.1770000","DurationForVideoObject":"PT5M1S","Description":null,"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":"00:09.750","Text":"we have to find the direction cosines and the direction angles for this."},{"Start":"00:09.750 ","End":"00:14.210","Text":"This concept of direction cosines doesn\u0027t apply in 2D,"},{"Start":"00:14.210 ","End":"00:16.410","Text":"we 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":"00:29.985","Text":"This will be the vector v, the 1 in red,"},{"Start":"00:29.985 ","End":"00:36.525","Text":"and let\u0027s assume it has components v_1,"},{"Start":"00:36.525 ","End":"00:45.495","Text":"v_2, v_3, or v_1i plus v_2j plus v_3k."},{"Start":"00:45.495 ","End":"00:55.665","Text":"One 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":"01:01.230","Text":"Alpha goes with 1, Beta goes with 2, Gamma goes with 3."},{"Start":"01:01.230 ","End":"01:06.570","Text":"Cosine Alpha is going to be v_1 over the magnitude of"},{"Start":"01:06.570 ","End":"01:15.090","Text":"v. Cosine of Beta will be v_2 over the magnitude of v,"},{"Start":"01:15.090 ","End":"01:23.400","Text":"and cosine of Gamma will be v_3 over the magnitude of v. Then here,"},{"Start":"01:23.400 ","End":"01:28.395","Text":"we have the v_1 is 1,"},{"Start":"01:28.395 ","End":"01:31.830","Text":"v_2 will be negative 2,"},{"Start":"01:31.830 ","End":"01:34.305","Text":"and v_3 will be 3."},{"Start":"01:34.305 ","End":"01:36.080","Text":"It doesn\u0027t matter if we use the i, j,"},{"Start":"01:36.080 ","End":"01:38.795","Text":"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":"01:53.420","Text":"Let\u0027s see the magnitude of v is just equal to the square root."},{"Start":"01:53.420 ","End":"01:56.390","Text":"Just add each component up and square,"},{"Start":"01:56.390 ","End":"02:00.030","Text":"1 squared, never mind the minus,"},{"Start":"02:00.030 ","End":"02:01.964","Text":"it will be plus 2 squared,"},{"Start":"02:01.964 ","End":"02:08.430","Text":"plus 3 squared which is 1 plus 4 plus 9,"},{"Start":"02:08.430 ","End":"02:11.595","Text":"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":"02:26.920","Text":"Here we\u0027ll have minus 2 over square root of 14,"},{"Start":"02:26.920 ","End":"02:31.925","Text":"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":"02:42.620","Text":"For the angles, we will need to use the calculator."},{"Start":"02:42.620 ","End":"02:45.645","Text":"Let me just write Alpha equals,"},{"Start":"02:45.645 ","End":"02:49.740","Text":"Beta equals, and Gamma equals."},{"Start":"02:49.740 ","End":"02:52.709","Text":"It didn\u0027t say degrees or radians,"},{"Start":"02:52.709 ","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":"03:07.990","Text":"The direction cosine is always between 0 and 180."},{"Start":"03:07.990 ","End":"03:09.695","Text":"Let\u0027s start with this."},{"Start":"03:09.695 ","End":"03:14.075","Text":"What you do is you compute this on the calculator and then"},{"Start":"03:14.075 ","End":"03:21.185","Text":"you do inverse cosine or shift cosine or however your calculator works."},{"Start":"03:21.185 ","End":"03:24.305","Text":"If you have the calculator set for degrees,"},{"Start":"03:24.305 ","End":"03:29.750","Text":"you get something like 74.49 something degrees,"},{"Start":"03:29.750 ","End":"03:32.990","Text":"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":"03:40.605","Text":"Now the other 1,"},{"Start":"03:40.605 ","End":"03:46.390","Text":"as I said we\u0027re expecting an obtuse angle, meaning bigger than 90 degrees."},{"Start":"03:46.390 ","End":"03:53.735","Text":"This comes out to be 122.31 something in degrees."},{"Start":"03:53.735 ","End":"03:55.775","Text":"If you want it in radians,"},{"Start":"03:55.775 ","End":"04:03.520","Text":"2.134 something, something, something radians."},{"Start":"04:03.520 ","End":"04:05.535","Text":"The last 1, Gamma,"},{"Start":"04:05.535 ","End":"04:10.470","Text":"the angle between the v and the z axis,"},{"Start":"04:10.470 ","End":"04:13.719","Text":"that will come out to be,"},{"Start":"04:13.790 ","End":"04:22.710","Text":"I make this 36.699 something"},{"Start":"04:22.710 ","End":"04:33.075","Text":"degrees and in radians, 0.640 something radians."},{"Start":"04:33.075 ","End":"04:35.820","Text":"The exact answer is not important,"},{"Start":"04:35.820 ","End":"04:38.264","Text":"it\u0027s the method that\u0027s important."},{"Start":"04:38.264 ","End":"04:42.500","Text":"Really what we do is just take each of"},{"Start":"04:42.500 ","End":"04:48.084","Text":"the components and divide by the magnitude of the vector,"},{"Start":"04:48.084 ","End":"04:52.335","Text":"and then we get the 3 cosines."},{"Start":"04:52.335 ","End":"04:57.545","Text":"Then the inverse cosine or arc cosine will give us the actual angles,"},{"Start":"04:57.545 ","End":"05:02.670","Text":"whatever we want, degrees or radians. That\u0027s it."}],"ID":10290},{"Watched":false,"Name":"Exercise 6","Duration":"8m 32s","ChapterTopicVideoID":27704,"CourseChapterTopicPlaylistID":57472,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/27704.jpeg","UploadDate":"2022-05-03T09:16:51.3370000","DurationForVideoObject":"PT8M32S","Description":null,"VideoComments":[],"Subtitles":[],"ID":28998}],"Thumbnail":null,"ID":57472},{"Name":"Vectors Cross Product","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors - Cross Product","Duration":"19m 5s","ChapterTopicVideoID":10110,"CourseChapterTopicPlaylistID":57473,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10110.jpeg","UploadDate":"2022-05-18T07:46:42.2430000","DurationForVideoObject":"PT19M5S","Description":null,"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":"00:06.150","Text":"and now we\u0027re going to learn a cross-product."},{"Start":"00:06.150 ","End":"00:08.985","Text":"There\u0027s some major differences."},{"Start":"00:08.985 ","End":"00:14.920","Text":"Cross-product is something that only works in 3D."},{"Start":"00:14.960 ","End":"00:19.860","Text":"Another major difference is that when we took"},{"Start":"00:19.860 ","End":"00:24.915","Text":"a vector and we did a dot-product with another vector,"},{"Start":"00:24.915 ","End":"00:28.320","Text":"what we got was a scalar, a number."},{"Start":"00:28.320 ","End":"00:32.505","Text":"But when we take a vector and we cross it,"},{"Start":"00:32.505 ","End":"00:35.940","Text":"this is how the cross-product looks just like a multiplication,"},{"Start":"00:35.940 ","End":"00:38.610","Text":"when 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":"00:46.660","Text":"Those are the major differences."},{"Start":"00:46.730 ","End":"00:49.229","Text":"Just like with the dot-product,"},{"Start":"00:49.229 ","End":"00:50.910","Text":"we gave sort of 2 definitions."},{"Start":"00:50.910 ","End":"00:55.190","Text":"1 was like a formula for how to compute this,"},{"Start":"00:55.190 ","End":"01:00.560","Text":"given these, and the other was 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":"01:17.120","Text":"When you have 1 vectors,"},{"Start":"01:17.120 ","End":"01:20.785","Text":"a and b, and I want to know what is a cross b,"},{"Start":"01:20.785 ","End":"01:28.805","Text":"what we do is we take a vector which is perpendicular to both,"},{"Start":"01:28.805 ","End":"01:38.930","Text":"and the size of this vector is determined by the formula that the magnitude of"},{"Start":"01:38.930 ","End":"01:44.510","Text":"a cross b is equal to the magnitude of a times"},{"Start":"01:44.510 ","End":"01:50.765","Text":"the magnitude of b 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":"02:07.430","Text":"Notice that there are actually 2 vectors which are perpendicular and have this size."},{"Start":"02:07.430 ","End":"02:10.655","Text":"It could face up or it could face down in this diagram."},{"Start":"02:10.655 ","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":"02:28.180","Text":"pointing towards a and the middle finger is pointing towards b,"},{"Start":"02:28.180 ","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":"02:46.760","Text":"That\u0027s the geometric side of it."},{"Start":"02:46.760 ","End":"02:52.350","Text":"Now let\u0027s get to the more formula side of how to compute it."},{"Start":"02:52.790 ","End":"02:57.815","Text":"Let\u0027s say that the vector a is given by,"},{"Start":"02:57.815 ","End":"03:01.190","Text":"as usual a_1, a_2, a_3."},{"Start":"03:01.190 ","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":"03:32.455","Text":"The first component is a_2 times b_3 minus a_3 b_2."},{"Start":"03:32.455 ","End":"03:34.835","Text":"I\u0027m going to try and show you there\u0027s a pattern here."},{"Start":"03:34.835 ","End":"03:39.569","Text":"In the first place, the first component,"},{"Start":"03:39.740 ","End":"03:43.725","Text":"the number after 1 is 2,"},{"Start":"03:43.725 ","End":"03:46.380","Text":"and then the number after that 3, and there\u0027s always,"},{"Start":"03:46.380 ","End":"03:48.870","Text":"an a and a b and an a and a b with a minus."},{"Start":"03:48.870 ","End":"03:51.250","Text":"The next 1, if I take a_2,"},{"Start":"03:51.250 ","End":"03:54.980","Text":"is I take what comes after 2 is 3,"},{"Start":"03:54.980 ","End":"03:57.540","Text":"and what comes after 3, well,"},{"Start":"03:57.540 ","End":"03:59.490","Text":"we wrap around back to 1 again,"},{"Start":"03:59.490 ","End":"04:00.810","Text":"and then we do the reverse,"},{"Start":"04:00.810 ","End":"04:03.090","Text":"the a_1 with the b_3."},{"Start":"04:03.090 ","End":"04:05.460","Text":"Then when we get to the third place,"},{"Start":"04:05.460 ","End":"04:08.580","Text":"after 3 comes 1 because we\u0027re going cyclically."},{"Start":"04:08.580 ","End":"04:13.920","Text":"It\u0027s a_1 and after that is b_2 minus a_2 b_1."},{"Start":"04:13.920 ","End":"04:17.270","Text":"It\u0027s tricky to remember."},{"Start":"04:17.270 ","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":"04:25.430","Text":"but I don\u0027t want to assume you know what a determinant is."},{"Start":"04:25.430 ","End":"04:30.900","Text":"At the end I\u0027ll show you a way to compute it using determinants."},{"Start":"04:31.820 ","End":"04:43.170","Text":"1 thing that\u0027s clear from this rule is that a cross b is not the same as b cross a."},{"Start":"04:43.170 ","End":"04:45.495","Text":"If I do b cross a,"},{"Start":"04:45.495 ","End":"04:47.460","Text":"the thumb point down,"},{"Start":"04:47.460 ","End":"04:50.330","Text":"I\u0027ll get exactly the same magnitude,"},{"Start":"04:50.330 ","End":"04:52.160","Text":"but in the downward direction."},{"Start":"04:52.160 ","End":"04:58.640","Text":"In fact, b cross a is going to be the negative,"},{"Start":"04:58.640 ","End":"05:01.650","Text":"negative vector of a cross b,"},{"Start":"05:01.650 ","End":"05:03.870","Text":"same size but opposite direction."},{"Start":"05:03.870 ","End":"05:07.160","Text":"The order is important with the cross-products as"},{"Start":"05:07.160 ","End":"05:11.375","Text":"opposed to dot-product where the order wasn\u0027t important."},{"Start":"05:11.375 ","End":"05:14.495","Text":"I\u0027d like to do a computational example."},{"Start":"05:14.495 ","End":"05:21.260","Text":"Let\u0027s take a to be the vector 2,"},{"Start":"05:21.260 ","End":"05:31.570","Text":"1 minus 1, and let\u0027s take the vector b to equal minus 3, 4,1."},{"Start":"05:31.570 ","End":"05:39.295","Text":"Let\u0027s see if we can compute what is the cross product of these 2 vectors."},{"Start":"05:39.295 ","End":"05:48.120","Text":"I think 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":"06:04.275","Text":"Well, maybe I\u0027ll write it 1 times 1,"},{"Start":"06:04.275 ","End":"06:07.360","Text":"and then a_3, b_2."},{"Start":"06:07.360 ","End":"06:12.050","Text":"This times this minus,"},{"Start":"06:12.050 ","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":"06:23.640","Text":"Next 1, a_3, b_1, this with this."},{"Start":"06:23.640 ","End":"06:33.915","Text":"It\u0027s this with this minus 1 times minus 3 minus a_1,"},{"Start":"06:33.915 ","End":"06:36.330","Text":"b_3 can always look at it here,"},{"Start":"06:36.330 ","End":"06:39.480","Text":"a_1, b_3, 2 times 1."},{"Start":"06:39.480 ","End":"06:44.410","Text":"I\u0027ll just to put it in brackets instead of the dot."},{"Start":"06:44.410 ","End":"06:49.680","Text":"Then the last component would be just a_1,"},{"Start":"06:49.680 ","End":"06:51.165","Text":"b_2 minus a_2, b_1."},{"Start":"06:51.165 ","End":"06:53.160","Text":"This times this."},{"Start":"06:53.160 ","End":"06:59.985","Text":"2 times 4 minus 1 times minus 3,"},{"Start":"06:59.985 ","End":"07:02.790","Text":"1 times minus 3."},{"Start":"07:02.790 ","End":"07:05.540","Text":"What is this come out to be?"},{"Start":"07:05.540 ","End":"07:13.630","Text":"Let\u0027s see, 1 plus 4 is 5."},{"Start":"07:14.960 ","End":"07:19.170","Text":"Well, the angular bracket."},{"Start":"07:19.170 ","End":"07:27.945","Text":"Next 1, plus 3 minus 2 is 1."},{"Start":"07:27.945 ","End":"07:35.115","Text":"The left 1 is 8 plus 3 is 11 and that\u0027s the answer."},{"Start":"07:35.115 ","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":"07:47.750","Text":"I\u0027d like to show you another property of the cross product."},{"Start":"07:47.750 ","End":"07:50.240","Text":"If we took 2 vectors,"},{"Start":"07:50.240 ","End":"07:55.785","Text":"let\u0027s say they\u0027re not 0, and if they\u0027re parallel,"},{"Start":"07:55.785 ","End":"07:59.780","Text":"parallel means could be in the same direction or opposite directions,"},{"Start":"07:59.780 ","End":"08:01.970","Text":"but on 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":"08:06.170","Text":"0 or a 180 degrees depending on"},{"Start":"08:06.170 ","End":"08:08.840","Text":"whether they\u0027re the same direction or precisely the opposite."},{"Start":"08:08.840 ","End":"08:11.540","Text":"If Theta is a 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":"08:22.070","Text":"If this is 0 and these 2 are non 0,"},{"Start":"08:22.070 ","End":"08:23.810","Text":"then sine Theta is 0."},{"Start":"08:23.810 ","End":"08:29.630","Text":"So theta has to be 0 or Pi or plus multiples of 2 Pi,"},{"Start":"08:29.630 ","End":"08:32.375","Text":"but in this range 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":"08:50.315","Text":"it\u0027s the 0 vector, unlike like the dot-product."},{"Start":"08:50.315 ","End":"08:59.730","Text":"I\u0027m assuming that a and b are both non 0 vectors."},{"Start":"08:59.730 ","End":"09:02.160","Text":"I got rid of that picture."},{"Start":"09:02.160 ","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":"09:11.180","Text":"but also in mathematics,"},{"Start":"09:11.180 ","End":"09:15.875","Text":"there\u0027s a concept called a normal vector to a plane."},{"Start":"09:15.875 ","End":"09:19.320","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":"09:28.365","Text":"but every plane, there\u0027s a direction called normal,"},{"Start":"09:28.365 ","End":"09:30.140","Text":"or it could be the opposite."},{"Start":"09:30.140 ","End":"09:33.980","Text":"It means it\u0027s perpendicular to all the plane,"},{"Start":"09:33.980 ","End":"09:36.440","Text":"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":"09:53.600","Text":"then 1 way of finding a normal vector would be to take"},{"Start":"09:53.600 ","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":"Beside you have the terms perpendicular and orthogonal,"},{"Start":"10:04.835 ","End":"10:06.710","Text":"but often we have a question where we\u0027re given"},{"Start":"10:06.710 ","End":"10:10.580","Text":"2 vectors and we want to find something that\u0027s perpendicular to both."},{"Start":"10:10.580 ","End":"10:14.885","Text":"The cross-product is a perfect way to do that."},{"Start":"10:14.885 ","End":"10:17.855","Text":"Let\u0027s take an example problem."},{"Start":"10:17.855 ","End":"10:21.500","Text":"Suppose I have 3 points in the plane,"},{"Start":"10:21.500 ","End":"10:23.210","Text":"and these are points as round brackets."},{"Start":"10:23.210 ","End":"10:27.610","Text":"P is 1, 0,"},{"Start":"10:27.610 ","End":"10:32.995","Text":"0, let\u0027s say Q is the point 1, 1, 1,"},{"Start":"10:32.995 ","End":"10:37.720","Text":"and R is the point 2 minus 1,"},{"Start":"10:37.720 ","End":"10:44.810","Text":"3, and we want to find a normal to the plane."},{"Start":"10:44.810 ","End":"10:49.380","Text":"I wrote it to these 3 points in a plane,"},{"Start":"10:49.380 ","End":"10:52.030","Text":"and I\u0027m going to find a vector orthogonal to the plane."},{"Start":"10:52.030 ","End":"10:55.270","Text":"I didn\u0027t use the word normal orthogonal or perpendicular."},{"Start":"10:55.270 ","End":"11:05.680","Text":"What we do is let suppose this is my P and this is my Q, and this is R,"},{"Start":"11:05.680 ","End":"11:09.180","Text":"inside the plane P Q and R,"},{"Start":"11:09.180 ","End":"11:18.469","Text":"what we do is we take the vector that goes from P to Q,"},{"Start":"11:18.469 ","End":"11:22.345","Text":"and we call that say a,"},{"Start":"11:22.345 ","End":"11:25.535","Text":"and the vector from P to R,"},{"Start":"11:25.535 ","End":"11:27.965","Text":"and call that let\u0027s say b,"},{"Start":"11:27.965 ","End":"11:33.800","Text":"and then a perpendicular vector that we want,"},{"Start":"11:33.800 ","End":"11:37.130","Text":"which would be perpendicular to these 2 vectors."},{"Start":"11:37.130 ","End":"11:39.095","Text":"It\u0027ll be perpendicular to the whole plane."},{"Start":"11:39.095 ","End":"11:45.090","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":"12:03.609","Text":"If you remember, the way we do this is we take the coordinates"},{"Start":"12:03.609 ","End":"12:07.600","Text":"of Q and subtract the coordinates of P. I want 1,"},{"Start":"12:07.600 ","End":"12:10.105","Text":"1, 1 less 1, 0, 0."},{"Start":"12:10.105 ","End":"12:14.080","Text":"So 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":"12:18.250","Text":"1 minus 0 is 1."},{"Start":"12:18.250 ","End":"12:21.820","Text":"B similarly joins P to"},{"Start":"12:21.820 ","End":"12:30.250","Text":"R. Similarly we take away the P from R,"},{"Start":"12:30.250 ","End":"12:33.505","Text":"so 2 minus 1 is 1,"},{"Start":"12:33.505 ","End":"12:36.280","Text":"minus 1 less 0 is minus 1,"},{"Start":"12:36.280 ","End":"12:39.055","Text":"and 3 less 0 is 3."},{"Start":"12:39.055 ","End":"12:46.850","Text":"All we have to do is to do the cross product of a with b."},{"Start":"12:46.860 ","End":"12:50.140","Text":"If we use the formula,"},{"Start":"12:50.140 ","End":"12:52.420","Text":"the formula has disappeared."},{"Start":"12:52.420 ","End":"12:55.975","Text":"I just copied it from above."},{"Start":"12:55.975 ","End":"13:01.340","Text":"Now we\u0027ve got a_2 b_3 minus a_3 b_2."},{"Start":"13:02.550 ","End":"13:07.060","Text":"This times this minus this times this,"},{"Start":"13:07.060 ","End":"13:09.115","Text":"I\u0027ll do it all in one stage."},{"Start":"13:09.115 ","End":"13:13.210","Text":"This times this is 3 minus minus 1,"},{"Start":"13:13.210 ","End":"13:16.000","Text":"3 minus minus 1 is 4."},{"Start":"13:16.000 ","End":"13:18.760","Text":"Next, I see I need 3 and 1,"},{"Start":"13:18.760 ","End":"13:23.500","Text":"so a_3 b_1 minus the other diagonal."},{"Start":"13:23.500 ","End":"13:25.750","Text":"So it\u0027s 1 times 1 is 1,"},{"Start":"13:25.750 ","End":"13:29.140","Text":"minus 0, so that\u0027s 1."},{"Start":"13:29.140 ","End":"13:36.880","Text":"For the last one I need 1 and 2, a_1 and b_2."},{"Start":"13:36.880 ","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":"13:47.140","Text":"Here we have a vector which is orthogonal to"},{"Start":"13:47.140 ","End":"13:52.000","Text":"the plane 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. Here they are."},{"Start":"13:59.065 ","End":"14:07.060","Text":"We assume that u and v and w here are vectors, any vectors."},{"Start":"14:07.060 ","End":"14:09.145","Text":"We already discussed this."},{"Start":"14:09.145 ","End":"14:12.370","Text":"We used a and b but the cross product,"},{"Start":"14:12.370 ","End":"14:14.320","Text":"if you do it switch the order,"},{"Start":"14:14.320 ","End":"14:19.015","Text":"it becomes minus the vector in the opposite direction."},{"Start":"14:19.015 ","End":"14:23.875","Text":"There\u0027s a distributive rule similar to what we had with the dot product,"},{"Start":"14:23.875 ","End":"14:26.950","Text":"just like with multiplication and addition in algebra,"},{"Start":"14:26.950 ","End":"14:28.615","Text":"that\u0027s the way to remember it."},{"Start":"14:28.615 ","End":"14:34.960","Text":"U cross v plus w. You do it cross with v separately, with u,"},{"Start":"14:34.960 ","End":"14:39.505","Text":"w separately and you add a constant,"},{"Start":"14:39.505 ","End":"14:41.530","Text":"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":"14:51.955","Text":"The last one is a strange-looking one."},{"Start":"14:51.955 ","End":"14:59.275","Text":"It turns out that if I put a dot here and a cross here or a cross here and a dot here,"},{"Start":"14:59.275 ","End":"15:00.520","Text":"I\u0027ve got the same answer."},{"Start":"15:00.520 ","End":"15:02.245","Text":"The answer will be a scalar."},{"Start":"15:02.245 ","End":"15:07.045","Text":"This is a vector and a vector dot a vector is a scalar, and same here."},{"Start":"15:07.045 ","End":"15:10.180","Text":"There\u0027s actually a nice way of computing this,"},{"Start":"15:10.180 ","End":"15:11.710","Text":"but you have to know determinants,"},{"Start":"15:11.710 ","End":"15:14.230","Text":"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":"15:28.345","Text":"I\u0027ll show you the diagram and then I\u0027ll explain."},{"Start":"15:28.345 ","End":"15:30.070","Text":"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":"15:41.800","Text":"When you take 3 vectors and you complete the parallelogram."},{"Start":"15:41.800 ","End":"15:45.265","Text":"Let\u0027s first take a look at just this front here."},{"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 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":"16:05.470","Text":"by the magnitude of vector a cross vector b."},{"Start":"16:05.470 ","End":"16:07.660","Text":"That\u0027s 1 use of a cross product."},{"Start":"16:07.660 ","End":"16:11.060","Text":"If I take the cross product and then I take its magnitude,"},{"Start":"16:11.060 ","End":"16:16.590","Text":"I can find the area of parallelogram of 2 vectors."},{"Start":"16:16.590 ","End":"16:20.460","Text":"The other thing is if I now forget about the area,"},{"Start":"16:20.460 ","End":"16:22.905","Text":"but talk about the volume."},{"Start":"16:22.905 ","End":"16:28.690","Text":"This whole parallelepiped, it is a 3-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":"16:41.305","Text":"a dot-product with"},{"Start":"16:41.305 ","End":"16:47.830","Text":"b cross c. Not quite,"},{"Start":"16:47.830 ","End":"16:49.660","Text":"this thing could be negative."},{"Start":"16:49.660 ","End":"16:53.584","Text":"We need the absolute value because the volume has to be positive."},{"Start":"16:53.584 ","End":"16:57.450","Text":"Now actually, this formula with the volume gives us"},{"Start":"16:57.450 ","End":"17:02.405","Text":"a way to find out if 3 vectors are in the same plane."},{"Start":"17:02.405 ","End":"17:05.530","Text":"If the 3 vectors are in the same plane then"},{"Start":"17:05.530 ","End":"17:09.850","Text":"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":"17:17.500","Text":"Let me take 3 vectors."},{"Start":"17:17.500 ","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":"17:52.479","Text":"First, we need to compute b cross"},{"Start":"17:52.479 ","End":"18:01.000","Text":"c. I did it for you."},{"Start":"18:01.000 ","End":"18:03.400","Text":"This is what it comes out to be."},{"Start":"18:03.400 ","End":"18:07.450","Text":"For example, the first one we would take from the formula,"},{"Start":"18:07.450 ","End":"18:13.855","Text":"a_2 b_3 minus a_3 b_2 minus 1 times 18,"},{"Start":"18:13.855 ","End":"18:18.040","Text":"and then less minus 36 from this times this,"},{"Start":"18:18.040 ","End":"18:20.260","Text":"it comes up plus 18 and so on."},{"Start":"18:20.260 ","End":"18:27.100","Text":"The next thing we need to do is to do a. with"},{"Start":"18:27.100 ","End":"18:34.195","Text":"b cross c. This comes out to be this thing above,"},{"Start":"18:34.195 ","End":"18:41.035","Text":"18, minus 36, 18 dot-product with c,"},{"Start":"18:41.035 ","End":"18:46.555","Text":"which is 0, minus 9, 18."},{"Start":"18:46.555 ","End":"18:49.675","Text":"At least one of them is a 0. It\u0027s this with this,"},{"Start":"18:49.675 ","End":"18:52.585","Text":"this with this, and this with this versus,"},{"Start":"18:52.585 ","End":"18:59.290","Text":"I\u0027ll 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":"19:04.990","Text":"Sorry, it\u0027s minus 18."},{"Start":"19:04.990 ","End":"19:08.560","Text":"Then minus 18 times 18."},{"Start":"19:08.560 ","End":"19:13.780","Text":"Anyway, it turns out this is 324 and this is 324 and this is 0."},{"Start":"19:13.780 ","End":"19:16.465","Text":"It ends up being 0,"},{"Start":"19:16.465 ","End":"19:19.375","Text":"so the answer is yes,"},{"Start":"19:19.375 ","End":"19:22.255","Text":"they are in the same plane."},{"Start":"19:22.255 ","End":"19:25.930","Text":"Now, I\u0027m basically done here."},{"Start":"19:25.930 ","End":"19:33.115","Text":"If you have studied determinants and you know what a determinant is,"},{"Start":"19:33.115 ","End":"19:38.830","Text":"then you\u0027re welcome to stay and I\u0027ll talk about some shortcut formulas of"},{"Start":"19:38.830 ","End":"19:45.250","Text":"finding the cross product as well as an easy way of finding this product."},{"Start":"19:45.250 ","End":"19:49.610","Text":"Stay or not, you are welcome."}],"ID":10291},{"Watched":false,"Name":"Vectors - Cross Product (continued)","Duration":"6m 42s","ChapterTopicVideoID":10111,"CourseChapterTopicPlaylistID":57473,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10111.jpeg","UploadDate":"2020-01-16T07:08:33.4570000","DurationForVideoObject":"PT6M42S","Description":null,"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":"00:07.400 ","End":"00:11.220","Text":"this formula has a simpler expression using"},{"Start":"00:11.220 ","End":"00:18.105","Text":"determinants and here\u0027s the formula and I\u0027m going to show you how we use this."},{"Start":"00:18.105 ","End":"00:23.670","Text":"I just copied the example we did earlier and I want to use the same example,"},{"Start":"00:23.670 ","End":"00:28.275","Text":"that way I can check 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":"00:34.365","Text":"what we would get would be the determinant,"},{"Start":"00:34.365 ","End":"00:39.290","Text":"it\u0027s always i, j,"},{"Start":"00:39.290 ","End":"00:44.370","Text":"and k and then on this row,"},{"Start":"00:44.370 ","End":"00:48.780","Text":"next I put the 2, 1,"},{"Start":"00:48.780 ","End":"00:55.000","Text":"minus 1 and then the minus 3, 4, 1."},{"Start":"00:55.550 ","End":"01:00.290","Text":"Even for determinants there is more than 1 way of doing it."},{"Start":"01:00.290 ","End":"01:04.650","Text":"I\u0027m going to use the method of cofactors."},{"Start":"01:07.280 ","End":"01:13.260","Text":"Let\u0027s see, we take the i or rather, you know what,"},{"Start":"01:13.260 ","End":"01:20.390","Text":"I\u0027ll move this to the right and here I put the cofactor of i,"},{"Start":"01:20.390 ","End":"01:22.760","Text":"which is like deleting the column and row with it,"},{"Start":"01:22.760 ","End":"01:24.590","Text":"so I\u0027ve got 1 minus 1, 4,"},{"Start":"01:24.590 ","End":"01:29.010","Text":"1 and then we alternate signs,"},{"Start":"01:29.010 ","End":"01:34.025","Text":"so it has to be minus something with j and then"},{"Start":"01:34.025 ","End":"01:40.740","Text":"plus the cofactor of k with k. In here I need to put,"},{"Start":"01:40.740 ","End":"01:44.220","Text":"let\u0027s see, for j, I cross this out and this out I got 2,"},{"Start":"01:44.220 ","End":"01:45.540","Text":"minus 1, minus 3,"},{"Start":"01:45.540 ","End":"01:48.360","Text":"1, 2, minus 1,"},{"Start":"01:48.360 ","End":"01:51.555","Text":"minus 3, 1. Then for k,"},{"Start":"01:51.555 ","End":"01:54.675","Text":"I need this so it\u0027s 2,"},{"Start":"01:54.675 ","End":"01:58.990","Text":"1, minus 3, 4."},{"Start":"02:00.470 ","End":"02:05.609","Text":"Now the determinant of a 2 by 2 is just diagonal less this diagonal,"},{"Start":"02:05.609 ","End":"02:12.480","Text":"so it\u0027s 1 minus minus 4, which is 5."},{"Start":"02:12.480 ","End":"02:15.105","Text":"This 1 is 5."},{"Start":"02:15.105 ","End":"02:17.800","Text":"Let\u0027s do the determinants, 2 plus 3. No, sorry,"},{"Start":"02:22.880 ","End":"02:25.755","Text":"2 minus this diagonal,"},{"Start":"02:25.755 ","End":"02:29.445","Text":"the diagonal is 3, 2 minus 3, minus 1, sorry,"},{"Start":"02:29.445 ","End":"02:38.610","Text":"and then 8 minus, minus 3 is 11. So what we end up getting is 5i"},{"Start":"02:38.610 ","End":"02:47.390","Text":"plus 1j and plus"},{"Start":"02:47.390 ","End":"02:50.970","Text":"that vector plus 11k."},{"Start":"02:52.180 ","End":"03:00.515","Text":"I would say that this and this are the same,"},{"Start":"03:00.515 ","End":"03:04.070","Text":"just a different form of it so we got the right answer."},{"Start":"03:04.070 ","End":"03:09.430","Text":"This is an easier formula if you know determinants."},{"Start":"03:09.430 ","End":"03:11.780","Text":"I\u0027m going to continue on a fresh page."},{"Start":"03:11.780 ","End":"03:20.854","Text":"I remember I said something when I introduced the concept of a dot b cross c,"},{"Start":"03:20.854 ","End":"03:24.800","Text":"it was in a formula and also we"},{"Start":"03:24.800 ","End":"03:28.895","Text":"used it to compute volume and I said there\u0027s an easier way to compute this."},{"Start":"03:28.895 ","End":"03:32.420","Text":"This has a formula using determinants,"},{"Start":"03:32.420 ","End":"03:37.865","Text":"that this is the determinant of a_1, a_2,"},{"Start":"03:37.865 ","End":"03:43.795","Text":"a_3, b_1, b_2, b_3,"},{"Start":"03:43.795 ","End":"03:48.750","Text":"and c_1, c_2, c_3."},{"Start":"03:48.750 ","End":"03:52.310","Text":"I didn\u0027t write it again but 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 and I\u0027m going to test it out on the example we used earlier."},{"Start":"04:02.089 ","End":"04:04.970","Text":"Here\u0027s the example we had earlier with"},{"Start":"04:04.970 ","End":"04:10.050","Text":"these 3 vectors so this time let\u0027s do it using a determinant."},{"Start":"04:10.720 ","End":"04:18.855","Text":"In our case, what we have is let\u0027s see 1, 4, minus 7,"},{"Start":"04:18.855 ","End":"04:21.390","Text":"and then 2, minus 1,"},{"Start":"04:21.390 ","End":"04:28.480","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 1st row,"},{"Start":"04:43.430 ","End":"04:46.550","Text":"which I usually use because there\u0027s a 0 in here and it\u0027ll save"},{"Start":"04:46.550 ","End":"04:51.000","Text":"us 1 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":"04:58.835","Text":"I\u0027ll write it out though it\u0027s 0 times the co-factor,"},{"Start":"04:58.835 ","End":"05:01.910","Text":"it doesn\u0027t really matter because it\u0027s 0,"},{"Start":"05:01.910 ","End":"05:04.475","Text":"but 4, 7, minus 1,"},{"Start":"05:04.475 ","End":"05:14.045","Text":"4 and then there\u0027s a minus in the middle and that would be minus,"},{"Start":"05:14.045 ","End":"05:17.509","Text":"minus 9 times its cofactor,"},{"Start":"05:17.509 ","End":"05:18.830","Text":"which is 1, 7, 2,"},{"Start":"05:18.830 ","End":"05:28.780","Text":"4 and then plus 18 times its cofactor, which is 1,"},{"Start":"05:28.780 ","End":"05:34.995","Text":"4, 2, minus 1, and this is equal to,"},{"Start":"05:34.995 ","End":"05:37.755","Text":"well, this is 0,"},{"Start":"05:37.755 ","End":"05:40.480","Text":"it doesn\u0027t matter what this is."},{"Start":"05:40.480 ","End":"05:45.050","Text":"This cofactor is; I just put a question mark,"},{"Start":"05:45.050 ","End":"05:47.485","Text":"I don\u0027t care because I\u0027m going to multiply it by 0,"},{"Start":"05:47.485 ","End":"05:49.315","Text":"oops, this is a minus 7,"},{"Start":"05:49.315 ","End":"05:57.575","Text":"4 plus 14, this is 18 and then minus 1,"},{"Start":"05:57.575 ","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":"06:19.610","Text":"I don\u0027t even have to compute what 9 times 18"},{"Start":"06:19.610 ","End":"06:24.875","Text":"is and I see that this actually is equal to 0,"},{"Start":"06:24.875 ","End":"06:32.520","Text":"which is what we got also the other way, so that\u0027s good."},{"Start":"06:33.170 ","End":"06:37.100","Text":"That\u0027s really it. For those of you who studied determinants,"},{"Start":"06:37.100 ","End":"06:40.325","Text":"there are some shortcut formulas."},{"Start":"06:40.325 ","End":"06:43.230","Text":"We\u0027re done with the cross-product."}],"ID":10292},{"Watched":false,"Name":"Exercise 1","Duration":"6m 8s","ChapterTopicVideoID":10112,"CourseChapterTopicPlaylistID":57473,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10112.jpeg","UploadDate":"2017-08-20T11:09:26.3530000","DurationForVideoObject":"PT6M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.805","Text":"The purpose of this exercise is just to practice computing the cross product."},{"Start":"00:05.805 ","End":"00:09.765","Text":"There\u0027s more than 1 way to do this."},{"Start":"00:09.765 ","End":"00:14.250","Text":"In number 1, 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":"00:49.800","Text":"and also here minus 2. 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":"01:17.970","Text":"b_1, b_2, b_3, etc., from here."},{"Start":"01:17.970 ","End":"01:23.715","Text":"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":"01:42.300","Text":"then we have 30 minus 21,"},{"Start":"01:42.300 ","End":"01:51.280","Text":"and then minus 12 plus 12."},{"Start":"01:53.630 ","End":"01:58.230","Text":"That is equal to, let\u0027s see, 6,"},{"Start":"01:58.230 ","End":"02:03.840","Text":"30 minus 21 is 9, and here 0."},{"Start":"02:03.840 ","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":"02:19.825","Text":"You\u0027re expected to remember that if you change the order of the cross product,"},{"Start":"02:19.825 ","End":"02:22.105","Text":"you have a minus sign added."},{"Start":"02:22.105 ","End":"02:25.210","Text":"It\u0027s minus a cross b."},{"Start":"02:25.210 ","End":"02:28.369","Text":"In this case, we just put a minus in front of everything,"},{"Start":"02:28.369 ","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":"02:39.785","Text":"In part 2, 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":"02:54.425","Text":"what you do is write a 3 by 3 determinant."},{"Start":"02:54.425 ","End":"02:59.610","Text":"Here we put i, j,"},{"Start":"02:59.610 ","End":"03:04.380","Text":"and k. On the next row we put the first 1,"},{"Start":"03:04.380 ","End":"03:06.540","Text":"which is the u."},{"Start":"03:06.540 ","End":"03:10.800","Text":"So 3, just the coefficients,"},{"Start":"03:10.800 ","End":"03:14.910","Text":"minus 1, and 5."},{"Start":"03:14.910 ","End":"03:17.690","Text":"Then the other 1, there is no i,"},{"Start":"03:17.690 ","End":"03:20.450","Text":"so that means it\u0027s a 0 here,"},{"Start":"03:20.450 ","End":"03:24.445","Text":"and then 4, and then minus 2."},{"Start":"03:24.445 ","End":"03:28.385","Text":"Then we use the method of cofactors."},{"Start":"03:28.385 ","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":"03:38.005","Text":"I\u0027ll fill it out in a moment."},{"Start":"03:38.005 ","End":"03:43.545","Text":"i, and then it\u0027ll be a minus another 2 by 2 determinant,"},{"Start":"03:43.545 ","End":"03:48.225","Text":"j, and it\u0027ll be a plus a 2 by 2 determinant,"},{"Start":"03:48.225 ","End":"03:50.490","Text":"k. What we do"},{"Start":"03:50.490 ","End":"03:58.830","Text":"to get the bit in front of the i, is just eliminate the row and column with the i."},{"Start":"03:58.830 ","End":"04:01.280","Text":"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":"04:14.945","Text":"We\u0027re left with 3, 5, 0, minus 2."},{"Start":"04:14.945 ","End":"04:18.680","Text":"3, 5, 0 minus 2."},{"Start":"04:18.680 ","End":"04:22.215","Text":"For k, we just get this corner here,"},{"Start":"04:22.215 ","End":"04:24.810","Text":"3, minus 1, 0, 4."},{"Start":"04:24.810 ","End":"04:28.440","Text":"3, minus 1, 0, 4."},{"Start":"04:28.440 ","End":"04:36.825","Text":"Then a 2 by 2 determinant is just this diagonals product minus this diagonal."},{"Start":"04:36.825 ","End":"04:43.505","Text":"Here we have 2 minus 20,"},{"Start":"04:43.505 ","End":"04:51.665","Text":"which gives us minus 18i minus, let\u0027s see."},{"Start":"04:51.665 ","End":"04:54.740","Text":"This times this is minus 6,"},{"Start":"04:54.740 ","End":"04:57.710","Text":"this times this is nothing,"},{"Start":"04:57.710 ","End":"05:00.200","Text":"so we have minus,"},{"Start":"05:00.200 ","End":"05:06.140","Text":"minus 6, 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":"05:20.250","Text":"That\u0027s the answer, and it\u0027s in i, j,"},{"Start":"05:20.250 ","End":"05:27.715","Text":"k form, which I would expect if the original question was given in i, j, k form."},{"Start":"05:27.715 ","End":"05:43.180","Text":"That\u0027s u cross v. If I want v cross u,"},{"Start":"05:43.180 ","End":"05:45.630","Text":"there\u0027s going to be more space,"},{"Start":"05:45.630 ","End":"05:52.700","Text":"then v cross u is just the negative of u cross v,"},{"Start":"05:52.700 ","End":"06:03.220","Text":"so it\u0027s going to be 18i minus 6j, minus 12k."},{"Start":"06:03.220 ","End":"06:08.260","Text":"Just practicing computation, and we\u0027re done."}],"ID":10293},{"Watched":false,"Name":"Exercise 2","Duration":"6m 51s","ChapterTopicVideoID":10113,"CourseChapterTopicPlaylistID":57473,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10113.jpeg","UploadDate":"2017-08-20T11:11:03.4670000","DurationForVideoObject":"PT6M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we want to find a vector that\u0027s"},{"Start":"00:04.170 ","End":"00:09.060","Text":"orthogonal or perpendicular to the plane containing these 3 points,"},{"Start":"00:09.060 ","End":"00:14.430","Text":"so just made up the example using the numbers from 1 through 9."},{"Start":"00:14.430 ","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":"00:27.920","Text":"Have a plane, and we have 3 points, P,"},{"Start":"00:27.920 ","End":"00:30.990","Text":"Q and R. Say P, Q and R,"},{"Start":"00:30.990 ","End":"00:35.265","Text":"label them PQR, or things to the scale here."},{"Start":"00:35.265 ","End":"00:41.765","Text":"In general, there is a plain single plane that goes through any 3 points."},{"Start":"00:41.765 ","End":"00:43.590","Text":"Yeah, there\u0027s an exception."},{"Start":"00:43.590 ","End":"00:46.340","Text":"If all 3 points happen to be on the same line,"},{"Start":"00:46.340 ","End":"00:51.380","Text":"that wouldn\u0027t work, but let\u0027s assume that we\u0027re not in an exceptional case."},{"Start":"00:51.380 ","End":"00:57.090","Text":"2 vectors that would be parallel to the plane can take any combination,"},{"Start":"00:57.090 ","End":"01:02.025","Text":"but let\u0027s say I use PQ, and PR."},{"Start":"01:02.025 ","End":"01:05.115","Text":"I can get these 2 vectors,"},{"Start":"01:05.115 ","End":"01:08.475","Text":"and then I\u0027ll take the cross product."},{"Start":"01:08.475 ","End":"01:11.575","Text":"I\u0027ll get a vector perpendicular to these 2,"},{"Start":"01:11.575 ","End":"01:17.935","Text":"and if a vector is perpendicular to 2 vectors in the plane that are not parallel,"},{"Start":"01:17.935 ","End":"01:21.580","Text":"then it\u0027s going to be perpendicular to the whole plane."},{"Start":"01:21.580 ","End":"01:23.620","Text":"Here\u0027s what we do. First of all,"},{"Start":"01:23.620 ","End":"01:28.815","Text":"let\u0027s see what vector PQ is, call that PQ."},{"Start":"01:28.815 ","End":"01:35.200","Text":"That would be the components"},{"Start":"01:35.200 ","End":"01:42.650","Text":"of Q minus the components of P. What we\u0027ll get would be 4 minus 1."},{"Start":"01:46.230 ","End":"01:51.340","Text":"In this exercise, we need to find a vector that\u0027s orthogonal,"},{"Start":"01:51.340 ","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 of vectors to help us here."},{"Start":"02:00.310 ","End":"02:02.670","Text":"First of all, there\u0027s a quick sketch,"},{"Start":"02:02.670 ","End":"02:07.690","Text":"and we have a plane that goes through 3 points,"},{"Start":"02:07.690 ","End":"02:11.260","Text":"this isn\u0027t always the case."},{"Start":"02:11.260 ","End":"02:15.940","Text":"It\u0027s true if the 3 points are not on a straight line,"},{"Start":"02:15.940 ","End":"02:17.460","Text":"which they\u0027re not here,"},{"Start":"02:17.460 ","End":"02:19.095","Text":"and the 3 points are not,"},{"Start":"02:19.095 ","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":"02:33.630","Text":"let say, from P to Q,"},{"Start":"02:33.630 ","End":"02:41.235","Text":"and from P to 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":"02:50.990","Text":"perpendicular to these 2, and will therefore be perpendicular to the whole plane."},{"Start":"02:50.990 ","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":"03:27.580","Text":"In other words, 5,3,1."},{"Start":"03:29.600 ","End":"03:34.040","Text":"The other one, I\u0027ll take PR, though you could take an RQ or"},{"Start":"03:34.040 ","End":"03:39.570","Text":"some other possibility. We\u0027re taking PR,"},{"Start":"03:40.480 ","End":"03:49.290","Text":"and that will equal, I\u0027ll just do the final answer, just taking R minus P coordinate wise,"},{"Start":"03:49.290 ","End":"03:52.295","Text":"so 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":"04:04.580","Text":"What we need now is the cross product,"},{"Start":"04:04.580 ","End":"04:09.355","Text":"PQ, cross product with PR,"},{"Start":"04:09.355 ","End":"04:12.155","Text":"to get a perpendicular to both of these,"},{"Start":"04:12.155 ","End":"04:13.550","Text":"and if it\u0027s perpendicular to"},{"Start":"04:13.550 ","End":"04:18.080","Text":"2 non-parallel vectors in a plane, perpendicular to the whole plane."},{"Start":"04:18.080 ","End":"04:20.360","Text":"What this will equal,"},{"Start":"04:20.360 ","End":"04:28.685","Text":"if we use the determinant notation with I, J, K,"},{"Start":"04:28.685 ","End":"04:32.250","Text":"here I put I, J, K,"},{"Start":"04:32.250 ","End":"04:35.840","Text":"here I put the first one,"},{"Start":"04:35.840 ","End":"04:38.480","Text":"which is 5, 3,"},{"Start":"04:38.480 ","End":"04:43.375","Text":"1, and here 6, 6, 6."},{"Start":"04:43.375 ","End":"04:49.835","Text":"What I\u0027ll do is, there\u0027s several techniques for doing this,"},{"Start":"04:49.835 ","End":"04:52.475","Text":"I\u0027ll do the cofactor method."},{"Start":"04:52.475 ","End":"04:57.710","Text":"What goes with I is you erase the I column and"},{"Start":"04:57.710 ","End":"05:03.650","Text":"row, and we\u0027re left with the determinant 3,1,6,6."},{"Start":"05:03.650 ","End":"05:06.065","Text":"This is the coefficient of I."},{"Start":"05:06.065 ","End":"05:07.985","Text":"Then the middle one gets a minus."},{"Start":"05:07.985 ","End":"05:09.560","Text":"The last one is plus again."},{"Start":"05:09.560 ","End":"05:12.880","Text":"We get minus something J,"},{"Start":"05:12.880 ","End":"05:16.425","Text":"and then plus something K,"},{"Start":"05:16.425 ","End":"05:18.235","Text":"and the same idea for J,"},{"Start":"05:18.235 ","End":"05:22.500","Text":"I erase this, and this and I\u0027ve got 5,6,1,6."},{"Start":"05:23.470 ","End":"05:29.580","Text":"For the K, I get these 4,5,3,6,6."},{"Start":"05:30.530 ","End":"05:35.975","Text":"This comes out to be, let\u0027s see,"},{"Start":"05:35.975 ","End":"05:39.725","Text":"determinant of a 2 by 2 is this diagonal,"},{"Start":"05:39.725 ","End":"05:43.310","Text":"minus this diagonal, product minus product,"},{"Start":"05:43.310 ","End":"05:48.240","Text":"18 minus 6 is 12,"},{"Start":"05:49.090 ","End":"05:55.470","Text":"let\u0027s see, 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":"06:09.660","Text":"We could leave that as an answer,"},{"Start":"06:09.660 ","End":"06:13.620","Text":"or I could also use angular notation,"},{"Start":"06:13.620 ","End":"06:18.795","Text":"angular bracket 12, minus 24, 12."},{"Start":"06:18.795 ","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":"06:30.940","Text":"Scalar times a vector keeps its direction."},{"Start":"06:30.940 ","End":"06:33.510","Text":"I could also take,"},{"Start":"06:33.510 ","End":"06:35.519","Text":"as an answer, 1,"},{"Start":"06:35.519 ","End":"06:39.320","Text":"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":10294},{"Watched":false,"Name":"Exercise 3","Duration":"9m ","ChapterTopicVideoID":10114,"CourseChapterTopicPlaylistID":57473,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10114.jpeg","UploadDate":"2017-08-20T11:13:42.0300000","DurationForVideoObject":"PT9M","Description":null,"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":"00:08.900","Text":"we\u0027re given 3 vectors in 3D and"},{"Start":"00:08.900 ","End":"00:14.295","Text":"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":"00:24.705","Text":"In general, 3 vectors will form what is called a parallelepiped,"},{"Start":"00:24.705 ","End":"00:27.630","Text":"and 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":"00:38.860","Text":"Now, if these 3 vectors happen to be in the same plane,"},{"Start":"00:38.860 ","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":"00:49.145","Text":"We have a formula that gives us the volume of this,"},{"Start":"00:49.145 ","End":"00:51.515","Text":"it uses the cross and the dot-product."},{"Start":"00:51.515 ","End":"00:59.070","Text":"The volume of this parallelepiped is equal to"},{"Start":"00:59.120 ","End":"01:09.380","Text":"a. b cross c. You could actually take these in any order."},{"Start":"01:09.380 ","End":"01:12.200","Text":"But you first have to do the cross-product and you get"},{"Start":"01:12.200 ","End":"01:15.875","Text":"a vector and then a vector dot with a vector gives you a scalar."},{"Start":"01:15.875 ","End":"01:18.304","Text":"Actually this could come out negative,"},{"Start":"01:18.304 ","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":"01:36.650 ","End":"01:40.150","Text":"We\u0027ve talked about this in the theory part,"},{"Start":"01:40.150 ","End":"01:44.245","Text":"so let me just write it that for part 1,"},{"Start":"01:44.245 ","End":"01:51.940","Text":"that a. b cross c"},{"Start":"01:51.940 ","End":"01:58.270","Text":"is equal to the determinant and you just take the components of a,"},{"Start":"01:58.270 ","End":"02:01.735","Text":"3, minus 2, 5,"},{"Start":"02:01.735 ","End":"02:03.880","Text":"and then for b, 6,"},{"Start":"02:03.880 ","End":"02:12.490","Text":"minus 4, 7, and for c, 1, 0, 1."},{"Start":"02:12.490 ","End":"02:14.795","Text":"We just want to see if this is 0 or not."},{"Start":"02:14.795 ","End":"02:16.550","Text":"If you actually want the volume,"},{"Start":"02:16.550 ","End":"02:18.950","Text":"at the end you have to take absolute value because it"},{"Start":"02:18.950 ","End":"02:21.795","Text":"comes out negative throughout the minus."},{"Start":"02:21.795 ","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":"02:39.560","Text":"and as a way of expanding using co-factors,"},{"Start":"02:39.560 ","End":"02:43.415","Text":"you expand according to the most convenient row or column."},{"Start":"02:43.415 ","End":"02:46.400","Text":"The third row looks very good because it has a"},{"Start":"02:46.400 ","End":"02:49.775","Text":"0 in it and the numbers are small, 2 or 1\u0027s."},{"Start":"02:49.775 ","End":"02:56.780","Text":"What this will be,"},{"Start":"02:56.780 ","End":"03:03.140","Text":"we take the 1 times the determinant"},{"Start":"03:03.140 ","End":"03:08.685","Text":"of what\u0027s left if you delete its row and column."},{"Start":"03:08.685 ","End":"03:14.300","Text":"So It\u0027s times minus 2, 5, minus 4,"},{"Start":"03:14.300 ","End":"03:20.870","Text":"7, and then minus 0 times,"},{"Start":"03:20.870 ","End":"03:22.520","Text":"well, it doesn\u0027t matter what it is."},{"Start":"03:22.520 ","End":"03:27.050","Text":"Something, something, something, something 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":"03:33.350","Text":"and then plus 1 times,"},{"Start":"03:33.350 ","End":"03:36.470","Text":"we\u0027re running out of space,"},{"Start":"03:36.470 ","End":"03:37.970","Text":"I\u0027ll move it over here,"},{"Start":"03:37.970 ","End":"03:39.725","Text":"and here I move down here."},{"Start":"03:39.725 ","End":"03:49.790","Text":"This 1 times the determinant of 3 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":"04:04.425","Text":"We won\u0027t get into technical details assuming you\u0027ve studied determinants."},{"Start":"04:04.425 ","End":"04:12.260","Text":"Then let\u0027s see what this equals 1 times minus 14,"},{"Start":"04:12.260 ","End":"04:16.985","Text":"minus 20 is minus 34."},{"Start":"04:16.985 ","End":"04:20.725","Text":"I didn\u0027t need the 1, I\u0027ll just put minus 34."},{"Start":"04:20.725 ","End":"04:24.165","Text":"This 1 is just minus 0,"},{"Start":"04:24.165 ","End":"04:25.865","Text":"it doesn\u0027t matter what this is,"},{"Start":"04:25.865 ","End":"04:27.680","Text":"and here plus 1 times,"},{"Start":"04:27.680 ","End":"04:34.140","Text":"we 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":"04:39.425","Text":"It was a minus, minus a minus, wait a minute."},{"Start":"04:39.425 ","End":"04:44.300","Text":"It was minus 14, minus, minus 20,"},{"Start":"04:44.300 ","End":"04:50.160","Text":"so it\u0027s minus 14 plus 20 so this is 6, I\u0027m sorry."},{"Start":"04:51.740 ","End":"04:57.200","Text":"Now here again we have a negative takeaway a negative,"},{"Start":"04:57.200 ","End":"05:04.955","Text":"we have minus 12, minus minus 12."},{"Start":"05:04.955 ","End":"05:12.970","Text":"That\u0027s actually 0, and so the answer is 6."},{"Start":"05:12.970 ","End":"05:20.010","Text":"But the main point is that this 6 is not equal to 0,"},{"Start":"05:20.010 ","End":"05:24.830","Text":"so we can say that in this case that a, b,"},{"Start":"05:24.830 ","End":"05:30.420","Text":"and c are not in the same plane,"},{"Start":"05:30.420 ","End":"05:33.525","Text":"let me just write that not in 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":"05:45.410","Text":"I got rid of the picture that way I can have room to do number 2 over here,"},{"Start":"05:45.410 ","End":"05:49.790","Text":"and it\u0027s the same idea, different letters."},{"Start":"05:49.790 ","End":"05:58.620","Text":"What I need is u. v cross w,"},{"Start":"05:58.620 ","End":"06:01.930","Text":"and let\u0027s see what determinant this gives us."},{"Start":"06:01.930 ","End":"06:06.795","Text":"Just copy the coordinates, I\u0027ve got 1,"},{"Start":"06:06.795 ","End":"06:14.620","Text":"4, minus 7 and then 2, minus 1, 4."},{"Start":"06:15.530 ","End":"06:19.130","Text":"Then you don\u0027t have any i here,"},{"Start":"06:19.130 ","End":"06:21.200","Text":"so that\u0027s a 0 here,"},{"Start":"06:21.200 ","End":"06:26.065","Text":"and I\u0027ve got minus 9, 18."},{"Start":"06:26.065 ","End":"06:32.560","Text":"Again I\u0027m going to use this method of co-factors."},{"Start":"06:32.600 ","End":"06:39.830","Text":"I\u0027ll take the first column or the last row,"},{"Start":"06:39.830 ","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":"06:46.220","Text":"I\u0027m going to expand by the first column."},{"Start":"06:46.220 ","End":"06:53.840","Text":"I\u0027ve got 1 times the determinant of this,"},{"Start":"06:53.840 ","End":"06:56.095","Text":"which is minus 1,"},{"Start":"06:56.095 ","End":"07:01.575","Text":"4, minus 9, 18."},{"Start":"07:01.575 ","End":"07:04.660","Text":"Then it alternates the sign,"},{"Start":"07:04.660 ","End":"07:06.040","Text":"this is a minus."},{"Start":"07:06.040 ","End":"07:12.240","Text":"Then 2 times what\u0027s left is this which is 4,"},{"Start":"07:12.240 ","End":"07:17.805","Text":"minus 7, minus 9, 18."},{"Start":"07:17.805 ","End":"07:26.020","Text":"Then plus 0 times whatever it doesn\u0027t matter because it\u0027s a 0."},{"Start":"07:27.350 ","End":"07:31.644","Text":"Maybe for the practice, I will put the numbers in, it\u0027s 4,"},{"Start":"07:31.644 ","End":"07:35.695","Text":"minus 7, minus 1,"},{"Start":"07:35.695 ","End":"07:39.805","Text":"4, even though we know it doesn\u0027t matter, but okay."},{"Start":"07:39.805 ","End":"07:46.550","Text":"This is equal to, down here I have minus 18,"},{"Start":"07:46.550 ","End":"07:55.455","Text":"minus minus 36, so that\u0027s minus 18 plus 36, that\u0027s 18."},{"Start":"07:55.455 ","End":"08:03.945","Text":"Now this determinant is lets see 4 times 18 is 72."},{"Start":"08:03.945 ","End":"08:08.025","Text":"This times this is plus 63,"},{"Start":"08:08.025 ","End":"08:13.755","Text":"so 72 minus 63 is 9."},{"Start":"08:13.755 ","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 would be 16 minus 7 is 9."},{"Start":"08:33.060 ","End":"08:38.980","Text":"Anyway this is 0, I\u0027ve got 18 minus 18, so it\u0027s 0."},{"Start":"08:38.980 ","End":"08:48.615","Text":"In this case we do have a 0 so these 3 vectors are in the same plane,"},{"Start":"08:48.615 ","End":"08:53.900","Text":"or maybe emphasize, are in the same plane."},{"Start":"08:53.900 ","End":"08:55.925","Text":"So part 1 not,"},{"Start":"08:55.925 ","End":"09:01.050","Text":"part 2 they are, and we\u0027re done."}],"ID":10295},{"Watched":false,"Name":"Exercise 4","Duration":"8m 12s","ChapterTopicVideoID":27705,"CourseChapterTopicPlaylistID":57473,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/27705.jpeg","UploadDate":"2022-05-03T09:19:51.5700000","DurationForVideoObject":"PT8M12S","Description":null,"VideoComments":[],"Subtitles":[],"ID":28999}],"Thumbnail":null,"ID":57473}]

[{"ID":57470,"Videos":[10282,10283]},{"ID":57471,"Videos":[10277,10278,10279,10280,10281]},{"ID":57472,"Videos":[10284,10285,10286,10287,10288,10289,10290,28998]},{"ID":57473,"Videos":[10291,10292,10293,10294,10295,28999]}];

[10277,10278,10279,10280,10281];

1.1

1

Get unlimited access to **1500 subjects** including **personalized courses**

Start your free trial
We couldn't find any results for

Upload your syllabus now and our team will create a customized course especially for you!

Alert

and we will create a personalized course (just for you) in less than **48 hours...**