1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high-quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:19,500 at ocw.mit.edu. 8 00:00:33,169 --> 00:00:34,470 PROFESSOR: Hi. 9 00:00:34,470 --> 00:00:39,450 Today, we finish up the last of the basic vector operations. 10 00:00:39,450 --> 00:00:41,270 Today, what we're going to discuss 11 00:00:41,270 --> 00:00:45,330 is something called the vector product or the cross product. 12 00:00:45,330 --> 00:00:49,450 And it's a new type of multiplication, whereby we now 13 00:00:49,450 --> 00:00:53,890 multiply two vectors to obtain a vector. 14 00:00:53,890 --> 00:00:56,150 You see, the dot product multiplied two vectors, 15 00:00:56,150 --> 00:00:57,820 but it gave a scalar. 16 00:00:57,820 --> 00:01:01,070 This is why the dot product was called a scalar product. 17 00:01:01,070 --> 00:01:04,510 Now, the idea is, again, we can go to the physical world 18 00:01:04,510 --> 00:01:06,270 to seek motivation. 19 00:01:06,270 --> 00:01:10,520 And my only problem with that is that some of the examples that 20 00:01:10,520 --> 00:01:13,430 motivated this are not quite as elementary 21 00:01:13,430 --> 00:01:18,860 as the work equals force times distance formula of last time. 22 00:01:18,860 --> 00:01:21,370 The cross product does come up in sort 23 00:01:21,370 --> 00:01:24,960 of trivial applications, in terms of moments, 24 00:01:24,960 --> 00:01:28,960 angular momentum, in terms of mechanics, 25 00:01:28,960 --> 00:01:33,350 and in more advanced situations, in terms of the Maxwell 26 00:01:33,350 --> 00:01:37,090 equations where, you may remember, in physics courses, 27 00:01:37,090 --> 00:01:40,200 going through this routine of the force 28 00:01:40,200 --> 00:01:43,280 and the flux and the intensity and what have you. 29 00:01:43,280 --> 00:01:46,960 And at any rate, rather than waste time-- and I mean, 30 00:01:46,960 --> 00:01:50,070 I use the word waste here judiciously, 31 00:01:50,070 --> 00:01:52,050 because, for the purpose of our course, 32 00:01:52,050 --> 00:01:56,320 it is not nearly as crucial to motivate why we invented 33 00:01:56,320 --> 00:01:58,620 the cross product physically as it 34 00:01:58,620 --> 00:02:03,900 is to define it mathematically and to use it structurally. 35 00:02:03,900 --> 00:02:05,980 You see, what I want to do is simply 36 00:02:05,980 --> 00:02:08,300 get to the crux of the situation, 37 00:02:08,300 --> 00:02:11,710 discuss the concept of a cross product, 38 00:02:11,710 --> 00:02:15,830 and simply point out that if A and B are vectors, 39 00:02:15,830 --> 00:02:21,060 by A cross B-- written this way, A cross B-- we mean a vector. 40 00:02:21,060 --> 00:02:22,870 Now, here's the kicker again. 41 00:02:22,870 --> 00:02:27,180 As soon as I say vector, that means to define it, 42 00:02:27,180 --> 00:02:31,820 I have to tell you three ingredients, namely magnitude, 43 00:02:31,820 --> 00:02:34,070 direction, and sense. 44 00:02:34,070 --> 00:02:34,740 Right? 45 00:02:34,740 --> 00:02:38,660 Now, for the magnitude of A cross B, 46 00:02:38,660 --> 00:02:43,290 it's simply the magnitude of A times the magnitude of B 47 00:02:43,290 --> 00:02:48,687 times the magnitude of the sine of the angle between A and B. 48 00:02:48,687 --> 00:02:50,270 And I'm going to come back and comment 49 00:02:50,270 --> 00:02:52,530 on that in a few moments, about the difference 50 00:02:52,530 --> 00:02:56,840 between the sine here versus the cosine of the previous lecture. 51 00:02:56,840 --> 00:02:59,560 Again, physically, there's a motivation for this. 52 00:02:59,560 --> 00:03:03,010 As far as we're concerned, all we have to do is say, 53 00:03:03,010 --> 00:03:05,380 this is what the magnitude is defined to be. 54 00:03:05,380 --> 00:03:06,130 This is the game. 55 00:03:06,130 --> 00:03:08,300 We define it this way. 56 00:03:08,300 --> 00:03:12,620 Next thing I have to do is tell you its direction. 57 00:03:12,620 --> 00:03:15,670 And the direction of A cross B is, by definition, 58 00:03:15,670 --> 00:03:21,110 perpendicular to both A and B. All right? 59 00:03:21,110 --> 00:03:23,080 Another way of saying that is in the event 60 00:03:23,080 --> 00:03:26,500 that A and B are not parallel, then the vectors A and B 61 00:03:26,500 --> 00:03:28,150 determine a plane. 62 00:03:28,150 --> 00:03:30,100 And what we're saying is that A cross B 63 00:03:30,100 --> 00:03:33,740 is in a vector perpendicular to the plane determined 64 00:03:33,740 --> 00:03:36,580 by A and B. All right? 65 00:03:36,580 --> 00:03:38,880 By the way, if A and B are parallel, 66 00:03:38,880 --> 00:03:42,260 they don't determine a plane, because they coincide. 67 00:03:42,260 --> 00:03:44,180 They have only one direction. 68 00:03:44,180 --> 00:03:47,270 But if A and B are parallel, the angle between them 69 00:03:47,270 --> 00:03:51,845 is 0 or 180, depending on whether their sense is 70 00:03:51,845 --> 00:03:52,760 the same or not. 71 00:03:52,760 --> 00:03:57,100 Once the angle is either 0 or 180, the sine of that angle 72 00:03:57,100 --> 00:03:58,410 is 0. 73 00:03:58,410 --> 00:04:02,750 Once one of the factors here is 0, the whole product is 0. 74 00:04:02,750 --> 00:04:06,740 So in the event that A and B do not determine the plane, 75 00:04:06,740 --> 00:04:09,080 the magnitude of A cross B is 0, which 76 00:04:09,080 --> 00:04:11,300 means that A cross B is the zero vector. 77 00:04:11,300 --> 00:04:13,290 And that's a pretty trivial situation. 78 00:04:13,290 --> 00:04:15,610 So we just won't even worry about that. 79 00:04:15,610 --> 00:04:20,440 And thirdly, I must now give you the sense of A cross B. 80 00:04:20,440 --> 00:04:23,460 And the sense is the usual right-hand rule, 81 00:04:23,460 --> 00:04:26,730 in other words, the direction in which a right-handed thread 82 00:04:26,730 --> 00:04:31,240 screw would turn as A is rotated into B-- 83 00:04:31,240 --> 00:04:33,310 and now I have to be careful because 84 00:04:33,310 --> 00:04:38,410 of this sine situation-- through the smaller angle. 85 00:04:38,410 --> 00:04:39,880 You see, look it. 86 00:04:39,880 --> 00:04:42,360 Notice that this sense is going to depend 87 00:04:42,360 --> 00:04:45,060 on what-- if you rotate A into B, 88 00:04:45,060 --> 00:04:50,620 you get the opposite sense than if you rotated B into A. 89 00:04:50,620 --> 00:04:54,130 And not only that, but if you rotate A into B 90 00:04:54,130 --> 00:04:56,350 through the larger of the two angles, 91 00:04:56,350 --> 00:04:59,250 notice that rotating A into B through one angle-- 92 00:04:59,250 --> 00:05:02,460 say the smaller one-- gives you a different sense 93 00:05:02,460 --> 00:05:05,760 than rotating A into B through the larger of the two angles. 94 00:05:05,760 --> 00:05:08,310 So there's a certain ambiguity that comes up here. 95 00:05:08,310 --> 00:05:11,080 And the ambiguity occurs because of the fact 96 00:05:11,080 --> 00:05:13,980 that we're using the sine rather than the cosine. 97 00:05:13,980 --> 00:05:16,740 In other words, if you reverse the sense of an angle, 98 00:05:16,740 --> 00:05:20,130 if you change its sine, the algebraic sine, 99 00:05:20,130 --> 00:05:23,670 the trigonometric sine, s-i-n changes. 100 00:05:23,670 --> 00:05:24,480 OK? 101 00:05:24,480 --> 00:05:27,680 And similarly, the sine of an angle 102 00:05:27,680 --> 00:05:33,070 is the negative of the sine of 360 degrees minus that angle. 103 00:05:33,070 --> 00:05:36,030 But that, at any rate, is the definition 104 00:05:36,030 --> 00:05:37,410 of the cross product. 105 00:05:37,410 --> 00:05:40,590 Given two vectors, their cross product is what? 106 00:05:40,590 --> 00:05:44,720 A vector whose magnitude is the product of the two magnitudes 107 00:05:44,720 --> 00:05:48,020 times the magnitude of the sine of the angle between the two 108 00:05:48,020 --> 00:05:49,040 vectors. 109 00:05:49,040 --> 00:05:51,440 The cross product is perpendicular to each 110 00:05:51,440 --> 00:05:52,850 of the two vectors. 111 00:05:52,850 --> 00:05:56,100 And it's sense is the right-hand rule 112 00:05:56,100 --> 00:05:58,530 as the first is rotated into the second 113 00:05:58,530 --> 00:06:01,370 through the smaller of the two angles. 114 00:06:03,910 --> 00:06:06,010 This is important because, you see, 115 00:06:06,010 --> 00:06:09,380 what happens is that as important as the cross product 116 00:06:09,380 --> 00:06:13,220 is from a physical point of view, arithmetically, 117 00:06:13,220 --> 00:06:14,350 it's a nuisance. 118 00:06:14,350 --> 00:06:18,230 We can take very few liberties with the cross product. 119 00:06:18,230 --> 00:06:20,440 It turns out that almost every rule 120 00:06:20,440 --> 00:06:24,400 that we would like to be true about numerical multiplication 121 00:06:24,400 --> 00:06:27,860 is false for vector cross products. 122 00:06:27,860 --> 00:06:31,650 For example, among other things, A cross B 123 00:06:31,650 --> 00:06:34,250 is not equal to B cross A. 124 00:06:34,250 --> 00:06:38,500 In fact, A cross B is the negative of B cross A. 125 00:06:38,500 --> 00:06:40,710 And the reason for this is simply, 126 00:06:40,710 --> 00:06:43,790 as you check this thing through, A cross B and B 127 00:06:43,790 --> 00:06:47,920 cross A certainly have the same magnitude. 128 00:06:47,920 --> 00:06:50,430 They have the same direction, because each is 129 00:06:50,430 --> 00:06:52,840 perpendicular to both A and B. 130 00:06:52,840 --> 00:06:55,230 But notice that the sense is different. 131 00:06:55,230 --> 00:06:59,440 Namely, A cross B tells us to rotate A 132 00:06:59,440 --> 00:07:02,680 into B through the smaller of the two angles. 133 00:07:02,680 --> 00:07:04,470 And in our particular diagram that 134 00:07:04,470 --> 00:07:07,020 is what, the counterclockwise direction? 135 00:07:07,020 --> 00:07:11,260 On the other hand, B cross A says, rotate B into A 136 00:07:11,260 --> 00:07:13,350 through the smaller of the two angles. 137 00:07:13,350 --> 00:07:17,800 And if we do that, notice that we get the clockwise direction 138 00:07:17,800 --> 00:07:22,030 so that B cross A and A cross B have the same magnitude. 139 00:07:22,030 --> 00:07:23,480 They have the same direction. 140 00:07:23,480 --> 00:07:25,470 But they have the opposite sense. 141 00:07:25,470 --> 00:07:29,260 That is, A cross B is the negative of B cross A. 142 00:07:29,260 --> 00:07:31,710 And what that means, you see, in computations 143 00:07:31,710 --> 00:07:34,700 is that every time you change the order of A and B 144 00:07:34,700 --> 00:07:38,560 in a cross product, you must remember to change the sign. 145 00:07:38,560 --> 00:07:40,600 In fact, what's particularly embarrassing is 146 00:07:40,600 --> 00:07:42,340 if you forgot to do this. 147 00:07:42,340 --> 00:07:46,070 If this particular term is going to be added on to something, 148 00:07:46,070 --> 00:07:47,600 and you forget to change its sign, 149 00:07:47,600 --> 00:07:50,820 you're going to be off by more than just the error of a sign. 150 00:07:50,820 --> 00:07:54,650 In other words, if something was supposed to be plus 4, 151 00:07:54,650 --> 00:07:57,450 and you have it as minus 4 or vice versa, the error 152 00:07:57,450 --> 00:08:00,540 is going to be 8 if you add it rather than subtract it, 153 00:08:00,540 --> 00:08:01,450 you see. 154 00:08:01,450 --> 00:08:04,390 So the error gets hidden in other computations. 155 00:08:04,390 --> 00:08:07,190 In other words, you can change the order of the factors, 156 00:08:07,190 --> 00:08:09,830 but you must remember to change the sign. 157 00:08:09,830 --> 00:08:13,830 Another interesting thing is that unlike in the dot product, 158 00:08:13,830 --> 00:08:17,380 where A dot (B dot C) didn't even make sense, 159 00:08:17,380 --> 00:08:23,430 it is true that A cross (B cross C) and (A cross B) cross C 160 00:08:23,430 --> 00:08:25,067 do makes sense. 161 00:08:25,067 --> 00:08:26,650 And by the way, what are these things? 162 00:08:26,650 --> 00:08:27,700 They're vectors. 163 00:08:27,700 --> 00:08:32,830 B cross C is a vector, which is perpendicular to both B and C. 164 00:08:32,830 --> 00:08:34,990 It's perpend-- you see. 165 00:08:34,990 --> 00:08:36,500 And A is a vector. 166 00:08:36,500 --> 00:08:38,210 And the cross product of two vectors 167 00:08:38,210 --> 00:08:39,780 is, again, a vector, et cetera. 168 00:08:39,780 --> 00:08:41,059 These are two vectors. 169 00:08:41,059 --> 00:08:43,580 But in general, these two vectors 170 00:08:43,580 --> 00:08:45,230 do not have to be equal. 171 00:08:45,230 --> 00:08:47,280 In fact, as I shall show you in a moment, 172 00:08:47,280 --> 00:08:48,710 these two vectors don't even have 173 00:08:48,710 --> 00:08:51,470 to be in the same plane, let alone being equal. 174 00:08:51,470 --> 00:08:53,530 They don't even have to be in parallel planes. 175 00:08:53,530 --> 00:08:56,960 And again, without making too much of a fuss over this, 176 00:08:56,960 --> 00:08:59,060 let's take a look and see if we can see why. 177 00:08:59,060 --> 00:09:02,130 Let's look at A cross (B cross C). 178 00:09:02,130 --> 00:09:04,510 By definition-- let's just even look at the direction. 179 00:09:04,510 --> 00:09:06,093 Let's not even worry about magnitudes. 180 00:09:06,093 --> 00:09:09,360 What is the direction of A cross (B cross C)? 181 00:09:09,360 --> 00:09:10,830 It must be what? 182 00:09:10,830 --> 00:09:14,270 It must be, on the one hand, perpendicular 183 00:09:14,270 --> 00:09:20,300 to A and on the other hand perpendicular to B cross C. OK? 184 00:09:20,300 --> 00:09:24,410 That's the definition of being a cross product. 185 00:09:24,410 --> 00:09:29,290 On the other hand, what about B cross C as a vector itself? 186 00:09:29,290 --> 00:09:34,070 B cross C is perpendicular to both B and C. 187 00:09:34,070 --> 00:09:35,490 So what do you have here? 188 00:09:35,490 --> 00:09:38,190 You have that whatever A cross (B cross C) 189 00:09:38,190 --> 00:09:43,260 is, it's perpendicular to B cross C, which, in turn, is 190 00:09:43,260 --> 00:09:46,090 perpendicular to both B and C. 191 00:09:46,090 --> 00:09:47,810 That sounds like a tongue twister. 192 00:09:47,810 --> 00:09:50,050 But if you decipher it, all it says 193 00:09:50,050 --> 00:09:54,480 is that A cross (B cross C) is perpendicular 194 00:09:54,480 --> 00:09:59,450 to the perpendicular to B and C. And I call that deciphering it. 195 00:09:59,450 --> 00:10:02,859 What is the perpendicular to the perpendicular? 196 00:10:02,859 --> 00:10:04,400 You say let's just write it this way. 197 00:10:04,400 --> 00:10:06,130 Here are two vectors in the plane. 198 00:10:06,130 --> 00:10:07,470 Here's a perpendicular. 199 00:10:07,470 --> 00:10:09,760 I now take a perpendicular to that. 200 00:10:09,760 --> 00:10:13,230 And that's going to be parallel to the original plane. 201 00:10:13,230 --> 00:10:14,910 This is written up in the text. 202 00:10:14,910 --> 00:10:17,830 It's written up as one of our learning exercises. 203 00:10:17,830 --> 00:10:22,350 But just for a quick run-through now, all I want you to see 204 00:10:22,350 --> 00:10:26,340 is that A cross (B cross C) is parallel 205 00:10:26,340 --> 00:10:29,200 to the plane determined by B and C. 206 00:10:29,200 --> 00:10:31,720 In fact, the easy way to memorize this 207 00:10:31,720 --> 00:10:34,520 is look at the vector that's in parentheses. 208 00:10:34,520 --> 00:10:36,570 And the cross product is what? 209 00:10:36,570 --> 00:10:39,800 It's parallel to the plane determined by the two 210 00:10:39,800 --> 00:10:41,570 vectors in parentheses. 211 00:10:41,570 --> 00:10:43,480 The only time these two vectors, by the way, 212 00:10:43,480 --> 00:10:46,070 won't determine a plane is if they're parallel. 213 00:10:46,070 --> 00:10:47,630 But in that case, the cross product 214 00:10:47,630 --> 00:10:49,120 would have been zero anyway. 215 00:10:49,120 --> 00:10:50,800 In other words, to summarize what 216 00:10:50,800 --> 00:10:55,520 I've written on the blackboards here, A cross (B cross C) 217 00:10:55,520 --> 00:11:00,150 is a vector which lies in the plane determined by B and C. 218 00:11:00,150 --> 00:11:02,300 After all, if the vector is parallel to the plane, 219 00:11:02,300 --> 00:11:03,980 you can assume it lies in the plane, 220 00:11:03,980 --> 00:11:07,070 because you can shift it parallel to itself. 221 00:11:07,070 --> 00:11:10,910 And correspondingly, (A cross B) cross 222 00:11:10,910 --> 00:11:14,850 C is a vector in the plane determined by A 223 00:11:14,850 --> 00:11:17,810 and B. In particular, there's no reason 224 00:11:17,810 --> 00:11:20,520 why the AB plane and the BC plane 225 00:11:20,520 --> 00:11:22,050 have to be the same plane. 226 00:11:22,050 --> 00:11:24,690 No law says that our three vectors, A, B, and C, all 227 00:11:24,690 --> 00:11:26,300 lie on the same plane. 228 00:11:26,300 --> 00:11:30,520 So in particular, this is why the associative property 229 00:11:30,520 --> 00:11:32,060 doesn't hold for a cross product. 230 00:11:32,060 --> 00:11:34,610 Because as soon as you shift the parentheses, 231 00:11:34,610 --> 00:11:40,150 you actually shift the plane in which the vector exists. 232 00:11:40,150 --> 00:11:40,650 OK? 233 00:11:40,650 --> 00:11:42,858 But again, I just want to go through these highlights 234 00:11:42,858 --> 00:11:44,760 to make sure that you don't miss them. 235 00:11:44,760 --> 00:11:48,410 But these are done in more detail in the notes. 236 00:11:48,410 --> 00:11:51,610 Before I start blasting the cross product too much, 237 00:11:51,610 --> 00:11:54,490 let me point out that certain properties that we 238 00:11:54,490 --> 00:11:58,100 like in numerical arithmetic are present in the cross product. 239 00:12:01,250 --> 00:12:03,680 For example, there is a-- I say this, 240 00:12:03,680 --> 00:12:06,750 there's a little structure. 241 00:12:06,750 --> 00:12:09,810 The distributive property holds for the cross product. 242 00:12:09,810 --> 00:12:12,440 In other words, if the vector A is 243 00:12:12,440 --> 00:12:15,430 crossed with the sum of the two vectors B and C-- 244 00:12:15,430 --> 00:12:19,310 and by the way, again notice, if B and C are vectors, 245 00:12:19,310 --> 00:12:20,345 B plus C is a vector. 246 00:12:20,345 --> 00:12:23,090 I'll put a v down here to indicate vector. 247 00:12:23,090 --> 00:12:24,820 A is a vector. 248 00:12:24,820 --> 00:12:27,680 And the cross product of two vectors is a vector. 249 00:12:27,680 --> 00:12:30,089 Notice that, again, with the cross product, 250 00:12:30,089 --> 00:12:31,630 all of the things that you're working 251 00:12:31,630 --> 00:12:33,240 with happen to be vectors. 252 00:12:33,240 --> 00:12:37,570 But my claim is that A crossed with (B plus C) is 253 00:12:37,570 --> 00:12:40,930 equal to A crossed with B plus the product 254 00:12:40,930 --> 00:12:46,540 of A crossed with C. That's a very nice structural property. 255 00:12:46,540 --> 00:12:50,140 Again, as usual, Cartesian coordinates 256 00:12:50,140 --> 00:12:51,670 have a lot to offer us. 257 00:12:51,670 --> 00:12:55,370 If we switch to Cartesian coordinates, again, as usual, 258 00:12:55,370 --> 00:12:59,380 writing A as a_1*i plus a_2*j plus a_3*k, 259 00:12:59,380 --> 00:13:03,510 and B as b_1*i plus b_2*j plus b_3*k, 260 00:13:03,510 --> 00:13:08,350 and multiplying term by term, we observe a rather interesting 261 00:13:08,350 --> 00:13:09,750 situation. 262 00:13:09,750 --> 00:13:12,030 You see, notice things like this. 263 00:13:12,030 --> 00:13:16,650 When you cross i with i, j with j, and k with k, 264 00:13:16,650 --> 00:13:18,980 you're always going to get the zero vector. 265 00:13:18,980 --> 00:13:20,650 And the reason for that is, we already 266 00:13:20,650 --> 00:13:24,320 saw that since the sine of the angle between two 267 00:13:24,320 --> 00:13:28,020 parallel vectors is 0, the magnitude of the cross product 268 00:13:28,020 --> 00:13:31,372 is 0, that means that the cross product is the zero vector. 269 00:13:31,372 --> 00:13:32,830 The question comes up, what happens 270 00:13:32,830 --> 00:13:36,400 when you cross i with j or j with i, i with k, 271 00:13:36,400 --> 00:13:38,820 k with j, et cetera? 272 00:13:38,820 --> 00:13:40,320 I think it's easy to see that when 273 00:13:40,320 --> 00:13:42,650 you cross i and j and j and i, you're going 274 00:13:42,650 --> 00:13:44,880 to get either k or minus k. 275 00:13:44,880 --> 00:13:48,870 The right-handed rule is set up very, very conveniently, 276 00:13:48,870 --> 00:13:51,550 again, to help us with a memory device. 277 00:13:51,550 --> 00:13:55,830 The easiest way to remember how these things go is i, j, and k 278 00:13:55,830 --> 00:13:57,930 were set up to form a right-handed rule, 279 00:13:57,930 --> 00:13:59,250 a right-handed system. 280 00:13:59,250 --> 00:14:01,550 Write i, j, and k sequentially. 281 00:14:01,550 --> 00:14:04,320 See, i, j, k, i, j, k, et cetera. 282 00:14:04,320 --> 00:14:08,290 Call the normal order the positive direction 283 00:14:08,290 --> 00:14:10,830 and the opposite order the negative direction. 284 00:14:10,830 --> 00:14:14,020 For example, when you want i cross j, just read 285 00:14:14,020 --> 00:14:17,990 this list as you go along: i followed by j 286 00:14:17,990 --> 00:14:20,530 gives k, which is in the right order-- 287 00:14:20,530 --> 00:14:24,370 see, i, j, k-- so i cross j is k. 288 00:14:24,370 --> 00:14:28,040 On the other hand, j cross i means what? 289 00:14:28,040 --> 00:14:30,290 j followed by i. 290 00:14:30,290 --> 00:14:32,330 j followed by i is this way. 291 00:14:32,330 --> 00:14:33,980 That also yields k. 292 00:14:33,980 --> 00:14:36,840 But you're now going in the negative direction. 293 00:14:36,840 --> 00:14:39,770 To summarize these results, i cross j 294 00:14:39,770 --> 00:14:44,386 is the same as minus j cross i, which is k. 295 00:14:44,386 --> 00:14:49,160 k cross i is the negative of i cross k, which is j. 296 00:14:49,160 --> 00:14:53,610 i cross i, j cross j, and k cross k are all 0. 297 00:14:53,610 --> 00:14:55,290 And notice the use here of zero vector, 298 00:14:55,290 --> 00:14:57,260 because we're dealing with vectors. 299 00:14:57,260 --> 00:14:59,600 And again, leaving the details to you. 300 00:14:59,600 --> 00:15:01,950 They're very simple to come up with. 301 00:15:01,950 --> 00:15:03,930 Just carry out the multiplication. 302 00:15:03,930 --> 00:15:06,790 And what you find is that A cross B 303 00:15:06,790 --> 00:15:09,280 is given by the following. 304 00:15:09,280 --> 00:15:15,880 The i component of the vector is a_2*b_3 minus a_3*b_2. 305 00:15:15,880 --> 00:15:21,330 The j component is a_3*b_1 minus a_1*b_3. 306 00:15:21,330 --> 00:15:27,290 And the k component is a_1*b_2 minus a_2*b_1. 307 00:15:27,290 --> 00:15:30,380 And if you want a convenient way of memorizing this, 308 00:15:30,380 --> 00:15:35,820 again, write 1, 2, 3 in their correct sequential order. 309 00:15:35,820 --> 00:15:40,080 And what you're saying is, each term in A cross B 310 00:15:40,080 --> 00:15:41,700 involves what? 311 00:15:41,700 --> 00:15:44,620 An a factor followed by a b factor. 312 00:15:44,620 --> 00:15:48,530 The subscripts involve the component, which isn't present. 313 00:15:48,530 --> 00:15:52,480 See, for the first component, i, the subscripts are 2 and 3. 314 00:15:52,480 --> 00:15:56,770 For the second component, j, the second component, 315 00:15:56,770 --> 00:15:58,930 the subscripts are 3 and 1. 316 00:15:58,930 --> 00:16:02,950 And for the third component, k, the subscripts are 1 and 2. 317 00:16:02,950 --> 00:16:05,930 And notice that the way the signs are determined 318 00:16:05,930 --> 00:16:09,000 is that if the subscripts occur in the correct sequential 319 00:16:09,000 --> 00:16:11,980 order, it's positive, otherwise negative. 320 00:16:11,980 --> 00:16:15,840 For example, 2 followed by 3 is the correct order. 321 00:16:15,840 --> 00:16:19,970 But 3 followed by 2 is the negative order. 322 00:16:19,970 --> 00:16:26,020 Notice, in this sequence, 1 follows 3, whereas 1 to 3 323 00:16:26,020 --> 00:16:27,420 is in the negative order. 324 00:16:27,420 --> 00:16:28,250 See? 325 00:16:28,250 --> 00:16:31,990 Just the same, this works out very, very nicely. 326 00:16:31,990 --> 00:16:33,530 And the reason I went through this-- 327 00:16:33,530 --> 00:16:35,071 it may seem like an awful lot of work 328 00:16:35,071 --> 00:16:38,990 to go through that to show you how logical this thing is-- 329 00:16:38,990 --> 00:16:43,030 is that something comes up in the textbook, which is not bad, 330 00:16:43,030 --> 00:16:47,610 but which I find as frightening if you have not had experience 331 00:16:47,610 --> 00:16:50,070 with something called determinants. 332 00:16:50,070 --> 00:16:52,830 You see, the author, Professor Thomas, 333 00:16:52,830 --> 00:16:55,870 elects to introduce a convenient device for memorizing 334 00:16:55,870 --> 00:17:00,610 this result. And he writes it as a 3 by 3 determinant. 335 00:17:00,610 --> 00:17:02,980 Now, let me just write down the 3 by 3 determinant, 336 00:17:02,980 --> 00:17:04,940 and then we'll talk about it later. 337 00:17:04,940 --> 00:17:08,500 I put convenient in quotation marks to emphasize to you 338 00:17:08,500 --> 00:17:11,950 that this is convenient only if you already know determinants. 339 00:17:11,950 --> 00:17:13,760 If you don't know determinants, this 340 00:17:13,760 --> 00:17:16,640 is not only not convenient, but it's frightening. 341 00:17:16,640 --> 00:17:18,200 We're always frightened of things 342 00:17:18,200 --> 00:17:19,770 that we don't understand too well. 343 00:17:19,770 --> 00:17:23,190 And I call it a memory device, because you do not 344 00:17:23,190 --> 00:17:25,490 have to have determinants to remember 345 00:17:25,490 --> 00:17:29,640 this particular result. In fact, if one 346 00:17:29,640 --> 00:17:33,790 were going to list, in order, the reasons for inventing 347 00:17:33,790 --> 00:17:36,830 determinants, and the man was reading off 348 00:17:36,830 --> 00:17:40,770 of very, very long list, and you were getting bored, 349 00:17:40,770 --> 00:17:44,160 as soon as he came to the reason that it was convenient to list 350 00:17:44,160 --> 00:17:47,040 the cross product, you could heave a sigh of relief, 351 00:17:47,040 --> 00:17:50,530 because you would know that he's near the end of the list. 352 00:17:50,530 --> 00:17:52,050 In other words, to make a long story 353 00:17:52,050 --> 00:17:56,110 short, of all the reasons for inventing determinants, 354 00:17:56,110 --> 00:17:59,450 using them for a cross product representation 355 00:17:59,450 --> 00:18:01,375 is perhaps the least significant. 356 00:18:01,375 --> 00:18:04,510 But you see, in Professor Thomas's approach, 357 00:18:04,510 --> 00:18:06,630 determinants are assumed to be known. 358 00:18:06,630 --> 00:18:09,800 And if you do know them, it's a good notation. 359 00:18:09,800 --> 00:18:11,670 If you don't, forget it. 360 00:18:11,670 --> 00:18:13,620 The trouble is, I tell you to forget it, 361 00:18:13,620 --> 00:18:15,119 and you won't forget it. 362 00:18:15,119 --> 00:18:16,160 It always works that way. 363 00:18:16,160 --> 00:18:18,743 When you tell the student not to forget something, he forgets. 364 00:18:18,743 --> 00:18:20,880 When you say, please ignore it and forget it, 365 00:18:20,880 --> 00:18:22,130 he doesn't forget. 366 00:18:22,130 --> 00:18:25,090 So since you may be a little bit rusty on determinants, 367 00:18:25,090 --> 00:18:28,840 let me give you a very, very brief review of determinants. 368 00:18:28,840 --> 00:18:32,660 In fact, it's not only very brief, it's a pseudo-review, 369 00:18:32,660 --> 00:18:35,060 because I will give you no logic behind this at all. 370 00:18:35,060 --> 00:18:37,310 I'm just going to tell you the recipe. 371 00:18:37,310 --> 00:18:40,060 First of all, by a 2 by 2 determinant, 372 00:18:40,060 --> 00:18:43,430 you mean a square array of numbers consisting 373 00:18:43,430 --> 00:18:45,470 of two rows and two columns. 374 00:18:45,470 --> 00:18:49,800 You use absolute value signs to indicate the determinant. 375 00:18:49,800 --> 00:18:53,183 And by definition, the 2 by 2 determinant a, b; 376 00:18:53,183 --> 00:19:00,720 c, d is the product of the two terms a and d, 377 00:19:00,720 --> 00:19:03,830 minus the product of the two terms b and c. 378 00:19:03,830 --> 00:19:07,530 In other words, if you multiply the upper left by the lower 379 00:19:07,530 --> 00:19:12,060 right and subtract from that the upper right times 380 00:19:12,060 --> 00:19:15,560 the lower left, that is, by definition, the value of the 2 381 00:19:15,560 --> 00:19:19,010 by 2 determinant, just by definition. 382 00:19:19,010 --> 00:19:20,720 By the way, in block seven of our course, 383 00:19:20,720 --> 00:19:22,780 we're going to do this in much more detail. 384 00:19:22,780 --> 00:19:25,380 For now I just want to give you enough of a hint, 385 00:19:25,380 --> 00:19:27,590 so that if you are frightened by determinants, 386 00:19:27,590 --> 00:19:30,250 you can see what Professor Thomas is talking about 387 00:19:30,250 --> 00:19:31,510 in the text here. 388 00:19:31,510 --> 00:19:34,080 Now, to expand a 3 by 3 determinant, 389 00:19:34,080 --> 00:19:37,120 one uses a little sign code. 390 00:19:37,120 --> 00:19:39,480 Starting at the upper left-hand corner, 391 00:19:39,480 --> 00:19:41,060 you put a little plus sign. 392 00:19:41,060 --> 00:19:45,430 Then you alternate plus, minus, plus, et cetera. 393 00:19:45,430 --> 00:19:48,220 The plus simply is a code to tell you, leave 394 00:19:48,220 --> 00:19:50,030 the sign of the term alone. 395 00:19:50,030 --> 00:19:53,210 And the minus tells you, change the sign of the term. 396 00:19:53,210 --> 00:19:54,910 What that means is this. 397 00:19:54,910 --> 00:19:58,180 By definition, this 3 by 3 determinant 398 00:19:58,180 --> 00:20:02,350 is obtained as follows-- take x_1 as it appears 399 00:20:02,350 --> 00:20:06,370 and strike out the row and column in which x_1 appears 400 00:20:06,370 --> 00:20:11,080 and multiply x_1 by the remaining 2 by 2 determinant. 401 00:20:11,080 --> 00:20:15,460 Then what you do is you factor out x_2 with a minus sign, 402 00:20:15,460 --> 00:20:18,690 strike out the row and column in which x_2 appears, 403 00:20:18,690 --> 00:20:22,000 and multiply x_2 by the remaining 2 404 00:20:22,000 --> 00:20:24,090 by 2 determinant that's left. 405 00:20:24,090 --> 00:20:30,000 Similarly, take out x_3 with the sign that it appears with, 406 00:20:30,000 --> 00:20:32,560 strike out the row and column in which it appears, 407 00:20:32,560 --> 00:20:35,840 and multiply by the remaining 2 by 2 determinant. 408 00:20:35,840 --> 00:20:39,790 Summarized, to expand this 3 by 3 determinant, 409 00:20:39,790 --> 00:20:46,110 it's x_1 times the determinant y_2, y3; z_2, z_3 minus x_2 410 00:20:46,110 --> 00:20:50,460 times the determinant y_1, y_3; z_1, z_3, 411 00:20:50,460 --> 00:20:55,920 plus x_3 times the determinant y_1, y_2; z_1, z_2. 412 00:20:55,920 --> 00:20:58,320 Now, that happens to have very practical application. 413 00:20:58,320 --> 00:21:01,160 As I say, for this particular situation, 414 00:21:01,160 --> 00:21:04,240 I just want to do it with you to have you see how it works. 415 00:21:04,240 --> 00:21:07,390 Let me come back to what we were doing before, 416 00:21:07,390 --> 00:21:10,190 and show you how, if we use this definition, 417 00:21:10,190 --> 00:21:13,410 we can mechanically write down A cross B in a hurry. 418 00:21:13,410 --> 00:21:14,760 What did the rule say? 419 00:21:14,760 --> 00:21:17,550 It says, to expand the 3 by 3 determinant, 420 00:21:17,550 --> 00:21:22,680 take out i, as it appears, and multiply by the 2 421 00:21:22,680 --> 00:21:25,760 by 2 determinant that's left, when you strike out the row 422 00:21:25,760 --> 00:21:27,440 and column that i appears in. 423 00:21:30,230 --> 00:21:31,630 OK? 424 00:21:31,630 --> 00:21:32,890 Then it says what? 425 00:21:32,890 --> 00:21:38,590 Take out j with a minus sign, strike out the row and column, 426 00:21:38,590 --> 00:21:42,420 multiply by the remaining 2 by 2 determinant. 427 00:21:45,240 --> 00:21:51,010 Then it says, factor out k and strike out the row and column 428 00:21:51,010 --> 00:21:52,350 that k appears in. 429 00:21:52,350 --> 00:21:57,570 Multiply by the remaining 2 by 2 determinant. 430 00:21:57,570 --> 00:22:00,270 Well lookit, by our rule of multiplying a 2 431 00:22:00,270 --> 00:22:03,030 by 2 determinant, what is this thing here? 432 00:22:03,030 --> 00:22:08,500 This is nothing more than a_2*b_3 minus a_3*b_2. 433 00:22:11,499 --> 00:22:13,790 I'm not going to go through this, because time is short 434 00:22:13,790 --> 00:22:15,920 and you can carry this operation out by yourself. 435 00:22:15,920 --> 00:22:18,200 What I would like you to see, however, 436 00:22:18,200 --> 00:22:22,810 is that this term here is precisely the same 437 00:22:22,810 --> 00:22:25,280 as the term we have here. 438 00:22:25,280 --> 00:22:28,890 In other words, if I carry out this mechanical recipe, 439 00:22:28,890 --> 00:22:31,585 I am going to be able to get the same answer that I 440 00:22:31,585 --> 00:22:35,070 got logically, only without having to worry about keeping 441 00:22:35,070 --> 00:22:36,660 track of things in my head. 442 00:22:36,660 --> 00:22:39,170 In other words, if I do know determinants, 443 00:22:39,170 --> 00:22:41,930 it is very convenient to write down 444 00:22:41,930 --> 00:22:46,050 the cross product of two vectors using Cartesian coordinates. 445 00:22:46,050 --> 00:22:48,985 And just to illustrate that, let me give you a few examples. 446 00:22:51,500 --> 00:22:53,720 I call this example one. 447 00:22:53,720 --> 00:22:56,520 Find the vector perpendicular to the plane, 448 00:22:56,520 --> 00:23:00,970 determined by the three points, (1, 2, 3); (5, 9, 4); 449 00:23:00,970 --> 00:23:03,100 and (7, 6, 8). 450 00:23:03,100 --> 00:23:07,430 Now, try to visualize this problem 451 00:23:07,430 --> 00:23:09,180 if you didn't know vectors. 452 00:23:09,180 --> 00:23:12,480 Try to visualize drawing this thing to scale, 453 00:23:12,480 --> 00:23:14,910 trying to imagine what slope the plane has 454 00:23:14,910 --> 00:23:17,445 and how you find a vector, or a line 455 00:23:17,445 --> 00:23:18,570 perpendicular to the plane. 456 00:23:18,570 --> 00:23:20,986 And by the way, I don't even need the word vector in here. 457 00:23:20,986 --> 00:23:23,600 Later I could say, find a direction. 458 00:23:23,600 --> 00:23:26,870 But we'll worry about that more next time. 459 00:23:26,870 --> 00:23:29,440 The thing that I want to emphasize here 460 00:23:29,440 --> 00:23:32,130 is, I just very quickly draw myself 461 00:23:32,130 --> 00:23:33,730 a little convenient device. 462 00:23:33,730 --> 00:23:35,200 I don't draw it to scale. 463 00:23:35,200 --> 00:23:37,410 I assume that the points A, B, and C are not 464 00:23:37,410 --> 00:23:38,760 in the same straight line. 465 00:23:38,760 --> 00:23:41,290 And if they are, it'll become very interesting 466 00:23:41,290 --> 00:23:44,690 that I simply won't get-- I'll get 0 for a cross product 467 00:23:44,690 --> 00:23:48,940 to tell me that they don't form two non-parallel vectors. 468 00:23:48,940 --> 00:23:52,670 But I mark off A, B, and C. And the first thing I do, 469 00:23:52,670 --> 00:23:58,160 as always, is I vectorize-- I draw the lines AB and AC, 470 00:23:58,160 --> 00:23:59,600 and I vectorize them. 471 00:23:59,600 --> 00:24:04,050 That forms for me, as usual, the vector AB and the vector AC. 472 00:24:04,050 --> 00:24:07,000 Remembering how I form a vector from two 473 00:24:07,000 --> 00:24:09,370 points in Cartesian coordinates, again, 474 00:24:09,370 --> 00:24:12,540 subtracting the coordinates of the point 475 00:24:12,540 --> 00:24:14,630 that I'm going-- from the point that I'm going to, 476 00:24:14,630 --> 00:24:17,300 I subtract away the coordinates of a point that I'm leaving. 477 00:24:17,300 --> 00:24:22,190 In other words, 5 minus 1, 9 minus 2, 4 minus 3. 478 00:24:22,190 --> 00:24:27,510 The vector AB, very simply, is 4i plus 7j plus k. 479 00:24:27,510 --> 00:24:31,650 And similarly, the vector AC, very conveniently, 480 00:24:31,650 --> 00:24:34,700 is 6i plus 4j plus 5k. 481 00:24:34,700 --> 00:24:39,132 You see, 7 minus 1, 6 minus 2, 8 minus 3. 482 00:24:39,132 --> 00:24:40,840 Now, I better stop checking these things. 483 00:24:40,840 --> 00:24:43,089 Because, with my luck, I've probably subtracted wrong, 484 00:24:43,089 --> 00:24:45,680 and you might not notice it if we just go through it rapidly. 485 00:24:45,680 --> 00:24:49,220 But look at how quickly I've got AB and AC now. 486 00:24:49,220 --> 00:24:50,770 Now, here's the beauty. 487 00:24:50,770 --> 00:24:52,150 What do I want? 488 00:24:52,150 --> 00:24:54,680 I want a vector perpendicular to the plane 489 00:24:54,680 --> 00:24:56,810 determined by A, B, and C. 490 00:24:56,810 --> 00:25:01,040 In particular, if I can find a vector perpendicular to both AB 491 00:25:01,040 --> 00:25:04,860 and AC, namely if a vector is perpendicular to two 492 00:25:04,860 --> 00:25:07,540 lines in a plane, it's perpendicular to the plane 493 00:25:07,540 --> 00:25:08,250 itself. 494 00:25:08,250 --> 00:25:11,280 If I can find a vector perpendicular to both AB 495 00:25:11,280 --> 00:25:13,240 and AC, I'm home free. 496 00:25:13,240 --> 00:25:14,960 That's the vector I'm looking for, 497 00:25:14,960 --> 00:25:17,150 or at least a vector that I'm looking for. 498 00:25:17,150 --> 00:25:20,090 Do I know, conveniently, a vector perpendicular 499 00:25:20,090 --> 00:25:21,560 to two given vectors? 500 00:25:21,560 --> 00:25:24,300 And the answer had darn well better hinge somehow 501 00:25:24,300 --> 00:25:27,770 on the title of today's lecture, being the cross product. 502 00:25:27,770 --> 00:25:31,760 Yes, by definition, if I cross AB and AC, whatever 503 00:25:31,760 --> 00:25:35,720 vector I get is going to be perpendicular to the plane 504 00:25:35,720 --> 00:25:37,470 determined by A, B, and C. 505 00:25:37,470 --> 00:25:39,800 And now what I'm saying is that the beauty 506 00:25:39,800 --> 00:25:43,710 of Cartesian coordinates is that I can now very quickly write 507 00:25:43,710 --> 00:25:46,940 down what AB cross AC is as a determinant. 508 00:25:46,940 --> 00:25:49,410 Namely, I write down i, j, k. 509 00:25:49,410 --> 00:25:51,870 Then I write down the components of AB. 510 00:25:51,870 --> 00:25:54,277 Then I write down the components of AC. 511 00:25:54,277 --> 00:25:57,370 See, 4, 7, 1; 6, 4, 5. 512 00:25:57,370 --> 00:25:58,860 And now I do what? 513 00:25:58,860 --> 00:26:03,090 I strike out the row and column in which i appears. 514 00:26:03,090 --> 00:26:05,770 And the remaining 2 by 2 determinant is what? 515 00:26:05,770 --> 00:26:11,640 7 times 5 minus 1 times 4-- in other words, 35 minus 4. 516 00:26:11,640 --> 00:26:14,950 I take out minus j, strike out the row and column 517 00:26:14,950 --> 00:26:16,440 in which j appears. 518 00:26:16,440 --> 00:26:19,570 The remaining 2 by 2 determinant is 4 times 519 00:26:19,570 --> 00:26:24,330 5, minus 1 times 6-- in other words, 20 minus 6. 520 00:26:24,330 --> 00:26:27,470 I strike out the row and column in which k appears. 521 00:26:27,470 --> 00:26:34,680 The remaining 2 by 2 determinant is 4 times 4 minus 7 times 6. 522 00:26:34,680 --> 00:26:36,800 That's 16 minus 42. 523 00:26:36,800 --> 00:26:39,370 And now simplifying, what do I have? 524 00:26:39,370 --> 00:26:47,230 I have 31i minus 14j minus 26k. 525 00:26:47,230 --> 00:26:52,920 In other words, the vector 31i minus 14j minus 26k 526 00:26:52,920 --> 00:26:55,070 is the vector that I'm looking for. 527 00:26:55,070 --> 00:26:58,290 It's a vector perpendicular to the plane. 528 00:26:58,290 --> 00:27:02,490 And what I would like you to see is how convenient 529 00:27:02,490 --> 00:27:06,990 this vector arithmetic is to studying space geometry. 530 00:27:06,990 --> 00:27:08,390 And in the next lesson, I'm going 531 00:27:08,390 --> 00:27:10,760 to say some more about space geometry. 532 00:27:10,760 --> 00:27:13,850 But for now I thought maybe another nice example 533 00:27:13,850 --> 00:27:16,850 before we finish up, and let it go at that. 534 00:27:16,850 --> 00:27:20,520 Let me just do one more example for you. 535 00:27:20,520 --> 00:27:22,520 And to simplify the computation, I 536 00:27:22,520 --> 00:27:25,040 might as well use the vectors I've already used. 537 00:27:25,040 --> 00:27:26,700 Let's take the same three points we had 538 00:27:26,700 --> 00:27:31,540 before, (1, 2, 3); (5, 9, 4); (7, 6, 8). 539 00:27:31,540 --> 00:27:35,110 And we're going to ask, what is the area of the triangle 540 00:27:35,110 --> 00:27:37,520 determined by these three points? 541 00:27:37,520 --> 00:27:39,910 And the subtlety that I wanted to bring out here 542 00:27:39,910 --> 00:27:43,790 was an excuse to show you how the magnitude 543 00:27:43,790 --> 00:27:46,860 of the cross product of two vectors can be interpreted. 544 00:27:46,860 --> 00:27:50,230 Namely, look at two vectors. 545 00:27:50,230 --> 00:27:50,730 All right? 546 00:27:50,730 --> 00:27:52,500 Call them x and y. 547 00:27:52,500 --> 00:27:54,810 Draw them in to scale from a common point. 548 00:27:54,810 --> 00:27:58,090 And let theta be the angle formed between these two 549 00:27:58,090 --> 00:28:00,120 vectors. 550 00:28:00,120 --> 00:28:03,390 Consider the parallelogram formed by these two vectors. 551 00:28:03,390 --> 00:28:06,040 The area of a parallelogram is what? 552 00:28:06,040 --> 00:28:08,640 It's the base times the height. 553 00:28:08,640 --> 00:28:10,210 Now, lookit. 554 00:28:10,210 --> 00:28:15,840 The base of this parallelogram is the magnitude of y. 555 00:28:15,840 --> 00:28:17,640 And the height of this parallelogram-- 556 00:28:17,640 --> 00:28:19,720 remember, this magnitude is x. 557 00:28:19,720 --> 00:28:22,250 So this height is x sine theta. 558 00:28:27,720 --> 00:28:32,880 This is the area of the parallelogram. 559 00:28:32,880 --> 00:28:35,280 But notice that this is also, by definition, 560 00:28:35,280 --> 00:28:37,750 the magnitude of x cross y. 561 00:28:37,750 --> 00:28:43,390 In other words, notice that this is, by definition, 562 00:28:43,390 --> 00:28:45,360 the magnitude of x cross y. x cross 563 00:28:45,360 --> 00:28:47,870 y is a vector whose magnitude is this. 564 00:28:47,870 --> 00:28:50,640 In other words, the magnitude of x cross y 565 00:28:50,640 --> 00:28:55,370 is the area of the parallelogram determined by x and y, 566 00:28:55,370 --> 00:28:58,680 which gives you a rather nice, interesting interpretation. 567 00:28:58,680 --> 00:29:02,360 Imagine, for example, that x and y don't change in length. 568 00:29:02,360 --> 00:29:06,710 Is it clear to you that as I change the angle between x 569 00:29:06,710 --> 00:29:09,400 and y, the area of the parallelogram 570 00:29:09,400 --> 00:29:12,030 formed by x and y changes? 571 00:29:12,030 --> 00:29:13,860 In other words, one extreme case is 572 00:29:13,860 --> 00:29:16,990 when the angle between x and y is 0, in which case 573 00:29:16,990 --> 00:29:18,970 the parallelogram has no area. 574 00:29:18,970 --> 00:29:22,860 Another extreme case is when the angle between x and y 575 00:29:22,860 --> 00:29:23,920 is 90 degrees. 576 00:29:23,920 --> 00:29:26,410 In other words, x and y are perpendicular to each other. 577 00:29:26,410 --> 00:29:29,220 But if the lengths of x and y remain 578 00:29:29,220 --> 00:29:32,900 fixed, as I alter the angle between them, 579 00:29:32,900 --> 00:29:36,880 I alter the area of the parallelogram determined 580 00:29:36,880 --> 00:29:37,550 by them. 581 00:29:37,550 --> 00:29:38,400 All right? 582 00:29:38,400 --> 00:29:41,220 I just thought that's a cute little device whereby 583 00:29:41,220 --> 00:29:44,630 you can visualize how the magnitude of the cross product 584 00:29:44,630 --> 00:29:46,340 of two vectors varies. 585 00:29:46,340 --> 00:29:50,090 Think of the two vectors at a common origin, 586 00:29:50,090 --> 00:29:55,990 and the magnitude is affected by the area of that parallelogram 587 00:29:55,990 --> 00:29:57,561 as the angle changes. 588 00:29:57,561 --> 00:29:58,060 OK? 589 00:29:58,060 --> 00:29:59,460 In other words, if you change the angle, 590 00:29:59,460 --> 00:30:01,130 the area of the parallelogram changes. 591 00:30:01,130 --> 00:30:03,750 But at any rate, getting back to our original problem, 592 00:30:03,750 --> 00:30:06,040 the corollary to that is the following. 593 00:30:06,040 --> 00:30:08,510 We already know what the magnitude of AB 594 00:30:08,510 --> 00:30:11,620 cross AC is from example one. 595 00:30:11,620 --> 00:30:13,900 But what does that mean now, geometrically? 596 00:30:13,900 --> 00:30:17,440 From what we've just seen, the magnitude of AB 597 00:30:17,440 --> 00:30:22,060 cross AC is the area of the parallelogram, which 598 00:30:22,060 --> 00:30:25,980 has AB and AC as edges. 599 00:30:25,980 --> 00:30:31,975 Notice that that parallelogram is precisely one-half-- I 600 00:30:31,975 --> 00:30:34,320 shouldn't say that-- the triangle that we're looking for 601 00:30:34,320 --> 00:30:37,060 is precisely one-half the parallelogram. 602 00:30:37,060 --> 00:30:40,360 In other words, notice that the area of the triangle 603 00:30:40,360 --> 00:30:44,420 is simply 1/2 the magnitude of AB cross AC. 604 00:30:44,420 --> 00:30:47,020 How do we find the magnitude of AB cross C? 605 00:30:47,020 --> 00:30:50,020 We square each of its components, 606 00:30:50,020 --> 00:30:54,480 add them, and extract the positive square root, as usual. 607 00:30:54,480 --> 00:30:55,950 Without going through the details, 608 00:30:55,950 --> 00:30:57,740 it's simply going to be what? 609 00:30:57,740 --> 00:31:00,960 This particular amount. 610 00:31:00,960 --> 00:31:04,820 And the rest is arithmetic's baby, so to speak. 611 00:31:04,820 --> 00:31:05,590 All right? 612 00:31:05,590 --> 00:31:09,930 Now, what I'm hoping is that by this stage of the game, 613 00:31:09,930 --> 00:31:12,970 you now have a better feeling as to what 614 00:31:12,970 --> 00:31:15,400 the cross product means. 615 00:31:15,400 --> 00:31:17,150 Going back to the beginning of my lecture, 616 00:31:17,150 --> 00:31:20,040 notice that I played down the physical applications. 617 00:31:20,040 --> 00:31:22,540 Because one of the things that I wanted you to see 618 00:31:22,540 --> 00:31:26,030 is that the cross product has tremendous application, 619 00:31:26,030 --> 00:31:30,180 just to space geometry alone, quite apart from anything else. 620 00:31:30,180 --> 00:31:32,240 Well, at any rate, what I'm going to do next time 621 00:31:32,240 --> 00:31:34,970 is-- now that we've talked about lines and planes 622 00:31:34,970 --> 00:31:39,290 and what have you-- to get down to the Cartesian representation 623 00:31:39,290 --> 00:31:41,440 of what lines and planes would look like. 624 00:31:41,440 --> 00:31:44,440 They play a very vital role in the study of functions 625 00:31:44,440 --> 00:31:45,530 of several variables. 626 00:31:45,530 --> 00:31:49,130 We'll see more about that as the course continues to unfold. 627 00:31:49,130 --> 00:31:52,740 So until next time, goodbye. 628 00:31:52,740 --> 00:31:55,100 Funding for the publication of this video 629 00:31:55,100 --> 00:31:59,990 was provided by the Gabriella and Paul Rosenbaum Foundation. 630 00:31:59,990 --> 00:32:04,160 Help OCW continue to provide free and open access to MIT 631 00:32:04,160 --> 00:32:08,578 courses by making a donation at ocw.mit.edu/donate.