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PROFESSOR: Hi.
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00:00:35,690 --> 00:00:39,080
In the last unit, we
introduced the notion
10
00:00:39,080 --> 00:00:44,210
of the structure of vectors in
terms of arrows in the plane.
11
00:00:44,210 --> 00:00:49,740
And today, we want to talk about
the added geometric property
12
00:00:49,740 --> 00:00:52,800
of what I could call
three-dimensional vectors
13
00:00:52,800 --> 00:00:55,040
or three-dimensional arrows.
14
00:00:55,040 --> 00:00:57,750
Now before I say that-- I
guess I've already said it--
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00:00:57,750 --> 00:00:59,470
but before I say
any more about it,
16
00:00:59,470 --> 00:01:01,220
let me point out,
of course, that this
17
00:01:01,220 --> 00:01:03,250
is a matter of semantics.
18
00:01:03,250 --> 00:01:08,010
Obviously an arrow, being a
line, has only one dimension,
19
00:01:08,010 --> 00:01:12,050
be it in the direction of the
x-axis, be it in the x-y plane,
20
00:01:12,050 --> 00:01:14,530
or be it in
three-dimensional space.
21
00:01:14,530 --> 00:01:17,010
When we say
"three-dimensional vectors,"
22
00:01:17,010 --> 00:01:21,270
we do not negate the fact that
we're talking about arrows,
23
00:01:21,270 --> 00:01:24,870
but rather that we are dealing
with a coordinate system which
24
00:01:24,870 --> 00:01:28,480
takes into consideration the
fact that the line is drawn
25
00:01:28,480 --> 00:01:30,640
through three-dimensional space.
26
00:01:30,640 --> 00:01:32,620
In other words, where
as I call the lecture
27
00:01:32,620 --> 00:01:36,230
three-dimensional vectors
or arrows-- and by the way,
28
00:01:36,230 --> 00:01:39,080
both in the notes
and as I'm lecturing,
29
00:01:39,080 --> 00:01:42,920
I will very often, whenever
I write the word "vector,"
30
00:01:42,920 --> 00:01:45,120
put "arrow" in
parentheses, whenever
31
00:01:45,120 --> 00:01:47,660
I write the word "arrow,"
put "vector" in parentheses
32
00:01:47,660 --> 00:01:50,190
so that you can see
the juxtaposition
33
00:01:50,190 --> 00:01:54,330
between these two, the
geometry versus the physical or
34
00:01:54,330 --> 00:01:55,850
mathematical concept.
35
00:01:55,850 --> 00:01:57,770
But the idea is
something like this.
36
00:01:57,770 --> 00:02:01,320
In three-dimensional space,
we have a natural extension
37
00:02:01,320 --> 00:02:02,980
of Cartesian coordinates.
38
00:02:02,980 --> 00:02:05,450
Rather than talk
about the x-y plane,
39
00:02:05,450 --> 00:02:09,699
we pick a third axis, a
third number line, which
40
00:02:09,699 --> 00:02:13,530
goes through the origin,
perpendicular to the x-y plane,
41
00:02:13,530 --> 00:02:17,010
and which has the sense
that if the x-axis is
42
00:02:17,010 --> 00:02:22,350
rotated into the y-axis through
the positive 90-degree angle
43
00:02:22,350 --> 00:02:25,420
here, that the z-axis
is in the direction
44
00:02:25,420 --> 00:02:28,950
in which a right-handed
screw would turn.
45
00:02:28,950 --> 00:02:30,410
I won't belabor this.
46
00:02:30,410 --> 00:02:32,370
It's done very, very
nicely in the textbook.
47
00:02:32,370 --> 00:02:35,710
And I'm sure that you have seen
this type of coordinate system
48
00:02:35,710 --> 00:02:36,360
before.
49
00:02:36,360 --> 00:02:37,818
If you haven't,
it's something that
50
00:02:37,818 --> 00:02:40,910
takes a matter of some 20
or 30 seconds to pick up.
51
00:02:40,910 --> 00:02:42,850
But at any rate,
what we're saying
52
00:02:42,850 --> 00:02:45,820
is, let's imagine this
three-dimensional coordinate
53
00:02:45,820 --> 00:02:48,930
system, three-dimensional
Cartesian coordinates.
54
00:02:48,930 --> 00:02:52,800
The convention is that
just as, in the plane,
55
00:02:52,800 --> 00:02:57,480
we label the point by its x and
y components, in three space,
56
00:02:57,480 --> 00:03:02,070
a point is labeled by its
x, y, and z components.
57
00:03:02,070 --> 00:03:06,070
So for example, if I take a
vector in three-space-- meaning
58
00:03:06,070 --> 00:03:06,570
what?
59
00:03:06,570 --> 00:03:09,580
It's a line that goes through
three-dimensional space-- I
60
00:03:09,580 --> 00:03:11,310
again shift it
parallel to itself,
61
00:03:11,310 --> 00:03:13,350
so it begins at my origin.
62
00:03:13,350 --> 00:03:16,340
And at the risk of causing
some confusion here,
63
00:03:16,340 --> 00:03:18,150
I still think it's worth doing.
64
00:03:18,150 --> 00:03:20,660
Let me call the
vector A, and let
65
00:03:20,660 --> 00:03:24,680
me call the point at which the
vector terminates the point A.
66
00:03:24,680 --> 00:03:29,290
And let's suppose that the point
A has coordinates A_1, A_2,
67
00:03:29,290 --> 00:03:30,130
A_3.
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00:03:30,130 --> 00:03:32,790
The reason that this
doesn't make much difference
69
00:03:32,790 --> 00:03:36,390
if you get confused is
that if the vector, which
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00:03:36,390 --> 00:03:39,150
originates at the
origin, terminates
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00:03:39,150 --> 00:03:41,850
at the point whose
coordinates a_1, a_2,
72
00:03:41,850 --> 00:03:46,810
and a_3, then the components
of that vector will have what?
73
00:03:46,810 --> 00:03:49,070
The i component will be a_1.
74
00:03:49,070 --> 00:03:51,480
The j component will be a_2.
75
00:03:51,480 --> 00:03:54,940
And by the way, we let
k be the unit vector
76
00:03:54,940 --> 00:03:56,910
in the positive z direction.
77
00:03:56,910 --> 00:04:00,100
And the k component will be a_3.
78
00:04:00,100 --> 00:04:03,260
To see this thing
pictorially, notice
79
00:04:03,260 --> 00:04:05,440
the three-dimensional
effect of the vector A.
80
00:04:05,440 --> 00:04:07,830
It means that if we drop a
perpendicular from the head
81
00:04:07,830 --> 00:04:15,300
of A to the x-y plane, then,
you see, this distance is a_1.
82
00:04:15,300 --> 00:04:17,300
This distance is a_2.
83
00:04:17,300 --> 00:04:19,540
And this height is a_3.
84
00:04:19,540 --> 00:04:23,810
And notice, again, in terms of
adding vectors head to tail et
85
00:04:23,810 --> 00:04:29,200
cetera, notice that as a vector,
this would be the vector a_1*i.
86
00:04:29,200 --> 00:04:33,460
This would be the vector a_2*j.
87
00:04:33,460 --> 00:04:37,420
And this would be
the vector a_3*k.
88
00:04:37,420 --> 00:04:41,350
In other words, the vector A
is the sum of the three vectors
89
00:04:41,350 --> 00:04:45,560
a_1*i, a_2*j, and a_3*k.
90
00:04:45,560 --> 00:04:47,430
In other words,
to summarize this,
91
00:04:47,430 --> 00:04:49,260
the vector A is simply what?
92
00:04:49,260 --> 00:04:52,660
a_1*i, plus a_2*j, plus a_3*k.
93
00:04:52,660 --> 00:04:55,630
And again, notice the
juxtapositioning, if you wish,
94
00:04:55,630 --> 00:04:58,910
between the components
of the vector
95
00:04:58,910 --> 00:05:02,810
and the coordinates
of the point.
96
00:05:02,810 --> 00:05:06,640
Also, we can get ahold of
the magnitude of this vector.
97
00:05:06,640 --> 00:05:09,300
Remember, the magnitude
is the length.
98
00:05:09,300 --> 00:05:12,270
And how can we figure out
the length of this vector?
99
00:05:12,270 --> 00:05:13,930
Notice that, by the
way, the geometry
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00:05:13,930 --> 00:05:16,275
gets much tougher in
three-dimensional space.
101
00:05:16,275 --> 00:05:18,650
But I want to show you something
interesting in a minute.
102
00:05:18,650 --> 00:05:21,190
Let's suffer through the
geometry for a minute.
103
00:05:21,190 --> 00:05:23,690
Let's see how we can find
the length of the vector A.
104
00:05:23,690 --> 00:05:27,670
Well, notice, by the way,
that A happens to be--
105
00:05:27,670 --> 00:05:30,660
or the magnitude of A
happens to be the hypotenuse
106
00:05:30,660 --> 00:05:32,160
of a right triangle.
107
00:05:32,160 --> 00:05:34,190
What right triangle is it?
108
00:05:34,190 --> 00:05:37,110
It's the right
triangle that joins
109
00:05:37,110 --> 00:05:42,010
the origin-- let me call
this point here B-- and A.
110
00:05:42,010 --> 00:05:45,270
In other words, triangle
OBA is a right triangle
111
00:05:45,270 --> 00:05:48,840
because OB is in the
x-y plane and AB is
112
00:05:48,840 --> 00:05:51,300
perpendicular to the x-y plane.
113
00:05:51,300 --> 00:05:55,320
Therefore, by the Pythagorean
theorem, the magnitude of A
114
00:05:55,320 --> 00:05:59,950
will be the square root of the
square of the magnitude of OB
115
00:05:59,950 --> 00:06:02,730
plus the square of
the magnitude of AB.
116
00:06:02,730 --> 00:06:10,750
On the other hand, notice that
OCB is also a right triangle.
117
00:06:10,750 --> 00:06:14,480
So by the Pythagorean
theorem, the length of OB
118
00:06:14,480 --> 00:06:20,510
is the square root of a_1
squared plus a_2 squared.
119
00:06:20,510 --> 00:06:23,800
And therefore, you see, by
the Pythagorean theorem,
120
00:06:23,800 --> 00:06:26,250
the square of the
magnitude of A is
121
00:06:26,250 --> 00:06:28,310
this squared plus this squared.
122
00:06:28,310 --> 00:06:30,820
In other words,
the magnitude of A
123
00:06:30,820 --> 00:06:34,550
is the square root of a_1
squared plus a_2 squared
124
00:06:34,550 --> 00:06:35,990
plus a_3 squared.
125
00:06:35,990 --> 00:06:38,150
In other words, in
Cartesian coordinates,
126
00:06:38,150 --> 00:06:40,250
to find the magnitude
of a vector,
127
00:06:40,250 --> 00:06:43,780
you need only take the
positive square root
128
00:06:43,780 --> 00:06:46,070
of the sum of the squares
of the components.
129
00:06:46,070 --> 00:06:48,750
It's a regular distance formula,
the Pythagorean theorem.
130
00:06:48,750 --> 00:06:50,930
By the way, just a
very, very quick review
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00:06:50,930 --> 00:06:52,970
from part one of our
course-- remember
132
00:06:52,970 --> 00:06:55,160
that technically
speaking, the square root
133
00:06:55,160 --> 00:06:57,130
is a double-valued function.
134
00:06:57,130 --> 00:07:00,820
Namely, any positive number
has two square roots.
135
00:07:00,820 --> 00:07:01,320
All right?
136
00:07:01,320 --> 00:07:04,200
In other words, the
square root of 16 is 4.
137
00:07:04,200 --> 00:07:06,150
It's also negative 4.
138
00:07:06,150 --> 00:07:09,470
But by convention, when
we don't indicate a sign,
139
00:07:09,470 --> 00:07:12,180
we're always referring to
the positive square root.
140
00:07:12,180 --> 00:07:13,280
And that's fine.
141
00:07:13,280 --> 00:07:15,220
Because after all,
the magnitude--
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00:07:15,220 --> 00:07:17,010
the length of an
arrow-- is always
143
00:07:17,010 --> 00:07:18,540
looked upon as being what?
144
00:07:18,540 --> 00:07:20,550
Some non-negative number.
145
00:07:20,550 --> 00:07:24,390
Notice that the negativeness
is taken care of by the sense.
146
00:07:24,390 --> 00:07:25,130
OK?
147
00:07:25,130 --> 00:07:26,900
And this is what's
rather interesting.
148
00:07:26,900 --> 00:07:28,380
I think it's worth noting.
149
00:07:28,380 --> 00:07:31,260
If we use Cartesian
coordinates in one dimension,
150
00:07:31,260 --> 00:07:34,480
all you have is the
x-axis, in which case
151
00:07:34,480 --> 00:07:38,720
any vector is some scale or
multiple of the unit vector i.
152
00:07:38,720 --> 00:07:42,190
In other words, A
is equal to a_1*i.
153
00:07:42,190 --> 00:07:45,750
In this case, the magnitude
of A is just what?
154
00:07:45,750 --> 00:07:48,470
It's the square
root of a_1 squared.
155
00:07:48,470 --> 00:07:50,890
And notice that means the
positive square root--
156
00:07:50,890 --> 00:07:53,330
notice, by the way, that
that's the usual meaning
157
00:07:53,330 --> 00:07:54,570
of absolute value.
158
00:07:54,570 --> 00:07:57,380
After all, if a_1
were already positive,
159
00:07:57,380 --> 00:08:00,640
taking the positive square
root of its square root--
160
00:08:00,640 --> 00:08:01,564
of its square.
161
00:08:01,564 --> 00:08:02,480
These tongue twisters.
162
00:08:02,480 --> 00:08:04,250
The positive square
root of the square
163
00:08:04,250 --> 00:08:05,920
would give you the
number back again.
164
00:08:05,920 --> 00:08:08,080
On the other hand,
if a_1 were negative,
165
00:08:08,080 --> 00:08:11,620
and then you square it, and then
take the positive square root,
166
00:08:11,620 --> 00:08:14,780
all you do is change
the sign of a_1.
167
00:08:14,780 --> 00:08:16,940
And since it was negative,
changing its sign
168
00:08:16,940 --> 00:08:18,420
makes it positive.
169
00:08:18,420 --> 00:08:21,020
So this is the usual
absolute value, all right?
170
00:08:21,020 --> 00:08:22,940
Now in terms of
the previous unit,
171
00:08:22,940 --> 00:08:25,240
we saw that in
two-dimensional space,
172
00:08:25,240 --> 00:08:27,620
using i and j as
our basic vectors,
173
00:08:27,620 --> 00:08:31,520
that if A was the
vector a_1*i plus a_2*j,
174
00:08:31,520 --> 00:08:34,970
then the magnitude of A, again,
by the Pythagorean theorem,
175
00:08:34,970 --> 00:08:40,130
was just a square root of
a_1 squared plus a_2 squared.
176
00:08:40,130 --> 00:08:42,330
And now we've seen
that structurally--
177
00:08:42,330 --> 00:08:46,010
even though geometrically
it was harder to show--
178
00:08:46,010 --> 00:08:50,230
that if A in three-dimensional
space is a_1*i plus a_2*j plus
179
00:08:50,230 --> 00:08:53,250
a_3*k, then A is what?
180
00:08:53,250 --> 00:08:55,190
The positive square
root of a_1 squared
181
00:08:55,190 --> 00:08:57,410
plus a_2 squared
plus a_3 squared.
182
00:08:57,410 --> 00:09:01,030
And notice you see this
structural resemblance.
183
00:09:01,030 --> 00:09:03,230
This is a beautiful
structural resemblance.
184
00:09:03,230 --> 00:09:05,610
Why is it a beautiful
structural resemblance?
185
00:09:05,610 --> 00:09:07,950
Because now, maybe we're
getting the feeling
186
00:09:07,950 --> 00:09:10,510
that once our rules and
recipes are made up,
187
00:09:10,510 --> 00:09:13,610
we don't have to worry about
how difficult the geometry is.
188
00:09:13,610 --> 00:09:15,510
In other words,
except for the fact
189
00:09:15,510 --> 00:09:17,910
that you have an extra
component to worry about here,
190
00:09:17,910 --> 00:09:20,320
there is no basic
difference structurally
191
00:09:20,320 --> 00:09:23,500
between finding the length of a
vector in one-dimensional space
192
00:09:23,500 --> 00:09:25,500
or in three-dimensional space.
193
00:09:25,500 --> 00:09:28,610
And in fact, another reason that
I'm harping on this is the fact
194
00:09:28,610 --> 00:09:31,740
that in a little while-- meaning
within the next few lessons--
195
00:09:31,740 --> 00:09:34,730
we are going to be talking
about some horrible thing called
196
00:09:34,730 --> 00:09:38,620
n-dimensional space, which
geometrically cannot be drawn,
197
00:09:38,620 --> 00:09:41,940
but which analytically preserves
the same structure that
198
00:09:41,940 --> 00:09:43,310
we're talking about here.
199
00:09:43,310 --> 00:09:46,520
You see, the whole bag is
still this idea of structure.
200
00:09:46,520 --> 00:09:49,580
We're trying to emphasize
that the recipes remain
201
00:09:49,580 --> 00:09:52,650
the same independently
of the dimension.
202
00:09:52,650 --> 00:09:56,970
And just to carry this idea a
few steps further, what we're
203
00:09:56,970 --> 00:10:00,620
saying is, remember in the last
unit, we talked about the fact
204
00:10:00,620 --> 00:10:02,780
that the nice thing
about i and j components
205
00:10:02,780 --> 00:10:05,070
was that the definition
for adding two vectors
206
00:10:05,070 --> 00:10:08,680
was that if they were in
terms of i and j components,
207
00:10:08,680 --> 00:10:10,860
you just add it
component by component?
208
00:10:10,860 --> 00:10:13,500
Well again, without
belaboring the point
209
00:10:13,500 --> 00:10:18,900
and leaving the details to
the text and to the exercises,
210
00:10:18,900 --> 00:10:21,650
let's keep in mind that
it's relatively easy to show
211
00:10:21,650 --> 00:10:24,780
that the same formal definition
of addition-- namely,
212
00:10:24,780 --> 00:10:27,680
placing the arrows head
to tail, et cetera--
213
00:10:27,680 --> 00:10:29,764
forgetting about Cartesian
coordinates, but notice
214
00:10:29,764 --> 00:10:31,180
the definition of
addition doesn't
215
00:10:31,180 --> 00:10:32,530
mention the coordinate system.
216
00:10:32,530 --> 00:10:35,300
It just says put the vectors
head to tail, et cetera.
217
00:10:35,300 --> 00:10:37,730
And what I'm saying
is it's easy to show,
218
00:10:37,730 --> 00:10:41,530
under those conditions, that
if A is the vector a_1*i plus
219
00:10:41,530 --> 00:10:46,640
a_2*j plus a_3*k, and B is the
vector b_1*i plus b_2*j plus
220
00:10:46,640 --> 00:10:49,710
b_3*k, then you still
add vectors the same way
221
00:10:49,710 --> 00:10:52,300
in three-space as
you did in two-space.
222
00:10:52,300 --> 00:10:55,220
Again, the geometric arguments
are a little bit more
223
00:10:55,220 --> 00:10:57,570
sophisticated, if only
because it's harder
224
00:10:57,570 --> 00:10:59,620
to draw three
dimensions to scale
225
00:10:59,620 --> 00:11:01,210
than it is two dimensions.
226
00:11:01,210 --> 00:11:02,350
But the idea is what?
227
00:11:02,350 --> 00:11:05,620
A plus B is just
obtained by adding what?
228
00:11:05,620 --> 00:11:08,740
The two i components,
the two j components,
229
00:11:08,740 --> 00:11:11,010
the two k components.
230
00:11:11,010 --> 00:11:13,740
And similarly, for
scalar multiplication,
231
00:11:13,740 --> 00:11:16,490
in the same way in terms
of i and j components,
232
00:11:16,490 --> 00:11:19,740
you simply multiplied each
complement by the scalar c.
233
00:11:19,740 --> 00:11:23,940
If c is any number and A is
still the vector a_1*i plus
234
00:11:23,940 --> 00:11:28,330
a_2*j plus a_3k, the scalar
multiple c times A turns out
235
00:11:28,330 --> 00:11:32,270
to be c*a_1*i plus
c*a_3*j plus c*a_3*k.
236
00:11:32,270 --> 00:11:35,470
That you just multiply
component by component.
237
00:11:35,470 --> 00:11:38,990
And by the way, don't lose
track of this important point,
238
00:11:38,990 --> 00:11:43,820
that the scalar multiple c*A
means the same thing in two
239
00:11:43,820 --> 00:11:45,770
dimensions as it did three.
240
00:11:45,770 --> 00:11:49,480
Because after all, when you draw
that vector, once that arrow
241
00:11:49,480 --> 00:11:54,100
is drawn, it's a one-dimensional
thing if you pick as your axis
242
00:11:54,100 --> 00:11:56,919
the line of action of
the particular arrow.
243
00:11:56,919 --> 00:11:58,460
In other words,
scalar multiplication
244
00:11:58,460 --> 00:11:59,450
means the same thing.
245
00:11:59,450 --> 00:12:02,280
What's happening is that
you're looking at it in terms
246
00:12:02,280 --> 00:12:03,820
of different components.
247
00:12:03,820 --> 00:12:06,780
You're looking at it in terms of
i, j, and k, rather than just i
248
00:12:06,780 --> 00:12:09,740
and j, or in terms of any
other coordinate system.
249
00:12:09,740 --> 00:12:11,930
But carrying this
out in particular,
250
00:12:11,930 --> 00:12:15,180
notice then, the connection
between the vector minus B--
251
00:12:15,180 --> 00:12:19,360
negative B-- and the
vector negative 1 times
252
00:12:19,360 --> 00:12:23,790
B. Namely, the vector negative
B, which has the same magnitude
253
00:12:23,790 --> 00:12:27,690
and direction as B,
but the opposite sense,
254
00:12:27,690 --> 00:12:29,560
is the same way as saying what?
255
00:12:29,560 --> 00:12:31,337
Multiply B by minus 1.
256
00:12:31,337 --> 00:12:33,920
Because that gives you a vector
which has the same magnitude--
257
00:12:33,920 --> 00:12:37,780
namely 1 times as much as
B-- the same direction,
258
00:12:37,780 --> 00:12:38,810
and the opposite sense.
259
00:12:38,810 --> 00:12:40,720
In other words, the
vector negative B,
260
00:12:40,720 --> 00:12:45,610
in three-dimensional space,
is minus b_1 i, minus b_2 j,
261
00:12:45,610 --> 00:12:47,440
minus b_3 k.
262
00:12:47,440 --> 00:12:52,490
And therefore, since A minus B
still means A plus negative B,
263
00:12:52,490 --> 00:12:56,370
the vector A minus B
is simply obtained how?
264
00:12:56,370 --> 00:12:58,680
You subtract the
same way as we did
265
00:12:58,680 --> 00:13:02,390
in ordinary one-dimensional
analytic geometry,
266
00:13:02,390 --> 00:13:03,610
in terms of a number line.
267
00:13:03,610 --> 00:13:06,390
In other words, you
subtract component
268
00:13:06,390 --> 00:13:08,800
by component-- subtracting what?
269
00:13:14,570 --> 00:13:18,430
The first vector minus the
corresponding component
270
00:13:18,430 --> 00:13:19,660
of the second vector.
271
00:13:19,660 --> 00:13:22,470
In other words, in Cartesian
coordinates, the vector A
272
00:13:22,470 --> 00:13:27,910
minus B-- once we know a_1, a_2,
and a_3 as the components of A,
273
00:13:27,910 --> 00:13:30,740
b_1, b_2, and b_3 as
the components of B,
274
00:13:30,740 --> 00:13:33,530
we just subtract
component by component
275
00:13:33,530 --> 00:13:35,770
to get this particular
result. And this is, I
276
00:13:35,770 --> 00:13:38,010
think, very important
to understand.
277
00:13:38,010 --> 00:13:40,900
By the same token,
as simple as this is,
278
00:13:40,900 --> 00:13:42,320
it's going to
cause-- if we're not
279
00:13:42,320 --> 00:13:45,230
careful-- great
misinterpretation.
280
00:13:45,230 --> 00:13:50,170
Namely, the vector arithmetic
that we're talking about
281
00:13:50,170 --> 00:13:53,420
does not depend on
Cartesian coordinates.
282
00:13:53,420 --> 00:13:56,580
Vectors or arrows, as
we're looking at them,
283
00:13:56,580 --> 00:13:58,970
and the arrows exist no
matter what coordinate system
284
00:13:58,970 --> 00:13:59,710
we're using.
285
00:13:59,710 --> 00:14:02,470
What is particularly
important, however,
286
00:14:02,470 --> 00:14:06,410
is that many of the luxuries
of using vector arithmetic
287
00:14:06,410 --> 00:14:08,890
hinge on using
Cartesian coordinates.
288
00:14:08,890 --> 00:14:10,590
Now what do I mean by that?
289
00:14:10,590 --> 00:14:14,340
Well, let me just
emphasize: Note the need
290
00:14:14,340 --> 00:14:16,950
for Cartesian coordinates.
291
00:14:16,950 --> 00:14:19,420
Let me talk instead,
you see, about something
292
00:14:19,420 --> 00:14:21,110
called polar coordinates.
293
00:14:21,110 --> 00:14:23,060
Now later on,
we're going to talk
294
00:14:23,060 --> 00:14:26,560
about polar coordinates
in more detail-- in fact,
295
00:14:26,560 --> 00:14:28,540
in very, very much more detail.
296
00:14:28,540 --> 00:14:30,930
But for the time
being, let's view polar
297
00:14:30,930 --> 00:14:34,270
coordinates simply as
a radar-type thing,
298
00:14:34,270 --> 00:14:37,590
as a range-and-bearing
type navigation.
299
00:14:37,590 --> 00:14:39,870
Namely, in polar
coordinates, the way
300
00:14:39,870 --> 00:14:42,600
you specify a vector
in the plane, say,
301
00:14:42,600 --> 00:14:49,020
is you specify its length, which
we'll call r, and the angle
302
00:14:49,020 --> 00:14:52,980
that it makes with the positive
x-axis, that we'll call theta.
303
00:14:52,980 --> 00:14:57,800
In other words, in polar
coordinates, the vector A,
304
00:14:57,800 --> 00:15:00,420
which I've labeled
r_2 comma theta_2 ,
305
00:15:00,420 --> 00:15:06,030
that would indicate that
the magnitude of A is r_2.
306
00:15:06,030 --> 00:15:09,340
And that the angle that A
makes with the positive x-axis
307
00:15:09,340 --> 00:15:13,050
is theta_2.
308
00:15:13,050 --> 00:15:16,510
And when I label the vector
B as r_1 comma theta_1,
309
00:15:16,510 --> 00:15:19,060
it's simply my way of
saying that the magnitude
310
00:15:19,060 --> 00:15:22,500
of the vector B is some
length, which we'll call r_1.
311
00:15:22,500 --> 00:15:26,790
And the angle that B makes
with the positive x-axis
312
00:15:26,790 --> 00:15:29,110
is some angle which
we'll call theta_1.
313
00:15:29,110 --> 00:15:30,160
OK?
314
00:15:30,160 --> 00:15:32,940
Perfectly well-defined
system, isn't it?
315
00:15:32,940 --> 00:15:36,010
In other words, if I tell
you the range and bearing,
316
00:15:36,010 --> 00:15:38,750
I've certainly given
you as much information
317
00:15:38,750 --> 00:15:41,610
as if I told you the
i and j components.
318
00:15:41,610 --> 00:15:44,940
Now here's the important point,
the real kicker to this thing.
319
00:15:44,940 --> 00:15:49,180
And that is that the vector
A minus B-- and A minus B
320
00:15:49,180 --> 00:15:50,030
is what?
321
00:15:50,030 --> 00:15:51,070
It's this vector here.
322
00:15:55,400 --> 00:15:57,840
And by the way, notice how
vector arithmetic goes.
323
00:15:57,840 --> 00:15:59,950
How could you tell
quickly whether this
324
00:15:59,950 --> 00:16:02,400
is A minus B or B minus A?
325
00:16:02,400 --> 00:16:05,040
We talked about this
in the previous unit,
326
00:16:05,040 --> 00:16:07,130
and had exercises
and discussion on it.
327
00:16:07,130 --> 00:16:10,600
But notice again, the
structure of vector arithmetic.
328
00:16:10,600 --> 00:16:14,380
If you look at addition,
what is this vector here?
329
00:16:14,380 --> 00:16:17,830
This is the vector which
must be added on to B
330
00:16:17,830 --> 00:16:22,610
to give the vector A. And that,
by definition, is called what?
331
00:16:22,610 --> 00:16:25,430
A minus B. A minus
B is what vector?
332
00:16:25,430 --> 00:16:30,200
The vector you must add on
to B to yield A. Subtraction
333
00:16:30,200 --> 00:16:32,020
is still the
inverse of addition.
334
00:16:32,020 --> 00:16:36,260
And hopefully by now, you should
be becoming much less fearful
335
00:16:36,260 --> 00:16:39,300
of this vector notation,
that structurally, it's
336
00:16:39,300 --> 00:16:43,810
the same as our first
unit of arithmetic.
337
00:16:43,810 --> 00:16:47,780
Well, numerical arithmetic,
OK, the more regular type,
338
00:16:47,780 --> 00:16:49,470
ordinary type of arithmetic.
339
00:16:49,470 --> 00:16:51,130
But here's the important point.
340
00:16:51,130 --> 00:16:55,070
This vector is still
A minus B. But I
341
00:16:55,070 --> 00:17:00,400
claim that it's trivial to
see that this vector is not
342
00:17:00,400 --> 00:17:03,800
the vector whose
magnitude is r_2 minus r_1
343
00:17:03,800 --> 00:17:07,720
and whose angle of bearing
is theta_2 minus theta_1.
344
00:17:07,720 --> 00:17:10,940
Now the word "trivial" is a very
misleading word in mathematics.
345
00:17:10,940 --> 00:17:13,420
One of our famous
mathematical anecdotes
346
00:17:13,420 --> 00:17:16,276
is the professor who said
that a proof was trivial.
347
00:17:16,276 --> 00:17:18,359
The student says, "It
doesn't seem trivial to me."
348
00:17:18,359 --> 00:17:20,810
Their professor says,
"Well, it is trivial."
349
00:17:20,810 --> 00:17:22,490
He didn't quite
see it in a hurry.
350
00:17:22,490 --> 00:17:23,400
He says, "Wait here."
351
00:17:23,400 --> 00:17:25,030
He ran down to his office.
352
00:17:25,030 --> 00:17:28,260
Came up three hours later
with a ream of papers
353
00:17:28,260 --> 00:17:30,880
that he had written on,
bathed in perspiration,
354
00:17:30,880 --> 00:17:31,637
and a big smile.
355
00:17:31,637 --> 00:17:32,720
And he said, "I was right.
356
00:17:32,720 --> 00:17:34,020
It was trivial."
357
00:17:34,020 --> 00:17:36,170
So just because what
I think is trivial
358
00:17:36,170 --> 00:17:38,130
may not be what you
think is trivial,
359
00:17:38,130 --> 00:17:43,060
let me go through this statement
in more computational detail.
360
00:17:43,060 --> 00:17:46,610
What I'm saying is, let v
denote the vector that we
361
00:17:46,610 --> 00:17:49,120
called A minus B before.
362
00:17:49,120 --> 00:17:52,670
Notice that in terms of what
the vectors A and B are,
363
00:17:52,670 --> 00:17:54,840
this length is r_1.
364
00:17:54,840 --> 00:17:56,650
This length is r_2.
365
00:17:56,650 --> 00:17:59,670
And notice that since this
whole angle was theta_2,
366
00:17:59,670 --> 00:18:03,630
and the angle from here to here
was theta_1, this angle in here
367
00:18:03,630 --> 00:18:06,140
must be the difference between
those two angles, which
368
00:18:06,140 --> 00:18:08,380
is theta_2 minus theta_1.
369
00:18:08,380 --> 00:18:11,320
Now notice that this
is still a triangle.
370
00:18:11,320 --> 00:18:13,120
The length of this
triangle is still
371
00:18:13,120 --> 00:18:16,850
expressible in terms of these
sides and the included angle.
372
00:18:16,850 --> 00:18:20,110
But you may recall from plane
trigonometry and our refresher
373
00:18:20,110 --> 00:18:22,630
of that in the first
part of our course-- part
374
00:18:22,630 --> 00:18:25,660
one-- that to find the
third side of a triangle,
375
00:18:25,660 --> 00:18:27,850
given two sides and
the included angle,
376
00:18:27,850 --> 00:18:30,010
one must use the Law of Cosines.
377
00:18:30,010 --> 00:18:33,490
In other words, the magnitude
of v has what property?
378
00:18:33,490 --> 00:18:36,860
That its square is equal to the
sum of the squares of these two
379
00:18:36,860 --> 00:18:40,870
sides minus twice the
product of these two
380
00:18:40,870 --> 00:18:44,220
lengths times the cosine
of the included angle.
381
00:18:44,220 --> 00:18:49,170
In other words, this is just--
I hope this shows up all right--
382
00:18:49,170 --> 00:18:51,650
this is the Law of Cosines.
383
00:18:51,650 --> 00:18:54,350
And if it's difficult to
read, don't bother reading it.
384
00:18:54,350 --> 00:18:56,720
It's still the Law of
Cosines, and hopefully you
385
00:18:56,720 --> 00:18:58,070
recognize it as such.
386
00:18:58,070 --> 00:19:00,590
In other words, to find
the magnitude of v,
387
00:19:00,590 --> 00:19:03,880
it's not just r_1
or r_2 minus r_1.
388
00:19:03,880 --> 00:19:08,380
It's the square root of r_1
squared plus r_2 squared minus
389
00:19:08,380 --> 00:19:11,790
2r_1*r_2 cosine
theta_2 minus theta_1.
390
00:19:11,790 --> 00:19:14,080
Can you use polar
coordinates if you want?
391
00:19:14,080 --> 00:19:15,700
The answer is you bet you can.
392
00:19:15,700 --> 00:19:18,540
But when you do use it, make
sure that whenever you're
393
00:19:18,540 --> 00:19:21,150
going to find the magnitude,
that you don't just
394
00:19:21,150 --> 00:19:22,470
subtract the two magnitudes.
395
00:19:22,470 --> 00:19:23,870
You have to use this.
396
00:19:23,870 --> 00:19:26,770
And by the way, notice I haven't
even gone into this part,
397
00:19:26,770 --> 00:19:28,780
because it's irrelevant
from the point of view
398
00:19:28,780 --> 00:19:30,610
that I'm trying
to emphasize now.
399
00:19:30,610 --> 00:19:33,700
This mess just gives you
the magnitude of a vector.
400
00:19:33,700 --> 00:19:36,040
It doesn't even tell you
what direction it's in.
401
00:19:36,040 --> 00:19:38,830
I have to still do a
heck of a lot of geometry
402
00:19:38,830 --> 00:19:43,060
if I want to find out
what this angle here
403
00:19:43,060 --> 00:19:46,130
is-- a lot of work involved.
404
00:19:46,130 --> 00:19:47,990
I can use polar
coordinates, but I
405
00:19:47,990 --> 00:19:50,770
lose some of the luxury
of Cartesian coordinates.
406
00:19:50,770 --> 00:19:55,010
In fact, notice that the
vector whose magnitude
407
00:19:55,010 --> 00:19:58,040
is r_2 minus r_1 and
whose angle of bearing
408
00:19:58,040 --> 00:20:01,750
is theta_2 minus theta_1,
can also be computed.
409
00:20:01,750 --> 00:20:05,850
Namely, how do we find
theta_2 minus theta_1?
410
00:20:05,850 --> 00:20:08,980
Well up here, we have
the angle theta_2.
411
00:20:08,980 --> 00:20:10,580
We have the angle theta_1.
412
00:20:10,580 --> 00:20:12,960
We computed theta_2
minus theta_1.
413
00:20:12,960 --> 00:20:17,210
I can now mark off that angle
here, theta_2 minus theta_1.
414
00:20:17,210 --> 00:20:20,820
I can now take a
circle of radius r_1--
415
00:20:20,820 --> 00:20:23,410
well, why even say a
circle-- mark off the length
416
00:20:23,410 --> 00:20:27,590
r_1 onto r_2, assuming r_2 is
the greater of the two lengths.
417
00:20:27,590 --> 00:20:30,690
That difference will
be r_2 minus r_1.
418
00:20:30,690 --> 00:20:33,290
In other words, if I
swing an arc over here,
419
00:20:33,290 --> 00:20:35,360
this would r_1 also.
420
00:20:35,360 --> 00:20:39,100
This distance here
would be r_2 minus r_1.
421
00:20:39,100 --> 00:20:42,990
I take that distance, and
mark it off down here.
422
00:20:42,990 --> 00:20:44,570
And what vector is this?
423
00:20:44,570 --> 00:20:47,100
This is the vector, which
in polar coordinates
424
00:20:47,100 --> 00:20:51,690
would have its magnitude
equal to r_2 minus r_1,
425
00:20:51,690 --> 00:20:55,980
and have its angle equal
to theta_2 minus theta_1.
426
00:20:55,980 --> 00:20:58,340
And all I want you
to see is no matter
427
00:20:58,340 --> 00:21:00,780
how you slice it, just
from this picture alone,
428
00:21:00,780 --> 00:21:06,530
this vector is not the
same as this vector.
429
00:21:06,530 --> 00:21:08,590
And by the way, don't make
the mistake of saying,
430
00:21:08,590 --> 00:21:10,230
gee, the way you've
draw them, they
431
00:21:10,230 --> 00:21:12,250
look like they could
be the same length.
432
00:21:12,250 --> 00:21:15,560
Remember that even if by
coincidence these two vectors
433
00:21:15,560 --> 00:21:19,000
had the same length,
you must remember
434
00:21:19,000 --> 00:21:22,850
that vector equality requires
not just the same magnitude,
435
00:21:22,850 --> 00:21:24,480
but the same direction.
436
00:21:24,480 --> 00:21:27,090
And even with the same
direction, the same sense.
437
00:21:27,090 --> 00:21:29,940
What should be
obvious is, at least
438
00:21:29,940 --> 00:21:33,420
based on this one picture, that
the direction of this vector
439
00:21:33,420 --> 00:21:37,450
certainly is not the same as
the direction of this vector.
440
00:21:37,450 --> 00:21:39,920
In other words, I guess I've
said this many times in part
441
00:21:39,920 --> 00:21:43,790
one, and even though it comes
out like phony facetiousness,
442
00:21:43,790 --> 00:21:45,430
I mean it quite sincerely.
443
00:21:45,430 --> 00:21:48,010
Given two vectors in
polar coordinates,
444
00:21:48,010 --> 00:21:52,690
one certainly has
the right to invent
445
00:21:52,690 --> 00:21:57,910
the vector r_2 minus r_1
comma theta_2 minus theta_1.
446
00:21:57,910 --> 00:22:00,230
You certainly have
the right to do that,
447
00:22:00,230 --> 00:22:03,100
but you don't have the right
to call that A minus B. What
448
00:22:03,100 --> 00:22:05,680
I meant by being facetious
is, you have the right
449
00:22:05,680 --> 00:22:08,060
to call it that, but
you're going to be wrong.
450
00:22:08,060 --> 00:22:10,920
Because A minus B has
already been defined.
451
00:22:10,920 --> 00:22:13,180
And all we have now
is a choice of what
452
00:22:13,180 --> 00:22:17,650
the answer looks like in a
particular coordinate system.
453
00:22:17,650 --> 00:22:19,880
And by the way, just
as a quick review,
454
00:22:19,880 --> 00:22:22,430
assuming that you've studied
different number bases at one
455
00:22:22,430 --> 00:22:25,220
time in your careers, this
has come up many times
456
00:22:25,220 --> 00:22:27,420
in the mathematics
curriculum under the heading
457
00:22:27,420 --> 00:22:30,680
of "Number-versus-Numeral,"
or more generally,
458
00:22:30,680 --> 00:22:34,330
I call it the heading of
"Name-versus-Concept."
459
00:22:34,330 --> 00:22:36,610
For example, consider
the number 6.
460
00:22:36,610 --> 00:22:40,450
6 is this many, and no matter
how you slice it again,
461
00:22:40,450 --> 00:22:41,890
this many is an even number.
462
00:22:41,890 --> 00:22:46,950
It breaks up into bundles
of 2 with none left over.
463
00:22:46,950 --> 00:22:47,960
OK?
464
00:22:47,960 --> 00:22:52,130
If you were to write 6 as a
base 5 numeral, it's what?
465
00:22:52,130 --> 00:22:55,260
One bundle of 5
with 1 left over.
466
00:22:55,260 --> 00:22:58,380
In other words, notice
that the number 6
467
00:22:58,380 --> 00:23:01,200
ends in an odd digit in base 5.
468
00:23:01,200 --> 00:23:03,800
In other words, the
test for evenness
469
00:23:03,800 --> 00:23:06,890
by looking at the last digit
depends on the number base.
470
00:23:06,890 --> 00:23:09,110
In other words, in an
even base, a number
471
00:23:09,110 --> 00:23:11,710
is even if its
units digit is even.
472
00:23:11,710 --> 00:23:14,800
In an odd base, a
number may or may not
473
00:23:14,800 --> 00:23:18,589
be even, depending on whether
its units digit is even.
474
00:23:18,589 --> 00:23:20,380
In other words, the
actual test that you're
475
00:23:20,380 --> 00:23:24,430
using-- the actual convenient
computational property--
476
00:23:24,430 --> 00:23:25,900
depends on the base.
477
00:23:25,900 --> 00:23:29,260
But the important point
is that 6 is always even,
478
00:23:29,260 --> 00:23:30,890
independently of a number base.
479
00:23:30,890 --> 00:23:34,050
In fact, the number
6 as six tally marks
480
00:23:34,050 --> 00:23:38,644
was invented long before
place-value enumeration.
481
00:23:38,644 --> 00:23:39,810
You see what I'm driving at?
482
00:23:39,810 --> 00:23:42,990
In other words, the number
concept stays the same.
483
00:23:42,990 --> 00:23:44,800
The concept stays the same.
484
00:23:44,800 --> 00:23:46,570
What the thing
looks like in terms
485
00:23:46,570 --> 00:23:48,670
of a particular set
of names, that's
486
00:23:48,670 --> 00:23:51,510
what depends on the
names that you choose.
487
00:23:51,510 --> 00:23:52,397
All right?
488
00:23:52,397 --> 00:23:54,230
I think that is very,
very important to see.
489
00:23:54,230 --> 00:23:56,760
The vector properties
do not change.
490
00:23:56,760 --> 00:23:59,450
The convenient
computational recipes
491
00:23:59,450 --> 00:24:02,320
do depend on what coordinate
system you're choosing.
492
00:24:02,320 --> 00:24:04,280
And that's one of the
reasons-- the same as
493
00:24:04,280 --> 00:24:07,640
in part one of this course--
why we find it very convenient,
494
00:24:07,640 --> 00:24:10,510
whenever possible, to use
Cartesian coordinates.
495
00:24:10,510 --> 00:24:11,305
OK?
496
00:24:11,305 --> 00:24:14,225
And let me just
summarize this some more.
497
00:24:17,240 --> 00:24:22,460
Structurally, this is
very important here.
498
00:24:22,460 --> 00:24:26,700
Structurally, arrow arithmetic
is the same for both two
499
00:24:26,700 --> 00:24:28,410
and three dimensions.
500
00:24:28,410 --> 00:24:30,890
For example, without
belaboring the point,
501
00:24:30,890 --> 00:24:33,040
whether we're dealing with
two-dimensional arrows
502
00:24:33,040 --> 00:24:36,260
or three-dimensional arrows,
A plus B is B plus A.
503
00:24:36,260 --> 00:24:41,230
A plus (B plus C) is equal
to (A plus B) plus C.
504
00:24:41,230 --> 00:24:44,520
and let me just say, et cetera,
and not belabor this particular
505
00:24:44,520 --> 00:24:48,180
point, that structurally, we
cannot tell the difference
506
00:24:48,180 --> 00:24:51,910
between two-dimensional vectors
and three-dimensional vectors.
507
00:24:51,910 --> 00:24:56,170
What we can tell the
difference between, I guess,
508
00:24:56,170 --> 00:24:57,860
is the geometry.
509
00:24:57,860 --> 00:25:01,070
The geometry may be more
difficult to visualize
510
00:25:01,070 --> 00:25:03,260
in three-space
than in two-space.
511
00:25:03,260 --> 00:25:07,710
But structurally, we
cannot tell the difference.
512
00:25:07,710 --> 00:25:09,840
And I guess this is what
this thing is all about,
513
00:25:09,840 --> 00:25:13,380
that as you read the textbook
and do your assignment,
514
00:25:13,380 --> 00:25:15,990
there is going to be a
tendency on your part
515
00:25:15,990 --> 00:25:18,370
to try to rely on diagrams.
516
00:25:18,370 --> 00:25:22,130
And boy, you have to be an
expert in descriptive geometry
517
00:25:22,130 --> 00:25:26,720
to be able to take a view of
something in three dimensions,
518
00:25:26,720 --> 00:25:31,430
and try to get its true lengths,
as you look along various axes
519
00:25:31,430 --> 00:25:33,080
and lines of sight.
520
00:25:33,080 --> 00:25:35,890
The beauty is going to
be that as we continue on
521
00:25:35,890 --> 00:25:40,160
with this course, we will
use geometry for motivation.
522
00:25:40,160 --> 00:25:42,780
But once we have motivated
things by geometry,
523
00:25:42,780 --> 00:25:47,010
we will extract those rules that
we can use even in cases where
524
00:25:47,010 --> 00:25:48,490
we can't see the picture.
525
00:25:48,490 --> 00:25:51,290
In much the same way, going
back to our first lecture
526
00:25:51,290 --> 00:25:54,380
when we talked about
why b to the 0 equals 1,
527
00:25:54,380 --> 00:25:57,910
we made up our
rules to make sure
528
00:25:57,910 --> 00:26:03,840
that they would conform to the
nice computational recipes that
529
00:26:03,840 --> 00:26:06,970
were prevalent in the
more simple cases.
530
00:26:06,970 --> 00:26:09,480
And we are going to play
this thing to the hilt.
531
00:26:09,480 --> 00:26:12,460
We're going to really
exploit this and explore it
532
00:26:12,460 --> 00:26:14,980
in future lectures, and in
fact, throughout the rest
533
00:26:14,980 --> 00:26:15,880
of this course.
534
00:26:15,880 --> 00:26:18,100
But more about that next time.
535
00:26:18,100 --> 00:26:22,140
And until next time, good bye.
536
00:26:22,140 --> 00:26:24,510
Funding for the
publication of this video
537
00:26:24,510 --> 00:26:29,390
was provided by the Gabriella
and Paul Rosenbaum Foundation.
538
00:26:29,390 --> 00:26:33,560
Help OCW continue to provide
free and open access to MIT
539
00:26:33,560 --> 00:26:37,978
courses by making a donation
at ocw.mit.edu/donate.