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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.