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 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:48,760 --> 00:00:49,650 PROFESSOR: Hi. 9 00:00:49,650 --> 00:00:52,380 Our unit today concerns something 10 00:00:52,380 --> 00:00:54,490 called the dot product. 11 00:00:54,490 --> 00:00:56,980 Now it's very easy to just mechanically 12 00:00:56,980 --> 00:00:59,080 give a definition of the dot product, 13 00:00:59,080 --> 00:01:02,380 but in keeping with the spirit both of our game 14 00:01:02,380 --> 00:01:05,030 and of our correlation between the real 15 00:01:05,030 --> 00:01:08,820 and the abstract world, let's keep in mind that when it came 16 00:01:08,820 --> 00:01:11,980 time to define vector addition, we chose, 17 00:01:11,980 --> 00:01:15,220 as our definition, a concept already used 18 00:01:15,220 --> 00:01:19,780 in the physical world-- namely, that of a resultant vector. 19 00:01:19,780 --> 00:01:21,840 And now what we would like to do is 20 00:01:21,840 --> 00:01:25,680 to introduce a further structure in our game of vectors. 21 00:01:25,680 --> 00:01:28,100 And by the way, keep in mind that I do not 22 00:01:28,100 --> 00:01:30,760 need any more physical interpretations 23 00:01:30,760 --> 00:01:33,550 to add more complexities to my game, 24 00:01:33,550 --> 00:01:35,970 but that perhaps if we do this, it 25 00:01:35,970 --> 00:01:38,490 makes the subject more meaningful, 26 00:01:38,490 --> 00:01:41,950 both to the applied person and to the theoretical person. 27 00:01:41,950 --> 00:01:45,380 The idea that I wanted to keep in mind for motivating 28 00:01:45,380 --> 00:01:50,500 today's lesson was the old high school idea of work 29 00:01:50,500 --> 00:01:55,720 equals force times distance in the elementary physics course. 30 00:01:55,720 --> 00:01:58,980 Well rather than talk on like this, let's just take a look 31 00:01:58,980 --> 00:02:01,380 and see what's going on. 32 00:02:01,380 --> 00:02:04,260 Let's say the lecture is called "The Dot Product," 33 00:02:04,260 --> 00:02:08,410 and our physical motivation is the recipe 34 00:02:08,410 --> 00:02:12,870 from elementary physics, work equals force times distance. 35 00:02:12,870 --> 00:02:16,340 Now this is a deceptive little formula. 36 00:02:16,340 --> 00:02:18,120 First of all, what it really meant was, 37 00:02:18,120 --> 00:02:21,460 when we learned this recipe, was that the force 38 00:02:21,460 --> 00:02:24,650 was taking place in the same direction 39 00:02:24,650 --> 00:02:26,740 as the object was moving. 40 00:02:26,740 --> 00:02:31,150 Now what happened next was the following situation. 41 00:02:31,150 --> 00:02:35,590 We have an object, say, being moved by a force. 42 00:02:35,590 --> 00:02:39,370 The object is being moved along a tabletop, say. 43 00:02:39,370 --> 00:02:43,440 The force is represented now by a vector, an arrow. 44 00:02:43,440 --> 00:02:45,440 And we're assuming enough friction 45 00:02:45,440 --> 00:02:49,310 here so that the block moves along the table 46 00:02:49,310 --> 00:02:50,540 and doesn't rise. 47 00:02:50,540 --> 00:02:54,100 And the question is, how much work is done on the object 48 00:02:54,100 --> 00:02:57,660 as the object moves, say, from this point to this point? 49 00:02:57,660 --> 00:02:59,830 Now without worrying about what motivates 50 00:02:59,830 --> 00:03:02,620 this thing physically, the important thing was, 51 00:03:02,620 --> 00:03:05,020 is that people observe that physically, 52 00:03:05,020 --> 00:03:07,320 the only thing that went into the work 53 00:03:07,320 --> 00:03:10,820 was the component of the force in the direction 54 00:03:10,820 --> 00:03:12,000 of the displacement. 55 00:03:12,000 --> 00:03:18,580 In other words, this was the force 56 00:03:18,580 --> 00:03:21,890 that one found had to be multiplied by the displacement. 57 00:03:21,890 --> 00:03:25,070 In other words, to find the work being done, 58 00:03:25,070 --> 00:03:30,420 one took not the magnitude of F, but rather the component of F 59 00:03:30,420 --> 00:03:31,830 in the direction of s. 60 00:03:31,830 --> 00:03:34,550 And in this diagram, that's just what? 61 00:03:34,550 --> 00:03:39,430 It's the magnitude of F times cosine theta. 62 00:03:39,430 --> 00:03:42,420 And that quantity, which was called the effective force, 63 00:03:42,420 --> 00:03:46,510 was then multiplied by the displacement. 64 00:03:46,510 --> 00:03:49,340 And to write that in more suggestive form, 65 00:03:49,340 --> 00:03:52,100 notice that the work was not the magnitude 66 00:03:52,100 --> 00:03:54,980 of the force times the magnitude of the displacement. 67 00:03:54,980 --> 00:03:57,780 That was only true in the special case 68 00:03:57,780 --> 00:03:59,640 where the force and the displacement 69 00:03:59,640 --> 00:04:01,020 were in the same direction. 70 00:04:01,020 --> 00:04:03,470 But that rather the work is what? 71 00:04:03,470 --> 00:04:06,460 It's the magnitude of the force times the magnitude 72 00:04:06,460 --> 00:04:08,790 of the displacement times the cosine 73 00:04:08,790 --> 00:04:13,520 of the angle between the force and the displacement. 74 00:04:13,520 --> 00:04:16,440 And notice by the way, the very special cases, 75 00:04:16,440 --> 00:04:20,440 that if F and s happen to be parallel and have 76 00:04:20,440 --> 00:04:27,270 the same sense, then the angle between F and s is 0. 77 00:04:27,270 --> 00:04:31,670 The cosine of 0 is 1, in which case that you have the work 78 00:04:31,670 --> 00:04:35,170 is the magnitude of the force times the displacement. 79 00:04:35,170 --> 00:04:38,050 The other extreme case magnitude-wise 80 00:04:38,050 --> 00:04:42,750 is if s and F happen to be perpendicular, in which case 81 00:04:42,750 --> 00:04:44,760 the angle, of course, is 90 degrees. 82 00:04:44,760 --> 00:04:48,150 The cosine of a 90-degree angle is 0. 83 00:04:48,150 --> 00:04:49,230 In which case, what? 84 00:04:49,230 --> 00:04:52,360 If the force was at right angles to the displacement, 85 00:04:52,360 --> 00:04:54,340 the work was 0. 86 00:04:54,340 --> 00:04:56,550 At any rate then, whether we understand 87 00:04:56,550 --> 00:04:59,120 this physical motivation or not is irrelevant. 88 00:04:59,120 --> 00:05:00,930 The important point is, if we want 89 00:05:00,930 --> 00:05:05,300 to keep this structure or this particular motivation 90 00:05:05,300 --> 00:05:09,230 in a structural form, we now generalize this as follows. 91 00:05:09,230 --> 00:05:14,000 We simply say, let A and B be any two vectors, arrows. 92 00:05:14,000 --> 00:05:18,370 And we will define A dot B to be the magnitude of A times 93 00:05:18,370 --> 00:05:23,320 the magnitude of B times the cosine of the angle between A 94 00:05:23,320 --> 00:05:27,310 and B. And I write this this way to indicate an ordering. 95 00:05:27,310 --> 00:05:31,360 In other words, don't think of A and B as being A and B. 96 00:05:31,360 --> 00:05:33,860 Think of A as denoting the first and B 97 00:05:33,860 --> 00:05:35,620 as denoting the second vector. 98 00:05:35,620 --> 00:05:37,650 And what we're saying is the first dotted 99 00:05:37,650 --> 00:05:40,550 with the second is the magnitude of the first, 100 00:05:40,550 --> 00:05:42,550 times the magnitude of the second, 101 00:05:42,550 --> 00:05:46,640 times the cosine of the angle as you rotate the first vector 102 00:05:46,640 --> 00:05:48,090 into the second. 103 00:05:48,090 --> 00:05:50,710 And the beauty of having a cosine over here 104 00:05:50,710 --> 00:05:52,100 is the fact that what? 105 00:05:52,100 --> 00:05:55,910 If you reverse the angle of rotation-- in other words, 106 00:05:55,910 --> 00:06:00,950 from B into A-- notice that you change the sign of the angle, 107 00:06:00,950 --> 00:06:04,820 but the cosine of theta is the same as cosine minus theta, 108 00:06:04,820 --> 00:06:07,570 so no harm is done this particular way. 109 00:06:07,570 --> 00:06:10,990 On the other hand, had we been dealing with sine of the angle, 110 00:06:10,990 --> 00:06:14,550 as we will in our next lecture, this will make a difference. 111 00:06:14,550 --> 00:06:17,380 But be this as it may, we define A dot 112 00:06:17,380 --> 00:06:20,630 B to be the magnitude of A times the magnitude of B 113 00:06:20,630 --> 00:06:24,560 times the cosine of the angle between A and B. All right? 114 00:06:24,560 --> 00:06:26,300 The two extreme cases being what? 115 00:06:26,300 --> 00:06:30,750 If A and B are perpendicular, the dot product is 0. 116 00:06:30,750 --> 00:06:34,360 If A and B are parallel, the dot product 117 00:06:34,360 --> 00:06:40,190 is equal to the magnitude of the product of the two magnitudes. 118 00:06:40,190 --> 00:06:43,460 Now the only difficult thing here, or what we sometimes 119 00:06:43,460 --> 00:06:45,570 call undesirable-- and maybe this 120 00:06:45,570 --> 00:06:50,280 is what separates the new three-dimensional geometry 121 00:06:50,280 --> 00:06:52,480 from the old three-dimensional geometry-- 122 00:06:52,480 --> 00:06:55,342 is that the cosine of an angle is particularly difficult 123 00:06:55,342 --> 00:06:57,300 to keep track of, especially when the lines are 124 00:06:57,300 --> 00:06:58,750 in three-dimensional space. 125 00:06:58,750 --> 00:07:00,890 How do you measure an angle this way? 126 00:07:00,890 --> 00:07:02,620 You see, in the plane, it's simple. 127 00:07:02,620 --> 00:07:04,050 You draw the thing to scale. 128 00:07:04,050 --> 00:07:06,860 But in three-space, this can be rather difficult. 129 00:07:06,860 --> 00:07:09,650 So what we would like to do is to eliminate 130 00:07:09,650 --> 00:07:10,720 this particular term. 131 00:07:10,720 --> 00:07:13,890 We would like to find an expression for A dot B 132 00:07:13,890 --> 00:07:18,620 that doesn't involve the cosine of an angle, at least directly. 133 00:07:18,620 --> 00:07:21,670 And to do this, we simply draw a little diagram. 134 00:07:21,670 --> 00:07:25,420 And notice that even if A and B are three-dimensional vectors, 135 00:07:25,420 --> 00:07:29,010 since A and B are lines, if they are not parallel, 136 00:07:29,010 --> 00:07:31,190 if they emanate at a common point, 137 00:07:31,190 --> 00:07:32,560 they form a plane of their own. 138 00:07:32,560 --> 00:07:34,640 Let's call that the plane of the blackboard. 139 00:07:34,640 --> 00:07:37,190 The third side of the triangle is either 140 00:07:37,190 --> 00:07:39,730 A minus B or B minus A, depending on where 141 00:07:39,730 --> 00:07:40,900 you put the arrowhead here. 142 00:07:40,900 --> 00:07:43,204 But we've already discussed that idea. 143 00:07:43,204 --> 00:07:44,870 And the interesting point here is notice 144 00:07:44,870 --> 00:07:50,050 that A minus B very subtly includes the angle theta. 145 00:07:50,050 --> 00:07:52,930 In other words, imagine the magnitudes of A and B 146 00:07:52,930 --> 00:07:54,070 to be fixed. 147 00:07:54,070 --> 00:07:59,040 And now fan out A and B. As you fan out A and B, 148 00:07:59,040 --> 00:08:00,270 what you're doing is what? 149 00:08:00,270 --> 00:08:01,790 Just changing this angle. 150 00:08:01,790 --> 00:08:05,540 As you change this angle, notice that A minus B changes. 151 00:08:05,540 --> 00:08:08,170 Namely, as these fan out, the vector 152 00:08:08,170 --> 00:08:10,940 that joins the two arrowheads here 153 00:08:10,940 --> 00:08:14,550 becomes a different vector, both in magnitude and direction. 154 00:08:14,550 --> 00:08:16,460 In other words, whether it looks it or not, 155 00:08:16,460 --> 00:08:19,070 one of the beauties of our vector notation 156 00:08:19,070 --> 00:08:21,630 is that the cosine of the angle theta 157 00:08:21,630 --> 00:08:25,400 is indirectly included in A minus B. 158 00:08:25,400 --> 00:08:28,590 But now you see, once we have this triangle here, 159 00:08:28,590 --> 00:08:31,550 notice that the Law of Cosines tells us 160 00:08:31,550 --> 00:08:35,370 how to relate the third side of a triangle in terms of two 161 00:08:35,370 --> 00:08:37,580 sides and the included angle. 162 00:08:37,580 --> 00:08:38,179 OK? 163 00:08:38,179 --> 00:08:41,309 And in fact, when we use the Law of Cosines, 164 00:08:41,309 --> 00:08:44,250 notice that one of the terms is going to be what? 165 00:08:44,250 --> 00:08:47,480 The product of the magnitudes of two sides 166 00:08:47,480 --> 00:08:49,950 times the cosine of the included angle. 167 00:08:49,950 --> 00:08:55,340 And that, roughly speaking, is just what we mean by A dot B. 168 00:08:55,340 --> 00:08:57,590 So without any further ado, what we 169 00:08:57,590 --> 00:09:00,040 do now is we just write the Law of Cosines 170 00:09:00,040 --> 00:09:01,840 down here, which says what? 171 00:09:01,840 --> 00:09:04,730 The magnitude of A minus B squared 172 00:09:04,730 --> 00:09:08,610 is equal to the magnitude of A squared plus the magnitude of B 173 00:09:08,610 --> 00:09:11,870 squared minus twice the magnitude of A times 174 00:09:11,870 --> 00:09:14,990 the magnitude of B times the cosine of the angle between A 175 00:09:14,990 --> 00:09:18,410 and B. And then, you see, we simply 176 00:09:18,410 --> 00:09:21,900 recognize that this term here is, by definition, 177 00:09:21,900 --> 00:09:25,880 A dot B. We can now take this equation 178 00:09:25,880 --> 00:09:28,260 and solve for A dot B-- which, by the way, 179 00:09:28,260 --> 00:09:30,230 this is very, very important to notice. 180 00:09:30,230 --> 00:09:32,030 I should have pointed this out sooner. 181 00:09:32,030 --> 00:09:34,440 But A dot B is a number. 182 00:09:34,440 --> 00:09:37,810 It's a product of two magnitudes times the cosine 183 00:09:37,810 --> 00:09:39,520 of an angle, which is a number. 184 00:09:39,520 --> 00:09:41,790 This is a numerical equation. 185 00:09:41,790 --> 00:09:46,460 We can therefore solve for A dot B. And we wind up with what? 186 00:09:46,460 --> 00:09:50,140 That A dot B is the magnitude of A minus B squared 187 00:09:50,140 --> 00:09:53,940 minus the magnitude of A squared minus the magnitude of B 188 00:09:53,940 --> 00:09:56,210 squared, all divided by 2. 189 00:09:56,210 --> 00:10:00,570 And the beauty now is that we have expressed A dot B solely 190 00:10:00,570 --> 00:10:03,200 in terms of magnitudes. 191 00:10:03,200 --> 00:10:05,110 And notice especially in Cartesian 192 00:10:05,110 --> 00:10:07,120 coordinates-- and I'll do that next-- 193 00:10:07,120 --> 00:10:09,270 but in terms of Cartesian coordinates, 194 00:10:09,270 --> 00:10:12,540 notice that magnitudes are particularly simple to find. 195 00:10:12,540 --> 00:10:15,090 We just subtract corresponding components 196 00:10:15,090 --> 00:10:16,850 and square, et cetera. 197 00:10:16,850 --> 00:10:18,520 But the important point is that even 198 00:10:18,520 --> 00:10:21,780 without Cartesian coordinates, this particular result, 199 00:10:21,780 --> 00:10:26,970 expressed as A dot B in terms of the magnitudes of A, B, and A 200 00:10:26,970 --> 00:10:29,690 minus B, and is a result which is 201 00:10:29,690 --> 00:10:32,320 independent of any coordinate system. 202 00:10:32,320 --> 00:10:35,570 However-- and this is done very simply in the text, 203 00:10:35,570 --> 00:10:38,190 reinforced in our exercises-- if you 204 00:10:38,190 --> 00:10:44,550 elect to write A, B and A minus B in Cartesian coordinates 205 00:10:44,550 --> 00:10:48,200 and use this particularly straightforward recipe, what 206 00:10:48,200 --> 00:10:51,770 we wind up with is a rather elegant result-- 207 00:10:51,770 --> 00:10:54,160 elegant in terms of simplicity, at least. 208 00:10:54,160 --> 00:10:56,740 And that is-- remember in Cartesian coordinates, 209 00:10:56,740 --> 00:11:01,062 we would write A as a_1*i plus a_2*j plus a_3*k. 210 00:11:01,062 --> 00:11:05,120 B would be b_1*i plus b_2*j plus b_3*k. 211 00:11:05,120 --> 00:11:06,450 And then the beauty is what? 212 00:11:06,450 --> 00:11:11,060 That A dot B turns out to be very simply and conveniently 213 00:11:11,060 --> 00:11:15,900 a_1*b_1 plus a_2*b_2 plus a_3*b_3. 214 00:11:15,900 --> 00:11:19,280 In other words, that to dot A and B, 215 00:11:19,280 --> 00:11:21,380 if the vectors are written in Cartesian 216 00:11:21,380 --> 00:11:23,170 coordinates-- and this is crucial. 217 00:11:23,170 --> 00:11:25,680 If this is not done in Cartesian coordinates, 218 00:11:25,680 --> 00:11:27,930 you can get into a heck of a mess. 219 00:11:27,930 --> 00:11:31,620 And I have deliberately made an exercise on this unit, 220 00:11:31,620 --> 00:11:35,040 get you into that mess, if you fall into that particular trap. 221 00:11:35,040 --> 00:11:37,970 But if we have Cartesian coordinates, 222 00:11:37,970 --> 00:11:41,880 it turns out that to dot two vectors, you simply do what? 223 00:11:41,880 --> 00:11:44,500 Multiply the two i components together. 224 00:11:44,500 --> 00:11:47,110 Multiply the two j components together. 225 00:11:47,110 --> 00:11:50,695 Multiply the two k components together, and add. 226 00:11:50,695 --> 00:11:51,740 All right? 227 00:11:51,740 --> 00:11:54,460 By the way, to show you why this works 228 00:11:54,460 --> 00:11:56,610 from a structural point of view, without belaboring 229 00:11:56,610 --> 00:11:58,830 this point right now, notice that if you 230 00:11:58,830 --> 00:12:01,730 were to multiply in the usual sense of the word 231 00:12:01,730 --> 00:12:04,520 "multiplication," form the dot product here, 232 00:12:04,520 --> 00:12:06,210 you would expect to get nine terms. 233 00:12:06,210 --> 00:12:08,820 In other words, each of the terms in A 234 00:12:08,820 --> 00:12:12,420 multiplies each of the three terms in B. 235 00:12:12,420 --> 00:12:15,002 So that altogether you would expect nine terms. 236 00:12:15,002 --> 00:12:16,460 The thing that's rather interesting 237 00:12:16,460 --> 00:12:21,400 here is that notice that i dot i, j dot j, and k dot 238 00:12:21,400 --> 00:12:23,580 k all happen to be 1. 239 00:12:23,580 --> 00:12:26,050 Because after all, the magnitudes of these vectors 240 00:12:26,050 --> 00:12:27,060 are each 1. 241 00:12:27,060 --> 00:12:30,580 The angle between our i and i is 0, j and j is 0. 242 00:12:30,580 --> 00:12:33,040 The angle between k and k is 0. 243 00:12:33,040 --> 00:12:37,710 So that i dot i, j dot j, and k dot k are all 1. 244 00:12:37,710 --> 00:12:40,940 Whereas on the other hand, when you take mixed terms, notice 245 00:12:40,940 --> 00:12:45,320 that because i and j, i and k, and j and k 246 00:12:45,320 --> 00:12:49,360 are all at right angles, i dot j, j dot k, 247 00:12:49,360 --> 00:12:52,270 i dot k are all going to be 0. 248 00:12:52,270 --> 00:12:55,520 And that therefore, those other six terms will drop out. 249 00:12:55,520 --> 00:12:58,140 In other words, structurally what's happening here 250 00:12:58,140 --> 00:13:01,210 is the fact that the three vectors that we're using here 251 00:13:01,210 --> 00:13:03,650 all happen to have unit length. 252 00:13:03,650 --> 00:13:07,170 And they happen to be mutually perpendicular. 253 00:13:07,170 --> 00:13:11,000 If they were not perpendicular, these mixed terms 254 00:13:11,000 --> 00:13:12,020 would appear in here. 255 00:13:12,020 --> 00:13:13,780 In other words, in general, when you 256 00:13:13,780 --> 00:13:17,100 dot two vectors in three-space, depending 257 00:13:17,100 --> 00:13:19,760 on the coordinate system, you can expect up 258 00:13:19,760 --> 00:13:22,370 to nine terms in your answer. 259 00:13:22,370 --> 00:13:24,430 But the beauty is that as long as we 260 00:13:24,430 --> 00:13:26,150 have Cartesian coordinates, there 261 00:13:26,150 --> 00:13:29,910 happens to be a particularly simple, beautiful recipe 262 00:13:29,910 --> 00:13:31,460 to compute A dot B. 263 00:13:31,460 --> 00:13:34,800 Now keep in mind, the A dot B that we're talking about here 264 00:13:34,800 --> 00:13:37,030 is the same one that we defined before. 265 00:13:37,030 --> 00:13:39,210 It's the magnitude of A times the magnitude 266 00:13:39,210 --> 00:13:41,830 of B times the cosine of the angle between A 267 00:13:41,830 --> 00:13:43,940 and B. All we're saying is that if we 268 00:13:43,940 --> 00:13:46,950 use Cartesian coordinates, we can compute it 269 00:13:46,950 --> 00:13:49,340 almost as fast as we can read. 270 00:13:49,340 --> 00:13:53,260 And let me show you that in terms of some examples. 271 00:13:53,260 --> 00:13:55,670 My first example is the following. 272 00:13:55,670 --> 00:13:57,790 Let's imagine that we have three points 273 00:13:57,790 --> 00:13:59,500 in Cartesian three-space. 274 00:13:59,500 --> 00:14:02,110 A is the point 1 comma 2 comma 3, 275 00:14:02,110 --> 00:14:04,680 B is the point 2 comma 4 comma 1, 276 00:14:04,680 --> 00:14:07,730 and C is the point 3 comma 0 comma 4. 277 00:14:07,730 --> 00:14:10,920 We draw the straight lines AB and AC, 278 00:14:10,920 --> 00:14:14,880 and we would like to find the angle BAC-- in other words, 279 00:14:14,880 --> 00:14:16,980 the angle theta. 280 00:14:16,980 --> 00:14:19,200 The first thing that we do-- and this 281 00:14:19,200 --> 00:14:22,490 is one of the beauties of how vectors are used in geometry-- 282 00:14:22,490 --> 00:14:25,470 is that we vectorize the lines A and B. 283 00:14:25,470 --> 00:14:27,070 We put arrowheads on them. 284 00:14:27,070 --> 00:14:29,450 That immediately makes them vectors. 285 00:14:29,450 --> 00:14:31,860 We already know from last time how 286 00:14:31,860 --> 00:14:34,950 to read the vectors AB and AC. 287 00:14:34,950 --> 00:14:39,910 Namely, AB is the vector i plus 2j-- see, just subtract 288 00:14:39,910 --> 00:14:41,700 component by component. 289 00:14:41,700 --> 00:14:43,180 2 minus 1 is 1. 290 00:14:43,180 --> 00:14:45,300 4 minus 2 is 2. 291 00:14:45,300 --> 00:14:48,220 1 minus 3 is minus 3, et cetera. 292 00:14:48,220 --> 00:14:53,890 So that the vector AB is the vector i plus 2j minus 2k. 293 00:14:53,890 --> 00:14:57,940 And the vector AC, working in a similar way, is 2i 294 00:14:57,940 --> 00:15:00,320 minus 2j plus k. 295 00:15:00,320 --> 00:15:03,230 Now the beauty is that we can compute these magnitudes very, 296 00:15:03,230 --> 00:15:04,870 very quickly by recipe. 297 00:15:04,870 --> 00:15:06,270 And we've just learned the recipe 298 00:15:06,270 --> 00:15:08,216 for finding A dot B in a hurry. 299 00:15:08,216 --> 00:15:09,632 I mean, well in this case, I don't 300 00:15:09,632 --> 00:15:14,010 mean A dot B. I mean the vector AB dotted with the vector AC. 301 00:15:14,010 --> 00:15:16,190 And going through the computational details 302 00:15:16,190 --> 00:15:19,310 here, we square the components of AB, 303 00:15:19,310 --> 00:15:21,460 extract the positive square root, 304 00:15:21,460 --> 00:15:26,150 and we find very easily that the magnitude of AB is 3. 305 00:15:26,150 --> 00:15:29,320 And the hardest part of these problems for me, as a teacher, 306 00:15:29,320 --> 00:15:33,010 is to find ones where I find the sum of 3 squares coming out 307 00:15:33,010 --> 00:15:34,130 to be a whole number. 308 00:15:34,130 --> 00:15:36,500 So I always use the vector [1, 2, 2], 309 00:15:36,500 --> 00:15:38,150 because that's a nice vector that way. 310 00:15:38,150 --> 00:15:43,740 Similarly, the vector AC also happens to have magnitude 3. 311 00:15:43,740 --> 00:15:44,720 OK? 312 00:15:44,720 --> 00:15:48,326 And to find AB dot AC, what do we do? 313 00:15:48,326 --> 00:15:49,950 Let's just come back here and make sure 314 00:15:49,950 --> 00:15:51,350 we know what we're doing now. 315 00:15:51,350 --> 00:15:54,260 We simply dot component by component. 316 00:15:54,260 --> 00:16:00,240 It's 1 times 2, plus 2 times minus 2, plus minus 2 times 1. 317 00:16:00,240 --> 00:16:04,550 In other words, AB dot AC is 2 minus 4 minus 2, 318 00:16:04,550 --> 00:16:06,490 which is minus 4. 319 00:16:06,490 --> 00:16:09,620 Now using our recipe, we see what? 320 00:16:09,620 --> 00:16:15,090 That cosine theta is AB dot AC divided 321 00:16:15,090 --> 00:16:18,150 by the product of the magnitudes of AB and AC, 322 00:16:18,150 --> 00:16:20,440 from which we very quickly conclude 323 00:16:20,440 --> 00:16:24,390 that the cosine of theta is minus 4/9. 324 00:16:24,390 --> 00:16:27,280 And if you're still mixed up as to what that minus sign means, 325 00:16:27,280 --> 00:16:30,490 just by way of a quick review of the inverse trigonometric 326 00:16:30,490 --> 00:16:34,110 functions, you locate the point minus 4 comma 327 00:16:34,110 --> 00:16:37,630 9 in the xy-plane, and your angle theta 328 00:16:37,630 --> 00:16:40,230 is this particular angle here, which 329 00:16:40,230 --> 00:16:42,740 means that in terms of principal values, 330 00:16:42,740 --> 00:16:45,730 if you look up the angle in the tables whose cosine is 331 00:16:45,730 --> 00:16:48,600 4/9, that will give you this angle here. 332 00:16:48,600 --> 00:16:50,530 Subtract that from 180. 333 00:16:50,530 --> 00:16:52,650 And that's the angle that you're looking for. 334 00:16:52,650 --> 00:16:54,120 But the beauty is what? 335 00:16:54,120 --> 00:16:59,720 That you can now find an angle between two lines in space 336 00:16:59,720 --> 00:17:02,770 without having to geometrically worry about what 337 00:17:02,770 --> 00:17:03,690 the angle looks like. 338 00:17:03,690 --> 00:17:06,839 The algebra in Cartesian coordinates takes care of this 339 00:17:06,839 --> 00:17:08,040 by itself. 340 00:17:08,040 --> 00:17:10,839 The same thing happens when you're looking for projections 341 00:17:10,839 --> 00:17:12,069 in three-dimensional space. 342 00:17:12,069 --> 00:17:14,720 Suppose you have a force and a displacement 343 00:17:14,720 --> 00:17:16,240 in three-dimensional space, and you 344 00:17:16,240 --> 00:17:21,390 want to project a force onto a line, a direction. 345 00:17:21,390 --> 00:17:25,359 And our next example shows how the dot product can 346 00:17:25,359 --> 00:17:27,170 be used to find projections. 347 00:17:27,170 --> 00:17:30,120 Namely here's a vector A, here's a vector B. 348 00:17:30,120 --> 00:17:33,262 And I would like to project the vector A onto the vector B. 349 00:17:33,262 --> 00:17:35,720 And I would like to find what the length of that projection 350 00:17:35,720 --> 00:17:36,450 is. 351 00:17:36,450 --> 00:17:39,390 Well, from elementary trigonometry, 352 00:17:39,390 --> 00:17:41,500 I know that the length of this projection 353 00:17:41,500 --> 00:17:45,360 is just the magnitude of A times the cosine of theta. 354 00:17:45,360 --> 00:17:48,580 And by the way, notice if theta were greater than 90 degrees, 355 00:17:48,580 --> 00:17:50,510 cosine theta would be negative. 356 00:17:50,510 --> 00:17:53,040 And the minus sign would not affect the length. 357 00:17:53,040 --> 00:17:56,550 It would simply tell us that the projection was 358 00:17:56,550 --> 00:18:00,090 in the opposite sense of B. That's all that would mean. 359 00:18:00,090 --> 00:18:02,960 But here's the interesting point. 360 00:18:02,960 --> 00:18:06,400 If you look at the magnitude of A times the cosine of theta, 361 00:18:06,400 --> 00:18:09,230 it almost looks like a dot product. 362 00:18:09,230 --> 00:18:12,960 After all, theta is the angle between A and B. 363 00:18:12,960 --> 00:18:15,680 And if the magnitude of B were in here, 364 00:18:15,680 --> 00:18:19,220 this would just be A dot B. But the magnitude of B 365 00:18:19,220 --> 00:18:20,520 isn't in here. 366 00:18:20,520 --> 00:18:23,590 Of course if the magnitude of B happened to be 1, 367 00:18:23,590 --> 00:18:25,280 that would be fine. 368 00:18:25,280 --> 00:18:27,830 But the magnitude of B might not be 1. 369 00:18:27,830 --> 00:18:29,820 And the most honest way to make it 1 370 00:18:29,820 --> 00:18:32,840 is to divide B by its magnitude. 371 00:18:32,840 --> 00:18:34,120 And what will that give you? 372 00:18:34,120 --> 00:18:37,080 If you divide any vector by its magnitude, 373 00:18:37,080 --> 00:18:39,790 that automatically gives you a unit vector 374 00:18:39,790 --> 00:18:43,729 having the same direction and sense as the vector that you 375 00:18:43,729 --> 00:18:44,270 started with. 376 00:18:44,270 --> 00:18:46,880 In other words, let u sub B, which 377 00:18:46,880 --> 00:18:51,174 is B divided by its magnitude, be the unit vector 378 00:18:51,174 --> 00:18:52,590 in the direction-- and by the way, 379 00:18:52,590 --> 00:18:56,280 here direction includes sense-- in the direction of B. 380 00:18:56,280 --> 00:19:00,160 And notice that the unit vector in the direction of B 381 00:19:00,160 --> 00:19:02,620 has the same direction as B itself. 382 00:19:02,620 --> 00:19:05,740 Therefore, to find the angle between u sub B 383 00:19:05,740 --> 00:19:09,830 and A is the same as finding the angle between B and A. 384 00:19:09,830 --> 00:19:11,580 In other words, the kicker now seems 385 00:19:11,580 --> 00:19:14,860 to be that I take this length, which 386 00:19:14,860 --> 00:19:17,590 is the magnitude of A times cosine theta, 387 00:19:17,590 --> 00:19:19,440 and rewrite that as follows. 388 00:19:19,440 --> 00:19:22,610 It's the magnitude of A-- and now remembering 389 00:19:22,610 --> 00:19:27,420 that u sub B has unit length, I just 390 00:19:27,420 --> 00:19:30,310 throw that in as a factor-- and theta, 391 00:19:30,310 --> 00:19:32,620 being the angle between A and B, is also 392 00:19:32,620 --> 00:19:37,160 the angle between A and u sub B. But this, by definition, is 393 00:19:37,160 --> 00:19:40,030 A dot u sub B. 394 00:19:40,030 --> 00:19:44,030 You see, in other words, to find the projection of A onto B, 395 00:19:44,030 --> 00:19:47,390 all you have to do is dot A with the unit 396 00:19:47,390 --> 00:19:51,010 vector in the direction of B. In fact, to summarize 397 00:19:51,010 --> 00:19:53,980 that without the u sub B in there, all I'm saying 398 00:19:53,980 --> 00:19:58,650 is given two vectors A and B, if you dot A and B 399 00:19:58,650 --> 00:20:01,480 and then divide by the magnitude of B, that 400 00:20:01,480 --> 00:20:06,350 will be the projection of A in the direction of B. OK? 401 00:20:06,350 --> 00:20:09,760 The projection of A in the direction of B. And of course, 402 00:20:09,760 --> 00:20:11,260 if you want it the other way around, 403 00:20:11,260 --> 00:20:13,330 you have to reverse the roles of A and B. 404 00:20:13,330 --> 00:20:16,560 The beauty of this unit, in Cartesian coordinates, 405 00:20:16,560 --> 00:20:20,370 is how easy it is to compute A dot B in Cartesian coordinates. 406 00:20:20,370 --> 00:20:23,620 Oh, another example that you might be interested in, that I 407 00:20:23,620 --> 00:20:26,190 think is very interesting, and that's 408 00:20:26,190 --> 00:20:28,130 the special case where A and B already 409 00:20:28,130 --> 00:20:30,030 happen to be unit vectors. 410 00:20:30,030 --> 00:20:32,870 If A and B already happen to be unit vectors, 411 00:20:32,870 --> 00:20:36,950 then if we use our recipe for the formula for A dot B, 412 00:20:36,950 --> 00:20:40,920 we observe that, in this case, by definition of unit vectors, 413 00:20:40,920 --> 00:20:43,730 both the magnitudes of A and B are 1. 414 00:20:43,730 --> 00:20:47,020 And we find that A dot B is then the cosine 415 00:20:47,020 --> 00:20:50,270 of the angle between A and B. Which 416 00:20:50,270 --> 00:20:53,890 means that if A and B happen to be unit vectors, as soon as you 417 00:20:53,890 --> 00:20:56,490 dot them, you have automatically found 418 00:20:56,490 --> 00:21:00,620 the cosine of the angle between the two vectors, which 419 00:21:00,620 --> 00:21:03,930 suggests a rather general type of approach. 420 00:21:03,930 --> 00:21:08,250 Given any two vectors, divide each by the magnitude, right? 421 00:21:08,250 --> 00:21:10,200 That gives you unit vectors. 422 00:21:10,200 --> 00:21:12,220 Dot them, and that gives you the cosine 423 00:21:12,220 --> 00:21:14,620 of the angle between them. 424 00:21:14,620 --> 00:21:15,350 You see? 425 00:21:15,350 --> 00:21:17,720 In particular, and here's an interesting thing. 426 00:21:17,720 --> 00:21:19,680 You know, I don't know if it's that funny, 427 00:21:19,680 --> 00:21:21,440 it just struck me as funny. 428 00:21:21,440 --> 00:21:24,540 Last night at supper as we were sitting down to eat, 429 00:21:24,540 --> 00:21:26,600 my four-year-old looked at me and said, 430 00:21:26,600 --> 00:21:28,750 "Dad, did they have baked potatoes 431 00:21:28,750 --> 00:21:30,500 when you was a little boy?" 432 00:21:30,500 --> 00:21:32,150 And you get the feeling sometimes 433 00:21:32,150 --> 00:21:34,560 that people think that the modern world really 434 00:21:34,560 --> 00:21:38,690 changed the old in certain basic ways that didn't happen at all. 435 00:21:38,690 --> 00:21:40,170 And one of the interesting points 436 00:21:40,170 --> 00:21:44,100 is that long before vector geometry was invented, 437 00:21:44,100 --> 00:21:47,390 people were doing three-dimensional geometry 438 00:21:47,390 --> 00:21:49,660 using non-vector methods. 439 00:21:49,660 --> 00:21:51,780 And one technique that happened to be used 440 00:21:51,780 --> 00:21:54,160 were things called directional cosines. 441 00:21:54,160 --> 00:21:57,650 Namely, suppose you were given a line in space. 442 00:21:57,650 --> 00:21:59,210 OK? 443 00:21:59,210 --> 00:22:01,610 As a vector, if you wish to look at it that way-- 444 00:22:01,610 --> 00:22:03,610 or if you didn't want to look at it as a vector, 445 00:22:03,610 --> 00:22:06,290 imagine the line parallel to the given line 446 00:22:06,290 --> 00:22:08,570 that goes through the origin. 447 00:22:08,570 --> 00:22:11,410 As soon as I know what that line looks like, 448 00:22:11,410 --> 00:22:14,770 I can compute the angle that it makes with the positive x-axis. 449 00:22:14,770 --> 00:22:18,720 I can compute the angle that it makes with the positive y-axis. 450 00:22:18,720 --> 00:22:22,130 I can compute the angle that it makes with the positive z-axis. 451 00:22:22,130 --> 00:22:26,070 And those three angles uniquely determine the position 452 00:22:26,070 --> 00:22:29,470 of the line in space, the direction of the line. 453 00:22:29,470 --> 00:22:30,960 OK? 454 00:22:30,960 --> 00:22:34,567 And those were called the directional angles. 455 00:22:34,567 --> 00:22:35,400 You understand that? 456 00:22:35,400 --> 00:22:38,530 That was the three-dimensional analog of slope. 457 00:22:38,530 --> 00:22:40,590 In other words, to find the slope of a line 458 00:22:40,590 --> 00:22:44,060 in three-dimensional space, draw the line parallel to that line 459 00:22:44,060 --> 00:22:46,950 that goes through the origin, and measure each of the three 460 00:22:46,950 --> 00:22:51,160 angles that that line makes with the positive x-, y-, 461 00:22:51,160 --> 00:22:52,780 and z-axes. 462 00:22:52,780 --> 00:22:55,100 And what the beauty was of the dot product was 463 00:22:55,100 --> 00:22:57,880 it just gave us a simpler way of doing that. 464 00:22:57,880 --> 00:23:03,790 Namely, if A is any vector, I divide A by its magnitude. 465 00:23:03,790 --> 00:23:06,870 That gives me the unit vector in the direction of A. 466 00:23:06,870 --> 00:23:09,730 If I now dot the unit vector in the direction of A 467 00:23:09,730 --> 00:23:12,400 with i-- and after all, what is i? 468 00:23:12,400 --> 00:23:14,360 i is the unit vector in the direction 469 00:23:14,360 --> 00:23:16,424 of the positive x-axis. 470 00:23:16,424 --> 00:23:17,840 Since these are both unit vectors, 471 00:23:17,840 --> 00:23:22,090 this would be the cosine of the angle between i and A. 472 00:23:22,090 --> 00:23:23,900 And that's just what? 473 00:23:23,900 --> 00:23:26,630 The cosine of the angle that A makes 474 00:23:26,630 --> 00:23:29,800 with the positive x-axis-- traditionally, 475 00:23:29,800 --> 00:23:31,810 that angle was called alpha. 476 00:23:31,810 --> 00:23:35,490 So u_A dot i is simply cosine alpha. 477 00:23:35,490 --> 00:23:38,880 Correspondingly, u_A dot j is the cosine 478 00:23:38,880 --> 00:23:41,080 of the angle between A and j. 479 00:23:41,080 --> 00:23:44,140 That's the cosine of the angle that A 480 00:23:44,140 --> 00:23:46,230 makes with the positive y-axis. 481 00:23:46,230 --> 00:23:48,230 That angle was called beta. 482 00:23:48,230 --> 00:23:49,960 This is cosine beta. 483 00:23:49,960 --> 00:23:53,660 And u sub A dot k is cosine gamma where 484 00:23:53,660 --> 00:23:57,040 gamma is the angle that A makes with the positive k direction. 485 00:23:57,040 --> 00:23:59,500 And these, being the cosines of the angles, 486 00:23:59,500 --> 00:24:01,840 these were called the directional cosines, 487 00:24:01,840 --> 00:24:03,910 and they yielded the slope of lines. 488 00:24:03,910 --> 00:24:08,110 But one must not believe that one needed vectors 489 00:24:08,110 --> 00:24:10,370 before he could do three-dimensional geometry. 490 00:24:10,370 --> 00:24:13,860 What did happen was that vector techniques greatly 491 00:24:13,860 --> 00:24:17,840 simplified many of the aspects of three-dimensional geometry. 492 00:24:17,840 --> 00:24:21,530 Well, let's leave this part for a moment 493 00:24:21,530 --> 00:24:26,190 and close for today by coming back to our game idea. 494 00:24:26,190 --> 00:24:30,680 Remember that ultimately, all we will ever use, once we get 495 00:24:30,680 --> 00:24:34,050 started with our game, all we will ever use 496 00:24:34,050 --> 00:24:36,920 are the structural properties. 497 00:24:36,920 --> 00:24:39,960 Now I've gone through these in the notes. 498 00:24:39,960 --> 00:24:41,990 I've gone through them-- well you 499 00:24:41,990 --> 00:24:43,495 go through them with me in the text 500 00:24:43,495 --> 00:24:45,180 or with yourselves in the text. 501 00:24:45,180 --> 00:24:49,760 Let me just point out certain properties of the dot product 502 00:24:49,760 --> 00:24:53,080 that are shared by regular arithmetic as well. 503 00:24:53,080 --> 00:24:56,440 For example, A dot B equals B dot A. 504 00:24:56,440 --> 00:24:58,380 The dot product is commutative. 505 00:24:58,380 --> 00:24:59,490 Why is that? 506 00:24:59,490 --> 00:25:01,580 Well think of what A dot B is. 507 00:25:01,580 --> 00:25:04,670 It's the magnitude of A times the magnitude of B 508 00:25:04,670 --> 00:25:07,600 times the cosine of the angle between A and B. 509 00:25:07,600 --> 00:25:09,400 But that's the same as what? 510 00:25:09,400 --> 00:25:11,840 The magnitude of B times the magnitude of A-- 511 00:25:11,840 --> 00:25:15,300 after all, numbers, we know are commutative when you multiply 512 00:25:15,300 --> 00:25:18,650 them-- and the cosine of the angle between A and B 513 00:25:18,650 --> 00:25:21,760 is the same as the cosine of the angle between B and A. 514 00:25:21,760 --> 00:25:27,320 So these are equal numerical quantities. 515 00:25:27,320 --> 00:25:29,730 Again, without going through the proof here, 516 00:25:29,730 --> 00:25:31,680 it turns out that if you want to dot 517 00:25:31,680 --> 00:25:34,330 a vector with the sum of two given vectors, 518 00:25:34,330 --> 00:25:35,840 the distributive property holds. 519 00:25:35,840 --> 00:25:42,650 Namely, A dot B plus C is A dot B plus A dot C. By the way, 520 00:25:42,650 --> 00:25:46,420 if you did want to prove this, all you would have to do, 521 00:25:46,420 --> 00:25:48,480 if you couldn't see it geometrically, 522 00:25:48,480 --> 00:25:50,330 is to argue as follows. 523 00:25:50,330 --> 00:25:51,450 You say, you know? 524 00:25:51,450 --> 00:25:54,420 The easiest way to add and dot vectors 525 00:25:54,420 --> 00:25:56,430 is in Cartesian coordinates. 526 00:25:56,430 --> 00:25:58,360 So let me prove that this result is 527 00:25:58,360 --> 00:26:00,650 true in Cartesian coordinates. 528 00:26:00,650 --> 00:26:04,660 Carry out the details, and if it works in Cartesian coordinates, 529 00:26:04,660 --> 00:26:08,130 since the result doesn't depend on the coordinate system, 530 00:26:08,130 --> 00:26:09,740 the result must be true, regardless 531 00:26:09,740 --> 00:26:11,130 of the coordinate system. 532 00:26:11,130 --> 00:26:15,130 But it's a very simple exercise to actually write A, B, and C 533 00:26:15,130 --> 00:26:18,220 in terms of i, j, k components, compute 534 00:26:18,220 --> 00:26:21,680 both sides of this expression, and show 535 00:26:21,680 --> 00:26:23,080 that they're numerically equal. 536 00:26:23,080 --> 00:26:26,140 I say "numerically" because it's crucial to notice 537 00:26:26,140 --> 00:26:29,980 that both expressions on either side of the equal sign 538 00:26:29,980 --> 00:26:31,080 are numbers. 539 00:26:31,080 --> 00:26:32,770 B plus C is a vector. 540 00:26:32,770 --> 00:26:33,750 A is a vector. 541 00:26:33,750 --> 00:26:36,940 When you dot two vectors, you get a number. 542 00:26:36,940 --> 00:26:40,030 Finally, a scalar multiple of a vector 543 00:26:40,030 --> 00:26:43,870 dotted with another vector has the property that you can leave 544 00:26:43,870 --> 00:26:45,600 the scalar multiple outside. 545 00:26:45,600 --> 00:26:48,870 In other words, you can first dot the two vectors 546 00:26:48,870 --> 00:26:50,930 and then multiply by the scalar. 547 00:26:50,930 --> 00:26:53,120 In other words, in a way, you don't 548 00:26:53,120 --> 00:26:55,160 have to worry about voice inflection 549 00:26:55,160 --> 00:26:56,910 when you have a scalar multiple. 550 00:26:56,910 --> 00:26:59,910 And we will talk more about these as we go along. 551 00:26:59,910 --> 00:27:02,290 However, what's very crucial is to notice 552 00:27:02,290 --> 00:27:05,510 that the dot product does have some difficulties not 553 00:27:05,510 --> 00:27:08,450 associated with ordinary multiplication. 554 00:27:08,450 --> 00:27:11,010 So I say, beware. 555 00:27:11,010 --> 00:27:12,940 For example, somebody might say, I 556 00:27:12,940 --> 00:27:15,970 wonder if the dot product is associative. 557 00:27:15,970 --> 00:27:19,270 I wonder if A dot B dotted with C 558 00:27:19,270 --> 00:27:23,070 is the same as A dotted with B dotted with C. 559 00:27:23,070 --> 00:27:25,350 And this is nonsensical. 560 00:27:25,350 --> 00:27:27,407 I wanted to do this with great dramatic gesture, 561 00:27:27,407 --> 00:27:29,990 but I probably would have broken two fingers against the board 562 00:27:29,990 --> 00:27:30,860 here. 563 00:27:30,860 --> 00:27:33,070 Let's just cross this out, so that you won't 564 00:27:33,070 --> 00:27:34,620 be inclined to remember that. 565 00:27:34,620 --> 00:27:37,660 This not only is false, it's stupid. 566 00:27:37,660 --> 00:27:39,480 And the reason that it's stupid is 567 00:27:39,480 --> 00:27:42,800 that it's nonsensical, that these things don't make sense. 568 00:27:42,800 --> 00:27:45,460 Namely, the dot product has been defined 569 00:27:45,460 --> 00:27:50,130 to be an operation between two vectors that yields a number. 570 00:27:50,130 --> 00:27:53,090 Notice that as soon as you dot A and B, 571 00:27:53,090 --> 00:27:54,440 you no longer have a vector. 572 00:27:54,440 --> 00:27:55,540 You have a number. 573 00:27:55,540 --> 00:27:58,400 And you cannot dot a number with a vector. 574 00:27:58,400 --> 00:28:01,036 In other words, neither (A dot B) 575 00:28:01,036 --> 00:28:06,110 dot C nor A dot (B dot C) is defined. 576 00:28:06,110 --> 00:28:10,270 Because you see, a number is never dotted with a vector. 577 00:28:10,270 --> 00:28:12,100 All right? 578 00:28:12,100 --> 00:28:15,860 And finally, a closing note, as we've already seen, 579 00:28:15,860 --> 00:28:18,880 if A is perpendicular to B, then the cosine 580 00:28:18,880 --> 00:28:21,760 of the angle between A and B is 0. 581 00:28:21,760 --> 00:28:24,470 And that says that if A is perpendicular to B, 582 00:28:24,470 --> 00:28:25,770 then A dot B is 0. 583 00:28:25,770 --> 00:28:28,730 In fact, this is one of the most common usages 584 00:28:28,730 --> 00:28:31,000 on the elementary level, of the dot product, 585 00:28:31,000 --> 00:28:33,440 is to prove that two vectors are perpendicular. 586 00:28:33,440 --> 00:28:36,740 But, whereas that's a nice property, 587 00:28:36,740 --> 00:28:39,900 what causes great hardship here is to notice the following, 588 00:28:39,900 --> 00:28:42,990 in particular, that if A dot B is 0, 589 00:28:42,990 --> 00:28:47,000 we cannot conclude that either A is the zero vector or B is 590 00:28:47,000 --> 00:28:48,010 the zero vector. 591 00:28:48,010 --> 00:28:50,490 For example, i dot j is 0. 592 00:28:50,490 --> 00:28:53,890 But neither i nor j is the zero vector. 593 00:28:53,890 --> 00:28:55,800 You see, with ordinary arithmetic, 594 00:28:55,800 --> 00:28:57,980 we had the cancellation rule, things 595 00:28:57,980 --> 00:29:01,100 that said if the product of two numbers is 0, at least one 596 00:29:01,100 --> 00:29:02,570 of the factors must be 0. 597 00:29:02,570 --> 00:29:06,010 With the dot product, this need not be true. 598 00:29:06,010 --> 00:29:08,302 The thing I want you to get from this lesson more 599 00:29:08,302 --> 00:29:10,260 than anything else, other than the applications 600 00:29:10,260 --> 00:29:13,190 that you can get from the book and from the exercises, 601 00:29:13,190 --> 00:29:15,810 is to learn to get a feeling for the structure. 602 00:29:15,810 --> 00:29:18,530 Don't be upset that certain vector properties are 603 00:29:18,530 --> 00:29:21,250 different than arithmetic properties, 604 00:29:21,250 --> 00:29:22,890 and certain ones are the same. 605 00:29:22,890 --> 00:29:26,920 Notice in terms of the game, we take our rules 606 00:29:26,920 --> 00:29:29,840 as they may apply and just carry them out 607 00:29:29,840 --> 00:29:32,230 towards inescapable conclusions. 608 00:29:32,230 --> 00:29:35,120 But I think that will become clearer as you read the text 609 00:29:35,120 --> 00:29:36,800 and do the exercises. 610 00:29:36,800 --> 00:29:40,710 And until next time, when we'll talk about a new vector 611 00:29:40,710 --> 00:29:42,870 product, let's just say, so long. 612 00:29:45,410 --> 00:29:47,780 Funding for the publication of this video 613 00:29:47,780 --> 00:29:52,660 was provided by the Gabriella and Paul Rosenbaum Foundation. 614 00:29:52,660 --> 00:29:56,830 Help OCW continue to provide free and open access to MIT 615 00:29:56,830 --> 00:30:01,248 courses by making a donation at ocw.mit.edu/donate.