1 00:00:00,510 --> 00:00:07,530 ‫So all that was for the vector's in three dimensions, which is exactly what we need for discourse. 2 00:00:08,230 --> 00:00:12,560 ‫However, you can also compute the cross product mathematically. 3 00:00:13,110 --> 00:00:14,360 ‫Let's see how to do it. 4 00:00:14,940 --> 00:00:24,480 ‫Suppose that you have a vector V1 and vector V two, then you would compute the one vector, cross the 5 00:00:24,480 --> 00:00:28,250 ‫two vector, or you can write it in this form. 6 00:00:29,010 --> 00:00:31,910 ‫Then you would compute the cross product like this. 7 00:00:32,580 --> 00:00:41,730 ‫You create a three by three matrix where the first row will be this one and these are the unit vectors. 8 00:00:42,480 --> 00:00:46,090 ‫Then the second row will be the one vector. 9 00:00:46,590 --> 00:00:56,040 ‫But if here it's as a column vector here you write it as a row vector and the third row will be the 10 00:00:56,040 --> 00:00:57,240 ‫V two vector. 11 00:00:57,250 --> 00:01:07,110 ‫But now as our role vector and by the way, the unit vectors I j and K, their purpose is to give you 12 00:01:07,110 --> 00:01:08,100 ‫a direction. 13 00:01:08,310 --> 00:01:10,550 ‫They're magnitudes are all one. 14 00:01:11,130 --> 00:01:17,910 ‫So I has a magnitude of one, J has a magnitude of one and K has a magnitude of one. 15 00:01:18,270 --> 00:01:22,450 ‫But what they do they tell you in which direction you're going. 16 00:01:22,860 --> 00:01:30,110 ‫So for example, this V one vector, I can also write it down like this. 17 00:01:30,660 --> 00:01:37,170 ‫So this vector here that I have written in the form of column vector, I can write it like this v one 18 00:01:37,170 --> 00:01:47,880 ‫vector equals V one X I plus V one Y J and plus V one Z, K. 19 00:01:48,420 --> 00:01:55,020 ‫This formulation here is equivalent to this formulation here. 20 00:01:55,650 --> 00:02:02,250 ‫And so this I J and K, their unit vectors, they are all equal to one. 21 00:02:02,610 --> 00:02:09,990 ‫But you know that if you see I then you know that this component is in the X direction and thanks to 22 00:02:09,990 --> 00:02:13,020 ‫J you know that this component is in the Y direction. 23 00:02:13,170 --> 00:02:18,210 ‫And thanks to K you know that this component is in the Z direction. 24 00:02:18,480 --> 00:02:24,230 ‫And we're going to talk about the unit vectors more later in this section as well. 25 00:02:24,930 --> 00:02:28,350 ‫And so let's go back to computing our cross product. 26 00:02:28,800 --> 00:02:33,120 ‫We have this three by three matrix that we have constructed like this. 27 00:02:33,240 --> 00:02:40,480 ‫And now what we do, we take the determinant of it, which looks like this here. 28 00:02:40,620 --> 00:02:51,000 ‫I list the I J and K unit vectors horizontally and I assign the X components of the V one and the two 29 00:02:51,000 --> 00:03:01,740 ‫vectors to I like this then the Y components of these two vectors I assigned to J and the Z components 30 00:03:01,740 --> 00:03:08,040 ‫of them I assign to K and then I compute the determinant of the matrix. 31 00:03:08,730 --> 00:03:13,950 ‫The formula for the determinant comes from linear algebra and it looks like this. 32 00:03:14,580 --> 00:03:23,970 ‫You take the eye vector and you cross the corresponding row and column and the two by two matrix that 33 00:03:23,970 --> 00:03:34,440 ‫is remaining, you take its determinant and that will be your eye or X component in your total cross 34 00:03:34,440 --> 00:03:35,040 ‫product. 35 00:03:35,550 --> 00:03:36,810 ‫So it will be like that. 36 00:03:37,170 --> 00:03:45,900 ‫You take the eye vector and then to compute the determinant of a two by two matrix, you first multiply 37 00:03:46,110 --> 00:03:55,650 ‫the one Y times V to Z and then you subtract V, one Z times V to Y. 38 00:03:56,340 --> 00:04:04,980 ‫Let's just call it a four now and then you do the same thing for J and you cross out the corresponding 39 00:04:04,980 --> 00:04:14,280 ‫row and column your remaining two by two matrix will be this one composed of these elements here. 40 00:04:14,940 --> 00:04:18,240 ‫But in case of J there is an exception. 41 00:04:18,660 --> 00:04:22,520 ‫You have to put minus in front of J. 42 00:04:23,400 --> 00:04:31,620 ‫So the J component of your total cross product will be this one, this entire thing. 43 00:04:32,190 --> 00:04:41,480 ‫And let's call this part B, and now the only thing that's left is to find the key part of this total 44 00:04:41,490 --> 00:04:44,700 ‫cross product, and that will be your exercise. 45 00:04:45,120 --> 00:04:52,590 ‫Why don't you find the key part for this cross product and you'll see the solution in the next video?