1
00:00:01,000 --> 00:00:03,000
The following content is
provided under a Creative
2
00:00:03,000 --> 00:00:05,000
Commons license.
Your support will help MIT
3
00:00:05,000 --> 00:00:08,000
OpenCourseWare continue to offer
high quality educational
4
00:00:08,000 --> 00:00:13,000
resources for free.
To make a donation or to view
5
00:00:13,000 --> 00:00:18,000
additional materials from
hundreds of MIT courses,
6
00:00:18,000 --> 00:00:23,000
visit MIT OpenCourseWare at
ocw.mit.edu.
7
00:00:23,000 --> 00:00:25,000
So let's start right away with
stuff that we will need to see
8
00:00:25,000 --> 00:00:28,000
before we can go on to more
advanced things.
9
00:00:28,000 --> 00:00:31,000
So, hopefully yesterday in
recitation, you heard a bit
10
00:00:31,000 --> 00:00:34,000
about vectors.
How many of you actually knew
11
00:00:34,000 --> 00:00:39,000
about vectors before that?
OK, that's the vast majority.
12
00:00:39,000 --> 00:00:42,000
If you are not one of those
people, well,
13
00:00:42,000 --> 00:00:45,000
hopefully you'll learn about
vectors right now.
14
00:00:45,000 --> 00:00:48,000
I'm sorry that the learning
curve will be a bit steeper for
15
00:00:48,000 --> 00:00:50,000
the first week.
But hopefully,
16
00:00:50,000 --> 00:00:55,000
you'll adjust fine.
If you have trouble with
17
00:00:55,000 --> 00:00:59,000
vectors, do go to your
recitation instructor's office
18
00:00:59,000 --> 00:01:03,000
hours for extra practice if you
feel the need to.
19
00:01:03,000 --> 00:01:09,000
You will see it's pretty easy.
So, just to remind you,
20
00:01:09,000 --> 00:01:18,000
a vector is a quantity that has
both a direction and a magnitude
21
00:01:18,000 --> 00:01:20,000
of length.
22
00:01:33,000 --> 00:01:38,000
So -- So, concretely the way
you draw a vector is by some
23
00:01:38,000 --> 00:01:40,000
arrow, like that,
OK?
24
00:01:40,000 --> 00:01:43,000
And so, it has a length,
and it's pointing in some
25
00:01:43,000 --> 00:01:45,000
direction.
And, so, now,
26
00:01:45,000 --> 00:01:49,000
the way that we compute things
with vectors,
27
00:01:49,000 --> 00:01:53,000
typically, as we introduce a
coordinate system.
28
00:01:53,000 --> 00:01:57,000
So, if we are in the plane,
x-y-axis, if we are in space,
29
00:01:57,000 --> 00:02:00,000
x-y-z axis.
So, usually I will try to draw
30
00:02:00,000 --> 00:02:04,000
my x-y-z axis consistently to
look like this.
31
00:02:04,000 --> 00:02:07,000
And then, I can represent my
vector in terms of its
32
00:02:07,000 --> 00:02:10,000
components along the coordinate
axis.
33
00:02:10,000 --> 00:02:13,000
So, that means when I have this
row, I can ask,
34
00:02:13,000 --> 00:02:15,000
how much does it go in the x
direction?
35
00:02:15,000 --> 00:02:17,000
How much does it go in the y
direction?
36
00:02:17,000 --> 00:02:20,000
How much does it go in the z
direction?
37
00:02:20,000 --> 00:02:25,000
And, so, let's call this a
vector A.
38
00:02:25,000 --> 00:02:29,000
So, it's more convention.
When we have a vector quantity,
39
00:02:29,000 --> 00:02:32,000
we put an arrow on top to
remind us that it's a vector.
40
00:02:32,000 --> 00:02:35,000
If it's in the textbook,
then sometimes it's in bold
41
00:02:35,000 --> 00:02:39,000
because it's easier to typeset.
If you've tried in your
42
00:02:39,000 --> 00:02:44,000
favorite word processor,
bold is easy and vectors are
43
00:02:44,000 --> 00:02:49,000
not easy.
So, the vector you can try to
44
00:02:49,000 --> 00:02:56,000
decompose terms of unit vectors
directed along the coordinate
45
00:02:56,000 --> 00:02:59,000
axis.
So, the convention is there is
46
00:02:59,000 --> 00:03:03,000
a vector that we call
***amp***lt;i***amp***gt;
47
00:03:03,000 --> 00:03:08,000
hat that points along the x
axis and has length one.
48
00:03:08,000 --> 00:03:10,000
There's a vector called
***amp***lt;j***amp***gt;
49
00:03:10,000 --> 00:03:12,000
hat that does the same along
the y axis,
50
00:03:12,000 --> 00:03:14,000
and the
***amp***lt;k***amp***gt;
51
00:03:14,000 --> 00:03:16,000
hat that does the same along
the z axis.
52
00:03:16,000 --> 00:03:20,000
And, so, we can express any
vector in terms of its
53
00:03:20,000 --> 00:03:24,000
components.
So, the other notation is
54
00:03:24,000 --> 00:03:29,000
***amp***lt;a1,
a2, a3 ***amp***gt;
55
00:03:29,000 --> 00:03:37,000
between these square brackets.
Well, in angular brackets.
56
00:03:37,000 --> 00:03:42,000
So, the length of a vector we
denote by, if you want,
57
00:03:42,000 --> 00:03:47,000
it's the same notation as the
absolute value.
58
00:03:47,000 --> 00:03:50,000
So, that's going to be a
number, as we say,
59
00:03:50,000 --> 00:03:54,000
now, a scalar quantity.
OK, so, a scalar quantity is a
60
00:03:54,000 --> 00:03:58,000
usual numerical quantity as
opposed to a vector quantity.
61
00:03:58,000 --> 00:04:08,000
And, its direction is sometimes
called dir A,
62
00:04:08,000 --> 00:04:13,000
and that can be obtained just
by scaling the vector down to
63
00:04:13,000 --> 00:04:17,000
unit length,
for example,
64
00:04:17,000 --> 00:04:26,000
by dividing it by its length.
So -- Well, there's a lot of
65
00:04:26,000 --> 00:04:32,000
notation to be learned.
So, for example,
66
00:04:32,000 --> 00:04:37,000
if I have two points,
P and Q, then I can draw a
67
00:04:37,000 --> 00:04:42,000
vector from P to Q.
And, that vector is called
68
00:04:42,000 --> 00:04:46,000
vector PQ, OK?
So, maybe we'll call it A.
69
00:04:46,000 --> 00:04:48,000
But, a vector doesn't really
have, necessarily,
70
00:04:48,000 --> 00:04:50,000
a starting point and an ending
point.
71
00:04:50,000 --> 00:04:54,000
OK, so if I decide to start
here and I go by the same
72
00:04:54,000 --> 00:04:57,000
distance in the same direction,
this is also vector A.
73
00:04:57,000 --> 00:05:04,000
It's the same thing.
So, a lot of vectors we'll draw
74
00:05:04,000 --> 00:05:08,000
starting at the origin,
but we don't have to.
75
00:05:08,000 --> 00:05:19,000
So, let's just check and see
how things went in recitation.
76
00:05:19,000 --> 00:05:23,000
So, let's say that I give you
the vector
77
00:05:23,000 --> 00:05:34,000
***amp***lt;3,2,1***amp***gt;.
And so, what do you think about
78
00:05:34,000 --> 00:05:46,000
the length of this vector?
OK, I see an answer forming.
79
00:05:46,000 --> 00:05:49,000
So, a lot of you are answering
the same thing.
80
00:05:49,000 --> 00:05:54,000
Maybe it shouldn't spoil it for
those who haven't given it yet.
81
00:05:54,000 --> 00:05:59,000
OK, I think the overwhelming
vote is in favor of answer
82
00:05:59,000 --> 00:06:02,000
number two.
I see some sixes, I don't know.
83
00:06:02,000 --> 00:06:06,000
That's a perfectly good answer,
too, but hopefully in a few
84
00:06:06,000 --> 00:06:10,000
minutes it won't be I don't know
anymore.
85
00:06:10,000 --> 00:06:17,000
So, let's see.
How do we find -- -- the length
86
00:06:17,000 --> 00:06:24,000
of a vector three,
two, one?
87
00:06:24,000 --> 00:06:30,000
Well, so, this vector,
A, it comes towards us along
88
00:06:30,000 --> 00:06:37,000
the x axis by three units.
It goes to the right along the
89
00:06:37,000 --> 00:06:42,000
y axis by two units,
and then it goes up by one unit
90
00:06:42,000 --> 00:06:46,000
along the z axis.
OK, so, it's pointing towards
91
00:06:46,000 --> 00:06:51,000
here.
That's pretty hard to draw.
92
00:06:51,000 --> 00:06:55,000
So, how do we get its length?
Well, maybe we can start with
93
00:06:55,000 --> 00:06:58,000
something easier,
the length of the vector in the
94
00:06:58,000 --> 00:07:01,000
plane.
So, observe that A is obtained
95
00:07:01,000 --> 00:07:04,000
from a vector,
B, in the plane.
96
00:07:04,000 --> 00:07:09,000
Say, B equals three (i) hat
plus two (j) hat.
97
00:07:09,000 --> 00:07:15,000
And then, we just have to,
still, go up by one unit,
98
00:07:15,000 --> 00:07:17,000
OK?
So, let me try to draw a
99
00:07:17,000 --> 00:07:20,000
picture in this vertical plane
that contains A and B.
100
00:07:20,000 --> 00:07:23,000
If I draw it in the vertical
plane,
101
00:07:23,000 --> 00:07:27,000
so, that's the Z axis,
that's not any particular axis,
102
00:07:27,000 --> 00:07:38,000
then my vector B will go here,
and my vector A will go above
103
00:07:38,000 --> 00:07:43,000
it.
And here, that's one unit.
104
00:07:43,000 --> 00:07:49,000
And, here I have a right angle.
So, I can use the Pythagorean
105
00:07:49,000 --> 00:07:57,000
theorem to find that length A^2
equals length B^2 plus one.
106
00:07:57,000 --> 00:08:00,000
Now, we are reduced to finding
the length of B.
107
00:08:00,000 --> 00:08:02,000
The length of B,
we can again find using the
108
00:08:02,000 --> 00:08:06,000
Pythagorean theorem in the XY
plane because here we have the
109
00:08:06,000 --> 00:08:09,000
right angle.
Here we have three units,
110
00:08:09,000 --> 00:08:12,000
and here we have two units.
OK, so, if you do the
111
00:08:12,000 --> 00:08:15,000
calculations,
you will see that,
112
00:08:15,000 --> 00:08:18,000
well, length of B is square
root of (3^2 2^2),
113
00:08:18,000 --> 00:08:23,000
that's 13.
So, the square root of 13 -- --
114
00:08:23,000 --> 00:08:32,000
and length of A is square root
of length B^2 plus one (square
115
00:08:32,000 --> 00:08:41,000
it if you want) which is going
to be square root of 13 plus one
116
00:08:41,000 --> 00:08:49,000
is the square root of 14,
hence, answer number two which
117
00:08:49,000 --> 00:08:54,000
almost all of you gave.
OK, so the general formula,
118
00:08:54,000 --> 00:09:02,000
if you follow it with it,
in general if we have a vector
119
00:09:02,000 --> 00:09:07,000
with components a1,
a2, a3,
120
00:09:07,000 --> 00:09:16,000
then the length of A is the
square root of a1^2 plus a2^2
121
00:09:16,000 --> 00:09:23,000
plus a3^2.
OK, any questions about that?
122
00:09:23,000 --> 00:09:29,000
Yes?
Yes.
123
00:09:29,000 --> 00:09:32,000
So, in general,
we indeed can consider vectors
124
00:09:32,000 --> 00:09:36,000
in abstract spaces that have any
number of coordinates.
125
00:09:36,000 --> 00:09:38,000
And that you have more
components.
126
00:09:38,000 --> 00:09:40,000
In this class,
we'll mostly see vectors with
127
00:09:40,000 --> 00:09:44,000
two or three components because
they are easier to draw,
128
00:09:44,000 --> 00:09:47,000
and because a lot of the math
that we'll see works exactly the
129
00:09:47,000 --> 00:09:50,000
same way whether you have three
variables or a million
130
00:09:50,000 --> 00:09:52,000
variables.
If we had a factor with more
131
00:09:52,000 --> 00:09:55,000
components, then we would have a
lot of trouble drawing it.
132
00:09:55,000 --> 00:09:58,000
But we could still define its
length in the same way,
133
00:09:58,000 --> 00:10:01,000
by summing the squares of the
components.
134
00:10:01,000 --> 00:10:04,000
So, I'm sorry to say that here,
multi-variable,
135
00:10:04,000 --> 00:10:07,000
multi will mean mostly two or
three.
136
00:10:07,000 --> 00:10:13,000
But, be assured that it works
just the same way if you have
137
00:10:13,000 --> 00:10:20,000
10,000 variables.
Just, calculations are longer.
138
00:10:20,000 --> 00:10:28,000
OK, more questions?
So, what else can we do with
139
00:10:28,000 --> 00:10:31,000
vectors?
Well, another thing that I'm
140
00:10:31,000 --> 00:10:35,000
sure you know how to do with
vectors is to add them to scale
141
00:10:35,000 --> 00:10:39,000
them.
So, vector addition,
142
00:10:39,000 --> 00:10:48,000
so, if you have two vectors,
A and B, then you can form,
143
00:10:48,000 --> 00:10:52,000
their sum, A plus B.
How do we do that?
144
00:10:52,000 --> 00:10:54,000
Well, first,
I should tell you,
145
00:10:54,000 --> 00:10:56,000
vectors, they have this double
life.
146
00:10:56,000 --> 00:10:59,000
They are, at the same time,
geometric objects that we can
147
00:10:59,000 --> 00:11:02,000
draw like this in pictures,
and there are also
148
00:11:02,000 --> 00:11:06,000
computational objects that we
can represent by numbers.
149
00:11:06,000 --> 00:11:09,000
So, every question about
vectors will have two answers,
150
00:11:09,000 --> 00:11:11,000
one geometric,
and one numerical.
151
00:11:11,000 --> 00:11:14,000
OK, so let's start with the
geometric.
152
00:11:14,000 --> 00:11:17,000
So, let's say that I have two
vectors, A and B,
153
00:11:17,000 --> 00:11:21,000
given to me.
And, let's say that I thought
154
00:11:21,000 --> 00:11:24,000
of drawing them at the same
place to start with.
155
00:11:24,000 --> 00:11:28,000
Well, to take the sum,
what I should do is actually
156
00:11:28,000 --> 00:11:33,000
move B so that it starts at the
end of A, at the head of A.
157
00:11:33,000 --> 00:11:38,000
OK, so this is, again, vector B.
So, observe,
158
00:11:38,000 --> 00:11:41,000
this actually forms,
now, a parallelogram,
159
00:11:41,000 --> 00:11:43,000
right?
So, this side is,
160
00:11:43,000 --> 00:11:48,000
again, vector A.
And now, if we take the
161
00:11:48,000 --> 00:11:57,000
diagonal of that parallelogram,
this is what we call A plus B,
162
00:11:57,000 --> 00:12:00,000
OK, so, the idea being that to
move along A plus B,
163
00:12:00,000 --> 00:12:03,000
it's the same as to move first
along A and then along B,
164
00:12:03,000 --> 00:12:09,000
or, along B, then along A.
A plus B equals B plus A.
165
00:12:09,000 --> 00:12:13,000
OK, now, if we do it
numerically,
166
00:12:13,000 --> 00:12:19,000
then all you do is you just add
the first component of A with
167
00:12:19,000 --> 00:12:23,000
the first component of B,
the second with the second,
168
00:12:23,000 --> 00:12:28,000
and the third with the third.
OK, say that A was
169
00:12:28,000 --> 00:12:31,000
***amp***lt;a1,
a2, a3***amp***gt;
170
00:12:31,000 --> 00:12:35,000
B was ***amp***lt;b1,
b2, b3***amp***gt;,
171
00:12:35,000 --> 00:12:40,000
then you just add this way.
OK, so it's pretty
172
00:12:40,000 --> 00:12:44,000
straightforward.
So, for example,
173
00:12:44,000 --> 00:12:48,000
I said that my vector over
there, its components are three,
174
00:12:48,000 --> 00:12:54,000
two, one.
But, I also wrote it as 3i 2j k.
175
00:12:54,000 --> 00:12:57,000
What does that mean?
OK, so I need to tell you first
176
00:12:57,000 --> 00:13:06,000
about multiplying by a scalar.
So, this is about addition.
177
00:13:06,000 --> 00:13:11,000
So, multiplication by a scalar,
it's very easy.
178
00:13:11,000 --> 00:13:15,000
If you have a vector,
A, then you can form a vector
179
00:13:15,000 --> 00:13:20,000
2A just by making it go twice as
far in the same direction.
180
00:13:20,000 --> 00:13:24,000
Or, we can make half A more
modestly.
181
00:13:24,000 --> 00:13:31,000
We can even make minus A,
and so on.
182
00:13:31,000 --> 00:13:35,000
So now, you see,
if I do the calculation,
183
00:13:35,000 --> 00:13:38,000
3i 2j k, well,
what does it mean?
184
00:13:38,000 --> 00:13:43,000
3i is just going to go along
the x axis, but by distance of
185
00:13:43,000 --> 00:13:47,000
three instead of one.
And then, 2j goes two units
186
00:13:47,000 --> 00:13:51,000
along the y axis,
and k goes up by one unit.
187
00:13:51,000 --> 00:13:54,000
Well, if you add these
together, you will go from the
188
00:13:54,000 --> 00:13:58,000
origin, then along the x axis,
then parallel to the y axis,
189
00:13:58,000 --> 00:14:02,000
and then up.
And, you will end up,
190
00:14:02,000 --> 00:14:05,000
indeed, at the endpoint of a
vector.
191
00:14:05,000 --> 00:14:19,000
OK, any questions at this point?
Yes?
192
00:14:19,000 --> 00:14:21,000
Exactly.
To add vectors geometrically,
193
00:14:21,000 --> 00:14:25,000
you just put the head of the
first vector and the tail of the
194
00:14:25,000 --> 00:14:30,000
second vector in the same place.
And then, it's head to tail
195
00:14:30,000 --> 00:14:35,000
addition.
Any other questions?
196
00:14:35,000 --> 00:14:41,000
Yes?
That's correct.
197
00:14:41,000 --> 00:14:43,000
If you subtract two vectors,
that just means you add the
198
00:14:43,000 --> 00:14:45,000
opposite of a vector.
So, for example,
199
00:14:45,000 --> 00:14:49,000
if I wanted to do A minus B,
I would first go along A and
200
00:14:49,000 --> 00:14:52,000
then along minus B,
which would take me somewhere
201
00:14:52,000 --> 00:14:55,000
over there, OK?
So, A minus B,
202
00:14:55,000 --> 00:15:01,000
if you want,
would go from here to here.
203
00:15:01,000 --> 00:15:08,000
OK, so hopefully you've kind of
seen that stuff either before in
204
00:15:08,000 --> 00:15:13,000
your lives, or at least
yesterday.
205
00:15:13,000 --> 00:15:23,000
So, I'm going to use that as an
excuse to move quickly forward.
206
00:15:23,000 --> 00:15:28,000
So, now we are going to learn a
few more operations about
207
00:15:28,000 --> 00:15:31,000
vectors.
And, these operations will be
208
00:15:31,000 --> 00:15:34,000
useful to us when we start
trying to do a bit of geometry.
209
00:15:34,000 --> 00:15:37,000
So, of course,
you've all done some geometry.
210
00:15:37,000 --> 00:15:40,000
But, we are going to see that
geometry can be done using
211
00:15:40,000 --> 00:15:42,000
vectors.
And, in many ways,
212
00:15:42,000 --> 00:15:44,000
it's the right language for
that,
213
00:15:44,000 --> 00:15:47,000
and in particular when we learn
about functions we really will
214
00:15:47,000 --> 00:15:51,000
want to use vectors more than,
maybe, the other kind of
215
00:15:51,000 --> 00:15:54,000
geometry that you've seen
before.
216
00:15:54,000 --> 00:15:56,000
I mean, of course,
it's just a language in a way.
217
00:15:56,000 --> 00:15:59,000
I mean, we are just
reformulating things that you
218
00:15:59,000 --> 00:16:02,000
have seen, you already know
since childhood.
219
00:16:02,000 --> 00:16:07,000
But, you will see that notation
somehow helps to make it more
220
00:16:07,000 --> 00:16:10,000
straightforward.
So, what is dot product?
221
00:16:10,000 --> 00:16:16,000
Well, dot product as a way of
multiplying two vectors to get a
222
00:16:16,000 --> 00:16:21,000
number, a scalar.
And, well, let me start by
223
00:16:21,000 --> 00:16:25,000
giving you a definition in terms
of components.
224
00:16:25,000 --> 00:16:29,000
What we do, let's say that we
have a vector,
225
00:16:29,000 --> 00:16:32,000
A, with components a1,
a2, a3, vector B with
226
00:16:32,000 --> 00:16:34,000
components b1,
b2, b3.
227
00:16:34,000 --> 00:16:38,000
Well, we multiply the first
components by the first
228
00:16:38,000 --> 00:16:43,000
components, the second by the
second, the third by the third.
229
00:16:43,000 --> 00:16:46,000
If you have N components,
you keep going.
230
00:16:46,000 --> 00:16:49,000
And, you sum all of these
together.
231
00:16:49,000 --> 00:16:55,000
OK, and important:
this is a scalar.
232
00:16:55,000 --> 00:16:59,000
OK, you do not get a vector.
You get a number.
233
00:16:59,000 --> 00:17:01,000
I know it sounds completely
obvious from the definition
234
00:17:01,000 --> 00:17:03,000
here,
but in the middle of the action
235
00:17:03,000 --> 00:17:07,000
when you're going to do
complicated problems,
236
00:17:07,000 --> 00:17:14,000
it's sometimes easy to forget.
So, that's the definition.
237
00:17:14,000 --> 00:17:17,000
What is it good for?
Why would we ever want to do
238
00:17:17,000 --> 00:17:20,000
that?
That's kind of a strange
239
00:17:20,000 --> 00:17:23,000
operation.
So, probably to see what it's
240
00:17:23,000 --> 00:17:27,000
good for, I should first tell
you what it is geometrically.
241
00:17:27,000 --> 00:17:29,000
OK, so what does it do
geometrically?
242
00:17:38,000 --> 00:17:42,000
Well, what you do when you
multiply two vectors in this
243
00:17:42,000 --> 00:17:45,000
way,
I claim the answer is equal to
244
00:17:45,000 --> 00:17:51,000
the length of A times the length
of B times the cosine of the
245
00:17:51,000 --> 00:17:59,000
angle between them.
So, I have my vector, A,
246
00:17:59,000 --> 00:18:04,000
and if I have my vector, B,
and I have some angle between
247
00:18:04,000 --> 00:18:06,000
them,
I multiply the length of A
248
00:18:06,000 --> 00:18:10,000
times the length of B times the
cosine of that angle.
249
00:18:10,000 --> 00:18:13,000
So, that looks like a very
artificial operation.
250
00:18:13,000 --> 00:18:16,000
I mean, why would want to do
that complicated multiplication?
251
00:18:16,000 --> 00:18:21,000
Well, the basic answer is it
tells us at the same time about
252
00:18:21,000 --> 00:18:25,000
lengths and about angles.
And, the extra bonus thing is
253
00:18:25,000 --> 00:18:29,000
that it's very easy to compute
if you have components,
254
00:18:29,000 --> 00:18:32,000
see, that formula is actually
pretty easy.
255
00:18:32,000 --> 00:18:39,000
So, OK, maybe I should first
tell you, how do we get this
256
00:18:39,000 --> 00:18:41,000
from that?
Because, you know,
257
00:18:41,000 --> 00:18:44,000
in math, one tries to justify
everything to prove theorems.
258
00:18:44,000 --> 00:18:45,000
So, if you want,
that's the theorem.
259
00:18:45,000 --> 00:18:47,000
That's the first theorem in
18.02.
260
00:18:47,000 --> 00:18:52,000
So, how do we prove the theorem?
How do we check that this is,
261
00:18:52,000 --> 00:18:55,000
indeed, correct using this
definition?
262
00:18:55,000 --> 00:19:06,000
So, in more common language,
what does this geometric
263
00:19:06,000 --> 00:19:11,000
definition mean?
Well, the first thing it means,
264
00:19:11,000 --> 00:19:14,000
before we multiply two vectors,
let's start multiplying a
265
00:19:14,000 --> 00:19:17,000
vector with itself.
That's probably easier.
266
00:19:17,000 --> 00:19:19,000
So, if we multiply a vector,
A, with itself,
267
00:19:19,000 --> 00:19:22,000
using this dot product,
so, by the way,
268
00:19:22,000 --> 00:19:24,000
I should point out,
we put this dot here.
269
00:19:24,000 --> 00:19:28,000
That's why it's called dot
product.
270
00:19:28,000 --> 00:19:33,000
So, what this tells us is we
should get the same thing as
271
00:19:33,000 --> 00:19:38,000
multiplying the length of A with
itself, so, squared,
272
00:19:38,000 --> 00:19:43,000
times the cosine of the angle.
But now, the cosine of an
273
00:19:43,000 --> 00:19:49,000
angle, of zero,
cosine of zero you all know is
274
00:19:49,000 --> 00:19:52,000
one.
OK, so that's going to be
275
00:19:52,000 --> 00:19:56,000
length A^2.
Well, doesn't stand a chance of
276
00:19:56,000 --> 00:19:57,000
being true?
Well, let's see.
277
00:19:57,000 --> 00:20:03,000
If we do AdotA using this
formula, we will get a1^2 a2^2
278
00:20:03,000 --> 00:20:07,000
a3^2.
That is, indeed,
279
00:20:07,000 --> 00:20:14,000
the square of the length.
So, check.
280
00:20:14,000 --> 00:20:18,000
That works.
OK, now, what about two
281
00:20:18,000 --> 00:20:23,000
different vectors?
Can we understand what this
282
00:20:23,000 --> 00:20:27,000
says, and how it relates to
that?
283
00:20:27,000 --> 00:20:33,000
So, let's say that I have two
different vectors,
284
00:20:33,000 --> 00:20:40,000
A and B, and I want to try to
understand what's going on.
285
00:20:40,000 --> 00:20:45,000
So, my claim is that we are
going to be able to understand
286
00:20:45,000 --> 00:20:49,000
the relation between this and
that in terms of the law of
287
00:20:49,000 --> 00:20:52,000
cosines.
So, the law of cosines is
288
00:20:52,000 --> 00:20:56,000
something that tells you about
the length of the third side in
289
00:20:56,000 --> 00:21:00,000
the triangle like this in terms
of these two sides,
290
00:21:00,000 --> 00:21:07,000
and the angle here.
OK, so the law of cosines,
291
00:21:07,000 --> 00:21:11,000
which hopefully you have seen
before, says that,
292
00:21:11,000 --> 00:21:14,000
so let me give a name to this
side.
293
00:21:14,000 --> 00:21:19,000
Let's call this side C,
and as a vector,
294
00:21:19,000 --> 00:21:29,000
C is A minus B.
It's minus B plus A.
295
00:21:29,000 --> 00:21:37,000
So, it's getting a bit
cluttered here.
296
00:21:37,000 --> 00:21:45,000
So, the law of cosines says
that the length of the third
297
00:21:45,000 --> 00:21:53,000
side in this triangle is equal
to length A2 plus length B2.
298
00:21:53,000 --> 00:21:56,000
Well, if I stopped here,
that would be Pythagoras,
299
00:21:56,000 --> 00:22:01,000
but I don't have a right angle.
So, I have a third term which
300
00:22:01,000 --> 00:22:07,000
is twice length A,
length B, cosine theta,
301
00:22:07,000 --> 00:22:10,000
OK?
Has everyone seen this formula
302
00:22:10,000 --> 00:22:13,000
sometime?
I hear some yeah's.
303
00:22:13,000 --> 00:22:16,000
I hear some no's.
Well, it's a fact about,
304
00:22:16,000 --> 00:22:19,000
I mean, you probably haven't
seen it with vectors,
305
00:22:19,000 --> 00:22:22,000
but it's a fact about the side
lengths in a triangle.
306
00:22:22,000 --> 00:22:27,000
And, well, let's say,
if you haven't seen it before,
307
00:22:27,000 --> 00:22:32,000
then this is going to be a
proof of the law of cosines if
308
00:22:32,000 --> 00:22:39,000
you believe this.
Otherwise, it's the other way
309
00:22:39,000 --> 00:22:43,000
around.
So, let's try to see how this
310
00:22:43,000 --> 00:22:47,000
relates to what I'm saying about
the dot product.
311
00:22:47,000 --> 00:22:54,000
So, I've been saying that
length C^2, that's the same
312
00:22:54,000 --> 00:22:56,000
thing as CdotC,
OK?
313
00:22:56,000 --> 00:23:01,000
That, we have checked.
Now, CdotC, well,
314
00:23:01,000 --> 00:23:06,000
C is A minus B.
So, it's A minus B,
315
00:23:06,000 --> 00:23:09,000
dot product,
A minus B.
316
00:23:09,000 --> 00:23:11,000
Now, what do we want to do in a
situation like that?
317
00:23:11,000 --> 00:23:16,000
Well, we want to expand this
into a sum of four terms.
318
00:23:16,000 --> 00:23:19,000
Are we allowed to do that?
Well, we have this dot product
319
00:23:19,000 --> 00:23:22,000
that's a mysterious new
operation.
320
00:23:22,000 --> 00:23:24,000
We don't really know.
Well, the answer is yes,
321
00:23:24,000 --> 00:23:27,000
we can do it.
You can check from this
322
00:23:27,000 --> 00:23:31,000
definition that it behaves in
the usual way in terms of
323
00:23:31,000 --> 00:23:34,000
expanding, vectoring,
and so on.
324
00:23:34,000 --> 00:23:49,000
So, I can write that as AdotA
minus AdotB minus BdotA plus
325
00:23:49,000 --> 00:23:55,000
BdotB.
So, AdotA is length A^2.
326
00:23:55,000 --> 00:23:56,000
Let me jump ahead to the last
term.
327
00:23:56,000 --> 00:24:01,000
BdotB is length B^2,
and then these two terms,
328
00:24:01,000 --> 00:24:04,000
well, they're the same.
You can check from the
329
00:24:04,000 --> 00:24:07,000
definition that AdotB and BdotA
are the same thing.
330
00:24:20,000 --> 00:24:24,000
Well, you see that this term,
I mean, this is the only
331
00:24:24,000 --> 00:24:30,000
difference between these two
formulas for the length of C.
332
00:24:30,000 --> 00:24:34,000
So, if you believe in the law
of cosines, then it tells you
333
00:24:34,000 --> 00:24:39,000
that, yes, this a proof that
AdotB equals length A length B
334
00:24:39,000 --> 00:24:41,000
cosine theta.
Or, vice versa,
335
00:24:41,000 --> 00:24:45,000
if you've never seen the law of
cosines, you are willing to
336
00:24:45,000 --> 00:24:49,000
believe this.
Then, this is the proof of the
337
00:24:49,000 --> 00:24:53,000
law of cosines.
So, the law of cosines,
338
00:24:53,000 --> 00:24:59,000
or this interpretation,
are equivalent to each other.
339
00:24:59,000 --> 00:25:07,000
OK, any questions?
Yes?
340
00:25:07,000 --> 00:25:12,000
So, in the second one there
isn't a cosine theta because I'm
341
00:25:12,000 --> 00:25:16,000
just expanding a dot product.
OK, so I'm just writing C
342
00:25:16,000 --> 00:25:19,000
equals A minus B,
and then I'm expanding this
343
00:25:19,000 --> 00:25:22,000
algebraically.
And then, I get to an answer
344
00:25:22,000 --> 00:25:24,000
that has an A.B.
So then, if I wanted to express
345
00:25:24,000 --> 00:25:27,000
that without a dot product,
then I would have to introduce
346
00:25:27,000 --> 00:25:31,000
a cosine.
And, I would get the same as
347
00:25:31,000 --> 00:25:34,000
that, OK?
So, yeah, if you want,
348
00:25:34,000 --> 00:25:38,000
the next step to recall the law
of cosines would be plug in this
349
00:25:38,000 --> 00:25:43,000
formula for AdotB.
And then you would have a
350
00:25:43,000 --> 00:25:58,000
cosine.
OK, let's keep going.
351
00:25:58,000 --> 00:26:03,000
OK, so what is this good for?
Now that we have a definition,
352
00:26:03,000 --> 00:26:06,000
we should figure out what we
can do with it.
353
00:26:06,000 --> 00:26:11,000
So, what are the applications
of dot product?
354
00:26:11,000 --> 00:26:14,000
Well, will this discover new
applications of dot product
355
00:26:14,000 --> 00:26:17,000
throughout the entire
semester,but let me tell you at
356
00:26:17,000 --> 00:26:20,000
least about those that are
readily visible.
357
00:26:20,000 --> 00:26:33,000
So, one is to compute lengths
and angles, especially angles.
358
00:26:33,000 --> 00:26:39,000
So, let's do an example.
Let's say that,
359
00:26:39,000 --> 00:26:44,000
for example,
I have in space,
360
00:26:44,000 --> 00:26:51,000
I have a point,
P, which is at (1,0,0).
361
00:26:51,000 --> 00:26:55,000
I have a point,
Q, which is at (0,1,0).
362
00:26:55,000 --> 00:26:58,000
So, it's at distance one here,
one here.
363
00:26:58,000 --> 00:27:03,000
And, I have a third point,
R at (0,0,2),
364
00:27:03,000 --> 00:27:07,000
so it's at height two.
And, let's say that I'm
365
00:27:07,000 --> 00:27:11,000
curious, and I'm wondering what
is the angle here?
366
00:27:11,000 --> 00:27:15,000
So, here I have a triangle in
space connect P,
367
00:27:15,000 --> 00:27:20,000
Q, and R, and I'm wondering,
what is this angle here?
368
00:27:20,000 --> 00:27:23,000
OK, so, of course,
one solution is to build a
369
00:27:23,000 --> 00:27:25,000
model and then go and measure
the angle.
370
00:27:25,000 --> 00:27:28,000
But, we can do better than that.
We can just find the angle
371
00:27:28,000 --> 00:27:32,000
using dot product.
So, how would we do that?
372
00:27:32,000 --> 00:27:38,000
Well, so, if we look at this
formula, we see,
373
00:27:38,000 --> 00:27:44,000
so, let's say that we want to
find the angle here.
374
00:27:44,000 --> 00:27:50,000
Well, let's look at the formula
for PQdotPR.
375
00:27:50,000 --> 00:27:56,000
Well, we said it should be
length PQ times length PR times
376
00:27:56,000 --> 00:27:59,000
the cosine of the angle,
OK?
377
00:27:59,000 --> 00:28:01,000
Now, what do we know,
and what do we not know?
378
00:28:01,000 --> 00:28:04,000
Well, certainly at this point
we don't know the cosine of the
379
00:28:04,000 --> 00:28:06,000
angle.
That's what we would like to
380
00:28:06,000 --> 00:28:08,000
find.
The lengths,
381
00:28:08,000 --> 00:28:11,000
certainly we can compute.
We know how to find these
382
00:28:11,000 --> 00:28:14,000
lengths.
And, this dot product we know
383
00:28:14,000 --> 00:28:17,000
how to compute because we have
an easy formula here.
384
00:28:17,000 --> 00:28:20,000
OK, so we can compute
everything else and then find
385
00:28:20,000 --> 00:28:25,000
theta.
So, I'll tell you what we will
386
00:28:25,000 --> 00:28:31,000
do is we will find theta -- --
in this way.
387
00:28:31,000 --> 00:28:34,000
We'll take the dot product of
PQ with PR, and then we'll
388
00:28:34,000 --> 00:28:36,000
divide by the lengths.
389
00:29:14,000 --> 00:29:27,000
OK, so let's see.
So, we said cosine theta is
390
00:29:27,000 --> 00:29:33,000
PQdotPR over length PQ length
PR.
391
00:29:33,000 --> 00:29:36,000
So, let's try to figure out
what this vector,
392
00:29:36,000 --> 00:29:39,000
PQ,
well, to go from P to Q,
393
00:29:39,000 --> 00:29:43,000
I should go minus one unit
along the x direction plus one
394
00:29:43,000 --> 00:29:46,000
unit along the y direction.
And, I'm not moving in the z
395
00:29:46,000 --> 00:29:49,000
direction.
So, to go from P to Q,
396
00:29:49,000 --> 00:29:54,000
I have to move by
***amp***lt;-1,1,0***amp***gt;.
397
00:29:54,000 --> 00:29:59,000
To go from P to R,
I go -1 along the x axis and 2
398
00:29:59,000 --> 00:30:04,000
along the z axis.
So, PR, I claim, is this.
399
00:30:04,000 --> 00:30:12,000
OK, then, the lengths of these
vectors, well,(-1)^2 (1)^2
400
00:30:12,000 --> 00:30:19,000
(0)^2, square root,
and then same thing with the
401
00:30:19,000 --> 00:30:24,000
other one.
OK, so, the denominator will
402
00:30:24,000 --> 00:30:30,000
become the square root of 2,
and there's a square root of 5.
403
00:30:30,000 --> 00:30:34,000
What about the numerator?
Well, so, remember,
404
00:30:34,000 --> 00:30:37,000
to do the dot product,
we multiply this by this,
405
00:30:37,000 --> 00:30:40,000
and that by that,
that by that.
406
00:30:40,000 --> 00:30:45,000
And, we add.
Minus 1 times minus 1 makes 1
407
00:30:45,000 --> 00:30:49,000
plus 1 times 0,
that's 0.
408
00:30:49,000 --> 00:30:55,000
Zero times 2 is 0 again.
So, we will get 1 over square
409
00:30:55,000 --> 00:30:59,000
root of 10.
That's the cosine of the angle.
410
00:30:59,000 --> 00:31:03,000
And, of course if we want the
actual angle,
411
00:31:03,000 --> 00:31:08,000
well, we have to take a
calculator, find the inverse
412
00:31:08,000 --> 00:31:12,000
cosine, and you'll find it's
about 71.5°.
413
00:31:12,000 --> 00:31:18,000
Actually, we'll be using mostly
radians, but for today,
414
00:31:18,000 --> 00:31:26,000
that's certainly more speaking.
OK, any questions about that?
415
00:31:26,000 --> 00:31:29,000
No?
OK, so in particular,
416
00:31:29,000 --> 00:31:32,000
I should point out one thing
that's really neat about the
417
00:31:32,000 --> 00:31:34,000
answer.
I mean, we got this number.
418
00:31:34,000 --> 00:31:37,000
We don't really know what it
means exactly because it mixes
419
00:31:37,000 --> 00:31:39,000
together the lengths and the
angle.
420
00:31:39,000 --> 00:31:41,000
But, one thing that's
interesting here,
421
00:31:41,000 --> 00:31:45,000
it's the sign of the answer,
the fact that we got a positive
422
00:31:45,000 --> 00:31:48,000
number.
So, if you think about it,
423
00:31:48,000 --> 00:31:50,000
the lengths are always
positive.
424
00:31:50,000 --> 00:31:56,000
So, the sign of a dot product
is the same as a sign of cosine
425
00:31:56,000 --> 00:32:00,000
theta.
So, in fact,
426
00:32:00,000 --> 00:32:13,000
the sign of AdotB is going to
be positive if the angle is less
427
00:32:13,000 --> 00:32:17,000
than 90°.
So, that means geometrically,
428
00:32:17,000 --> 00:32:21,000
my two vectors are going more
or less in the same direction.
429
00:32:21,000 --> 00:32:27,000
They make an acute angle.
It's going to be zero if the
430
00:32:27,000 --> 00:32:33,000
angle is exactly 90°,
OK, because that's when the
431
00:32:33,000 --> 00:32:39,000
cosine will be zero.
And, it will be negative if the
432
00:32:39,000 --> 00:32:43,000
angle is more than 90°.
So, that means they go,
433
00:32:43,000 --> 00:32:46,000
however, in opposite
directions.
434
00:32:46,000 --> 00:32:50,000
So, that's basically one way to
think about what dot product
435
00:32:50,000 --> 00:32:54,000
measures.
It measures how much the two
436
00:32:54,000 --> 00:32:58,000
vectors are going along each
other.
437
00:32:58,000 --> 00:33:02,000
OK, and that actually leads us
to the next application.
438
00:33:02,000 --> 00:33:05,000
So, let's see,
did I have a number one there?
439
00:33:05,000 --> 00:33:07,000
Yes.
So, if I had a number one,
440
00:33:07,000 --> 00:33:12,000
I must have number two.
The second application is to
441
00:33:12,000 --> 00:33:16,000
detect orthogonality.
It's to figure out when two
442
00:33:16,000 --> 00:33:21,000
things are perpendicular.
OK, so orthogonality is just a
443
00:33:21,000 --> 00:33:26,000
complicated word from Greek to
say things are perpendicular.
444
00:33:26,000 --> 00:33:34,000
So, let's just take an example.
Let's say I give you the
445
00:33:34,000 --> 00:33:41,000
equation x 2y 3z = 0.
OK, so that defines a certain
446
00:33:41,000 --> 00:33:46,000
set of points in space,
and what do you think the set
447
00:33:46,000 --> 00:33:52,000
of solutions look like if I give
you this equation?
448
00:33:52,000 --> 00:34:01,000
So far I see one,
two, three answers,
449
00:34:01,000 --> 00:34:06,000
OK.
So, I see various competing
450
00:34:06,000 --> 00:34:11,000
answers, but,
yeah, I see a lot of people
451
00:34:11,000 --> 00:34:18,000
voting for answer number four.
I see also some I don't knows,
452
00:34:18,000 --> 00:34:22,000
and some other things.
But, the majority vote seems to
453
00:34:22,000 --> 00:34:26,000
be a plane.
And, indeed that's the correct
454
00:34:26,000 --> 00:34:28,000
answer.
So, how do we see that it's a
455
00:34:28,000 --> 00:34:28,000
plane?
456
00:34:43,000 --> 00:34:49,000
So, I should say,
this is the equation of a
457
00:34:49,000 --> 00:34:52,000
plane.
So, there's many ways to see
458
00:34:52,000 --> 00:34:55,000
that, and I'm not going to give
you all of them.
459
00:34:55,000 --> 00:34:58,000
But, here's one way to think
about it.
460
00:34:58,000 --> 00:35:03,000
So, let's think geometrically
about how to express this
461
00:35:03,000 --> 00:35:09,000
condition in terms of vectors.
So, let's take the origin O,
462
00:35:09,000 --> 00:35:13,000
by convention is the point
(0,0,0).
463
00:35:13,000 --> 00:35:18,000
And, let's take a point,
P, that will satisfy this
464
00:35:18,000 --> 00:35:21,000
equation on it,
so, at coordinates x,
465
00:35:21,000 --> 00:35:24,000
y, z.
So, what does this condition
466
00:35:24,000 --> 00:35:28,000
here mean?
Well, it means the following
467
00:35:28,000 --> 00:35:32,000
thing.
So, let's take the vector, OP.
468
00:35:32,000 --> 00:35:37,000
OK, so vector OP,
of course, has components x,
469
00:35:37,000 --> 00:35:40,000
y, z.
Now, we can think of this as
470
00:35:40,000 --> 00:35:44,000
actually a dot product between
OP and a mysterious vector that
471
00:35:44,000 --> 00:35:47,000
won't remain mysterious for very
long,
472
00:35:47,000 --> 00:35:50,000
namely, the vector one,
two, three.
473
00:35:50,000 --> 00:35:59,000
OK, so, this condition is the
same as OP.A equals zero,
474
00:35:59,000 --> 00:36:03,000
right?
If I take the dot product
475
00:36:03,000 --> 00:36:09,000
OPdotA I get x times one plus y
times two plus z times three.
476
00:36:09,000 --> 00:36:14,000
But now, what does it mean that
the dot product between OP and A
477
00:36:14,000 --> 00:36:19,000
is zero?
Well, it means that OP and A
478
00:36:19,000 --> 00:36:25,000
are perpendicular.
OK, so I have this vector, A.
479
00:36:25,000 --> 00:36:28,000
I'm not going to be able to
draw it realistically.
480
00:36:28,000 --> 00:36:32,000
Let's say it goes this way.
Then, a point,
481
00:36:32,000 --> 00:36:37,000
P, solves this equation exactly
when the vector from O to P is
482
00:36:37,000 --> 00:36:40,000
perpendicular to A.
And, I claim that defines a
483
00:36:40,000 --> 00:36:41,000
plane.
For example,
484
00:36:41,000 --> 00:36:45,000
if it helps you to see it,
take a vertical vector.
485
00:36:45,000 --> 00:36:47,000
What does it mean to be
perpendicular to the vertical
486
00:36:47,000 --> 00:36:49,000
vector?
It means you are horizontal.
487
00:36:49,000 --> 00:36:56,000
It's the horizontal plane.
Here, it's a plane that passes
488
00:36:56,000 --> 00:37:05,000
through the origin and is
perpendicular to this vector,
489
00:37:05,000 --> 00:37:14,000
A.
OK, so what we get is a plane
490
00:37:14,000 --> 00:37:25,000
through the origin perpendicular
to A.
491
00:37:25,000 --> 00:37:29,000
And, in general,
what you should remember is
492
00:37:29,000 --> 00:37:35,000
that two vectors have a dot
product equal to zero if and
493
00:37:35,000 --> 00:37:41,000
only if that's equivalent to the
cosine of the angle between them
494
00:37:41,000 --> 00:37:46,000
is zero.
That means the angle is 90°.
495
00:37:46,000 --> 00:37:51,000
That means A and B are
perpendicular.
496
00:37:51,000 --> 00:37:57,000
So, we have a very fast way of
checking whether two vectors are
497
00:37:57,000 --> 00:38:01,000
perpendicular.
So, one additional application
498
00:38:01,000 --> 00:38:05,000
I think we'll see actually
tomorrow is to find the
499
00:38:05,000 --> 00:38:10,000
components of a vector along a
certain direction.
500
00:38:10,000 --> 00:38:13,000
So, I claim we can use this
intuition I gave about dot
501
00:38:13,000 --> 00:38:16,000
product telling us how much to
vectors go in the same direction
502
00:38:16,000 --> 00:38:19,000
to actually give a precise
meaning to the notion of
503
00:38:19,000 --> 00:38:22,000
component for vector,
not just along the x,
504
00:38:22,000 --> 00:38:27,000
y, or z axis,
but along any direction in
505
00:38:27,000 --> 00:38:31,000
space.
So, I think I should probably
506
00:38:31,000 --> 00:38:34,000
stop here.
But, I will see you tomorrow at
507
00:38:34,000 --> 00:38:38,000
2:00 here, and we'll learn more
about that and about cross
508
00:38:38,000 --> 00:38:44,000
products.