1 00:00:06,624 --> 00:00:08,790 DAVID JORDAN: Hello, and welcome back to recitation. 2 00:00:08,790 --> 00:00:13,740 In this problem, I'd like you to compute the area of a triangle. 3 00:00:13,740 --> 00:00:15,380 This triangle sits in space and it 4 00:00:15,380 --> 00:00:20,560 has its three vertices labeled here as P_1, P_2, and P_3. 5 00:00:20,560 --> 00:00:22,070 So we're going to compute this area, 6 00:00:22,070 --> 00:00:23,810 and we're going to do it using the cross product, which 7 00:00:23,810 --> 00:00:25,029 we learned about in lecture. 8 00:00:25,029 --> 00:00:27,070 So why don't you take some time to work this out, 9 00:00:27,070 --> 00:00:29,400 pause the video, and we'll check back in a minute 10 00:00:29,400 --> 00:00:30,310 and see how I did it. 11 00:00:39,373 --> 00:00:41,450 Hello and welcome back. 12 00:00:41,450 --> 00:00:43,950 So the first thing that I like to do with a problem 13 00:00:43,950 --> 00:00:47,140 like this is I like to draw a picture so I can kind of think 14 00:00:47,140 --> 00:00:48,530 about what's going on. 15 00:00:48,530 --> 00:00:59,720 So we have this triangle sitting out in space. 16 00:00:59,720 --> 00:01:02,890 And we know that we want to take a cross product in order 17 00:01:02,890 --> 00:01:07,866 to compute its area, but we need to be careful. 18 00:01:07,866 --> 00:01:09,365 Cross product, it doesn't make sense 19 00:01:09,365 --> 00:01:10,698 to take cross product of points. 20 00:01:10,698 --> 00:01:13,029 What makes sense is to take cross product of vectors. 21 00:01:13,029 --> 00:01:14,570 So the first thing that we need to do 22 00:01:14,570 --> 00:01:17,600 is build some vectors that describe this triangle. 23 00:01:17,600 --> 00:01:31,014 And the vectors that we need to build are P_1P_2 and P_1P_3. 24 00:01:31,014 --> 00:01:33,430 Since we're going to use these in a minute, let's go ahead 25 00:01:33,430 --> 00:01:34,960 and compute them now. 26 00:01:34,960 --> 00:01:44,700 So P_1P_2 is just the difference of P_2 minus P_1. 27 00:01:44,700 --> 00:01:48,390 So we get a 0 minus a negative 1. 28 00:01:48,390 --> 00:01:53,155 So we get 1, 2, and 1. 29 00:01:53,155 --> 00:01:55,530 Let me just check my notes to make sure I did that right. 30 00:01:55,530 --> 00:01:56,210 Good. 31 00:01:56,210 --> 00:02:04,030 And P_1P_3. 32 00:02:04,030 --> 00:02:06,500 We get again 0 minus a negative 1. 33 00:02:06,500 --> 00:02:11,540 So 1, we get minus 1, and then we get 1. 34 00:02:11,540 --> 00:02:13,120 Let me again check my notes. 35 00:02:13,120 --> 00:02:14,710 Very good. 36 00:02:14,710 --> 00:02:15,210 OK. 37 00:02:15,210 --> 00:02:16,800 So now that we have these vectors, 38 00:02:16,800 --> 00:02:20,180 we need to remember that if we take 39 00:02:20,180 --> 00:02:31,460 the absolute value of P_1P_2 cross product with P_1P_3, 40 00:02:31,460 --> 00:02:43,960 this will be equal to the area of the parallelogram they 41 00:02:43,960 --> 00:02:44,460 enclose. 42 00:02:49,730 --> 00:02:52,750 So let's get started by computing this cross product. 43 00:02:52,750 --> 00:02:57,690 So P_1P_2 cross P_2P_3. 44 00:03:01,940 --> 00:03:03,820 So, remember, to compute a cross product, 45 00:03:03,820 --> 00:03:08,260 we take the determinant of a matrix where we put in our unit 46 00:03:08,260 --> 00:03:11,290 normal vectors i, j, and k. 47 00:03:11,290 --> 00:03:14,740 And then we enter in, the remaining entries of the matrix 48 00:03:14,740 --> 00:03:16,240 are just the entries of our vectors. 49 00:03:16,240 --> 00:03:19,670 So we do [1, 2, 1]. 50 00:03:19,670 --> 00:03:23,750 And [1, -1, 1]. 51 00:03:23,750 --> 00:03:24,580 OK. 52 00:03:24,580 --> 00:03:27,535 And so we can compute this. 53 00:03:27,535 --> 00:03:34,910 And we get-- so the i component, we get 2 minus a negative 1. 54 00:03:34,910 --> 00:03:37,740 So we get 3. 55 00:03:37,740 --> 00:03:39,190 Now the j component. 56 00:03:39,190 --> 00:03:43,590 If we look at the cofactor matrix, it's just [1, 1; 1, 1], 57 00:03:43,590 --> 00:03:45,210 and that has determinant 0. 58 00:03:45,210 --> 00:03:48,310 So our middle component is just 0. 59 00:03:48,310 --> 00:03:50,180 And finally the k component. 60 00:03:50,180 --> 00:03:53,430 We get minus 1, minus another 2. 61 00:03:53,430 --> 00:03:58,180 So altogether, we get minus 3. 62 00:03:58,180 --> 00:04:05,010 So what that tells us now is that this quantity 63 00:04:05,010 --> 00:04:09,820 here, the magnitude of the cross product, 64 00:04:09,820 --> 00:04:15,190 is just 3 times the square root of 2, 65 00:04:15,190 --> 00:04:17,200 just looking at the length of this vector here. 66 00:04:19,900 --> 00:04:22,380 So we're almost done, but let's go back and look 67 00:04:22,380 --> 00:04:24,340 at what we had to start with. 68 00:04:24,340 --> 00:04:27,150 We were interested in the triangle over here 69 00:04:27,150 --> 00:04:31,980 which was enclosed by the vectors P_1P_2 70 00:04:31,980 --> 00:04:34,470 and the vectors P_1P_3. 71 00:04:34,470 --> 00:04:36,050 And what we just computed is actually 72 00:04:36,050 --> 00:04:43,210 the area of this parallelogram, which as you can see 73 00:04:43,210 --> 00:04:45,920 is twice the area of the triangle 74 00:04:45,920 --> 00:04:47,530 that we're actually interested in. 75 00:04:47,530 --> 00:04:55,540 So going back over here, we see that the area of our triangle 76 00:04:55,540 --> 00:04:59,480 is equal to 3 root 2, and we just 77 00:04:59,480 --> 00:05:02,410 need to divide by 2 to get the triangle. 78 00:05:02,410 --> 00:05:02,910 OK? 79 00:05:02,910 --> 00:05:04,406 And I'll leave it at that.