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 --> 00:00:30,000
Thank you.
Let's continue with vectors and
8
00:00:30,000 --> 00:00:37,000
operations of them.
Remember we saw the topic
9
00:00:37,000 --> 00:00:46,000
yesterday was dot product.
And remember the definition of
10
00:00:46,000 --> 00:00:51,000
dot product,
well, the dot product of two
11
00:00:51,000 --> 00:00:55,000
vectors is obtained by
multiplying the first component
12
00:00:55,000 --> 00:00:59,000
with the first component,
the second with the second and
13
00:00:59,000 --> 00:01:01,000
so on and summing these and you
get the scalar.
14
00:01:01,000 --> 00:01:05,000
And the geometric
interpretation of that is that
15
00:01:05,000 --> 00:01:09,000
you can also take the length of
A,
16
00:01:09,000 --> 00:01:16,000
take the length of B multiply
them and multiply that by the
17
00:01:16,000 --> 00:01:22,000
cosine of the angle between the
two vectors.
18
00:01:22,000 --> 00:01:34,000
We have seen several
applications of that.
19
00:01:34,000 --> 00:01:48,000
One application is to find
lengths and angles.
20
00:01:48,000 --> 00:01:52,000
For example,
you can use this relation to
21
00:01:52,000 --> 00:01:59,000
give you the cosine of the angle
between two vectors is the dot
22
00:01:59,000 --> 00:02:05,000
product divided by the product
of the lengths.
23
00:02:05,000 --> 00:02:14,000
Another application that we
have is to detect whether two
24
00:02:14,000 --> 00:02:21,000
vectors are perpendicular.
To decide if two vectors are
25
00:02:21,000 --> 00:02:28,000
perpendicular to each other,
all we have to do is compute
26
00:02:28,000 --> 00:02:34,000
our dot product and see if we
get zero.
27
00:02:34,000 --> 00:02:41,000
And one further application
that we did not have time to
28
00:02:41,000 --> 00:02:49,000
discuss yesterday that I will
mention very quickly is to find
29
00:02:49,000 --> 00:02:59,000
components of,
let's say, a vector A along a
30
00:02:59,000 --> 00:03:04,000
direction u.
So some unit vector.
31
00:03:04,000 --> 00:03:09,000
Let me explain.
Let's say that I have some
32
00:03:09,000 --> 00:03:11,000
direction.
For example,
33
00:03:11,000 --> 00:03:13,000
the horizontal axis on this
blackboard.
34
00:03:13,000 --> 00:03:16,000
But it could be any direction
in space.
35
00:03:16,000 --> 00:03:21,000
And, to describe this
direction, maybe I have a unit
36
00:03:21,000 --> 00:03:26,000
vector along this axis.
Let's say that I have any of a
37
00:03:26,000 --> 00:03:32,000
vector A and I want to find out
what is the component of A along
38
00:03:32,000 --> 00:03:36,000
u.
That means what is the length
39
00:03:36,000 --> 00:03:42,000
of this projection of A to the
given direction?
40
00:03:42,000 --> 00:03:55,000
This thing here is the
component of A along u.
41
00:03:55,000 --> 00:04:02,000
Well, how do we find that?
Well, we know that here we have
42
00:04:02,000 --> 00:04:07,000
a right angle.
So this component is just
43
00:04:07,000 --> 00:04:13,000
length A times cosine of the
angle between A and u.
44
00:04:13,000 --> 00:04:18,000
But now that means I should be
able to compute it very easily
45
00:04:18,000 --> 00:04:23,000
because that's the same as
length A times length u times
46
00:04:23,000 --> 00:04:27,000
cosine theta because u is a unit
vector.
47
00:04:27,000 --> 00:04:33,000
It is a unit vector.
That means this is equal to one.
48
00:04:33,000 --> 00:04:41,000
And so that's the same as the
dot product between A and u.
49
00:04:41,000 --> 00:04:43,000
That is very easy.
And, of course,
50
00:04:43,000 --> 00:04:47,000
the most of just cases of that
is say, for example,
51
00:04:47,000 --> 00:04:50,000
we want just to find the
component along i hat,
52
00:04:50,000 --> 00:04:53,000
the unit vector along the x
axis.
53
00:04:53,000 --> 00:04:57,000
Then you do the dot product
with i hat, which is 100.
54
00:04:57,000 --> 00:04:59,000
What you get is the first
component.
55
00:04:59,000 --> 00:05:01,000
And that is,
indeed, the x component of a
56
00:05:01,000 --> 00:05:04,000
vector.
Similarly, say you want the z
57
00:05:04,000 --> 00:05:08,000
component you do the dot product
with k that gives you the last
58
00:05:08,000 --> 00:05:14,000
component of your vector.
But the same works with a unit
59
00:05:14,000 --> 00:05:21,000
vector in any direction.
So what is an application of
60
00:05:21,000 --> 00:05:24,000
that?
Well, for example,
61
00:05:24,000 --> 00:05:30,000
in physics maybe you have seen
situations where you have a
62
00:05:30,000 --> 00:05:35,000
pendulum that swings.
You have maybe some mass at the
63
00:05:35,000 --> 00:05:41,000
end of the string and that mass
swings back and forth on a
64
00:05:41,000 --> 00:05:42,000
circle.
And to analyze this
65
00:05:42,000 --> 00:05:45,000
mechanically you want to use,
of course,
66
00:05:45,000 --> 00:05:50,000
Newton's Laws of Mechanics and
you want to use forces and so
67
00:05:50,000 --> 00:05:54,000
on,
but I claim that components of
68
00:05:54,000 --> 00:05:59,000
vectors are useful here to
understand what happens
69
00:05:59,000 --> 00:06:03,000
geometrically.
What are the forces exerted on
70
00:06:03,000 --> 00:06:10,000
this pendulum?
Well, there is its weight,
71
00:06:10,000 --> 00:06:21,000
which usually points downwards,
and there is the tension of the
72
00:06:21,000 --> 00:06:25,000
string.
And these two forces together
73
00:06:25,000 --> 00:06:30,000
are what explains how this
pendulum is going to move back
74
00:06:30,000 --> 00:06:33,000
and forth.
Now, you could try to
75
00:06:33,000 --> 00:06:36,000
understand the equations of
motion using x,
76
00:06:36,000 --> 00:06:39,000
y coordinates or x,
z or whatever you want to call
77
00:06:39,000 --> 00:06:41,000
them, let's say x,
y.
78
00:06:41,000 --> 00:06:47,000
But really what causes the
pendulum to swing back and forth
79
00:06:47,000 --> 00:06:52,000
and also to somehow stay a
constant distance are phenomenal
80
00:06:52,000 --> 00:06:56,000
relative to this circular
trajectory.
81
00:06:56,000 --> 00:06:59,000
For example,
maybe instead of taking
82
00:06:59,000 --> 00:07:03,000
components along the x and y
axis, we want to look at two
83
00:07:03,000 --> 00:07:09,000
other unit vectors.
We can look at a vector,
84
00:07:09,000 --> 00:07:15,000
let's call it T,
that is tangent to the
85
00:07:15,000 --> 00:07:18,000
trajectory.
Sorry. Can you read that?
86
00:07:18,000 --> 00:07:33,000
It's not very readable.
T is tangent to the trajectory.
87
00:07:33,000 --> 00:07:36,000
And, on the other hand,
we can introduce another
88
00:07:36,000 --> 00:07:42,000
vector.
Let's call that N.
89
00:07:42,000 --> 00:07:50,000
And that one is normal,
perpendicular to the
90
00:07:50,000 --> 00:07:55,000
trajectory.
And so now if you think about
91
00:07:55,000 --> 00:08:00,000
it you can look at the
components of the weight along
92
00:08:00,000 --> 00:08:06,000
the tangent direction and along
the normal direction.
93
00:08:06,000 --> 00:08:13,000
And so the component of F along
the tangent direction is what
94
00:08:13,000 --> 00:08:21,000
causes acceleration in the
direction along the trajectory.
95
00:08:21,000 --> 00:08:23,000
It is what causes the pendulum
to swing back and forth.
96
00:08:38,000 --> 00:08:45,000
And the component along N,
on the other hand.
97
00:08:45,000 --> 00:08:51,000
That is the part of the weight
that tends to pull our mass away
98
00:08:51,000 --> 00:08:54,000
from this point.
It is what is going to be
99
00:08:54,000 --> 00:08:56,000
responsible for the tension of
the string.
100
00:08:56,000 --> 00:09:02,000
It is why the string is taut
and not actually slack and with
101
00:09:02,000 --> 00:09:06,000
things moving all over the
place.
102
00:09:06,000 --> 00:09:18,000
That one is responsible for the
tension of a string.
103
00:09:18,000 --> 00:09:20,000
And now, of course,
if you want to compute things,
104
00:09:20,000 --> 00:09:23,000
well, maybe you will call this
angle theta and then you will
105
00:09:23,000 --> 00:09:27,000
express things explicitly using
sines and cosines and you will
106
00:09:27,000 --> 00:09:29,000
solve for the equations of
motion.
107
00:09:29,000 --> 00:09:32,000
That would be a very
interesting physics problem.
108
00:09:32,000 --> 00:09:35,000
But, to save time,
we are not going to do it.
109
00:09:35,000 --> 00:09:40,000
I'm sure you've seen that in
8.01 or similar classes.
110
00:09:40,000 --> 00:09:48,000
And so to find these components
we will just do dot products.
111
00:09:48,000 --> 00:09:56,000
Any questions?
No.
112
00:09:56,000 --> 00:10:01,000
OK.
Let's move onto our next topic.
113
00:10:01,000 --> 00:10:06,000
Here we have found things about
lengths, angles and stuff like
114
00:10:06,000 --> 00:10:10,000
that.
One important concept that we
115
00:10:10,000 --> 00:10:17,000
have not understood yet in terms
of vectors is area.
116
00:10:17,000 --> 00:10:25,000
Let's say that we want to find
the area of this pentagon.
117
00:10:25,000 --> 00:10:28,000
Well, how do we compute that
using vectors?
118
00:10:28,000 --> 00:10:32,000
Can we do it using vectors?
Yes we can.
119
00:10:32,000 --> 00:10:36,000
And that is going to be the
goal.
120
00:10:36,000 --> 00:10:42,000
The first thing we should do is
probably simplify the problem.
121
00:10:42,000 --> 00:10:44,000
We don't actually need to
bother with pentagons.
122
00:10:44,000 --> 00:10:48,000
All we need to know are
triangles because,
123
00:10:48,000 --> 00:10:51,000
for example,
you can cut that in three
124
00:10:51,000 --> 00:10:56,000
triangles and then sum the areas
of the triangles.
125
00:10:56,000 --> 00:11:05,000
Perhaps easier,
what is the area of a triangle?
126
00:11:05,000 --> 00:11:12,000
Let's start with a triangle in
the plane.
127
00:11:12,000 --> 00:11:16,000
Well, then we need two vectors
to describe it,
128
00:11:16,000 --> 00:11:20,000
say A and B here.
How do we find the area of a
129
00:11:20,000 --> 00:11:23,000
triangle?
Well, we all know base times
130
00:11:23,000 --> 00:11:25,000
height over two.
What is the base?
131
00:11:25,000 --> 00:11:30,000
What is the height?
The area of this triangle is
132
00:11:30,000 --> 00:11:35,000
going to be one-half of the
base, which is going to be the
133
00:11:35,000 --> 00:11:39,000
length of A.
And the height,
134
00:11:39,000 --> 00:11:47,000
well, if you call theta this
angle, then this is length B
135
00:11:47,000 --> 00:11:51,000
sine theta.
Now, that looks a lot like the
136
00:11:51,000 --> 00:11:54,000
formula we had there,
except for one little catch.
137
00:11:54,000 --> 00:11:58,000
This is a sine instead of a
cosine.
138
00:11:58,000 --> 00:12:03,000
How do we deal with that?
Well, what we could do is first
139
00:12:03,000 --> 00:12:10,000
find the cosine of the angle.
We know how to find the cosine
140
00:12:10,000 --> 00:12:17,000
of the angle using dot products.
Then solve for sine using sine
141
00:12:17,000 --> 00:12:22,000
square plus cosine square equals
one.
142
00:12:22,000 --> 00:12:25,000
And then plug that back into
here.
143
00:12:25,000 --> 00:12:28,000
Well, that works but it is kind
of a very complicated way of
144
00:12:28,000 --> 00:12:30,000
doing it.
So there is an easier way.
145
00:12:30,000 --> 00:12:34,000
And that is going to be
determinants,
146
00:12:34,000 --> 00:12:40,000
but let me explain how we get
to that maybe still doing
147
00:12:40,000 --> 00:12:45,000
elementary geometry and dot
products first.
148
00:12:45,000 --> 00:12:53,000
Let's see.
What we can do is instead of
149
00:12:53,000 --> 00:12:55,000
finding the sine of theta,
well,
150
00:12:55,000 --> 00:12:59,000
we're not good at finding sines
of angles but we are very good
151
00:12:59,000 --> 00:13:00,000
now at finding cosines of
angles.
152
00:13:00,000 --> 00:13:05,000
Maybe we can find another angle
whose cosine is the same as the
153
00:13:05,000 --> 00:13:09,000
sine of theta.
Well, you have already heard
154
00:13:09,000 --> 00:13:14,000
about complimentary angles and
how I take my vector A,
155
00:13:14,000 --> 00:13:18,000
my vector B here and I have an
angle theta.
156
00:13:18,000 --> 00:13:24,000
Well, let's say that I rotate
my vector A by 90 degrees to get
157
00:13:24,000 --> 00:13:34,000
a new vector A prime.
A prime is just A rotated by 90
158
00:13:34,000 --> 00:13:39,000
degrees.
Then the angle between these
159
00:13:39,000 --> 00:13:45,000
two guys, let's say theta prime,
well, theta prime is 90 degrees
160
00:13:45,000 --> 00:13:49,000
or pi over two gradients minus
theta.
161
00:13:49,000 --> 00:13:56,000
So, in particular,
cosine of theta prime is equal
162
00:13:56,000 --> 00:14:01,000
to sine of theta.
In particular,
163
00:14:01,000 --> 00:14:09,000
that means that length A,
length B, sine theta,
164
00:14:09,000 --> 00:14:13,000
which is what we would need to
know in order to find the area
165
00:14:13,000 --> 00:14:17,000
of this triangle is equal to,
well, A and A prime have the
166
00:14:17,000 --> 00:14:21,000
same length so let me replace
that by length of A prime.
167
00:14:21,000 --> 00:14:28,000
I am not changing anything,
length B, cosine theta prime.
168
00:14:28,000 --> 00:14:31,000
And now we have something that
is much easier for us.
169
00:14:31,000 --> 00:14:37,000
Because that is just A prime
dot B.
170
00:14:37,000 --> 00:14:40,000
That looks like a very good
plan.
171
00:14:40,000 --> 00:14:43,000
There is only one small thing
which is we don't know yet how
172
00:14:43,000 --> 00:14:48,000
to find this A prime.
Well, I think it is not very
173
00:14:48,000 --> 00:14:52,000
hard.
Let's see.
174
00:14:52,000 --> 00:14:58,000
Actually, why don't you guys do
the hard work?
175
00:14:58,000 --> 00:15:02,000
Let's say that I have a plane
vector A with two components a1,
176
00:15:02,000 --> 00:15:05,000
a2.
And I want to rotate it
177
00:15:05,000 --> 00:15:10,000
counterclockwise by 90 degrees.
It looks like maybe we should
178
00:15:10,000 --> 00:15:14,000
change some signs somewhere.
Maybe we should do something
179
00:15:14,000 --> 00:15:24,000
with the components.
Can you come up with an idea of
180
00:15:24,000 --> 00:15:34,000
what it might be?
I see a lot of people answering
181
00:15:34,000 --> 00:15:37,000
three.
I see some other answers,
182
00:15:37,000 --> 00:15:41,000
but the majority vote seems to
be number three.
183
00:15:41,000 --> 00:15:49,000
Minus a2 and a1.
I think I agree, so let's see.
184
00:15:49,000 --> 00:16:01,000
Let's say that we have this
vector A with components a1.
185
00:16:01,000 --> 00:16:05,000
So a1 is here.
And a2. So a2 is here.
186
00:16:05,000 --> 00:16:14,000
Let's rotate this box by 90
degrees counterclockwise.
187
00:16:14,000 --> 00:16:19,000
This box ends up there.
It's the same box just flipped
188
00:16:19,000 --> 00:16:23,000
on its side.
This thing here becomes a1 and
189
00:16:23,000 --> 00:16:31,000
this thing here becomes a2.
And that means our new vector A
190
00:16:31,000 --> 00:16:37,000
prime is going to be -- Well,
the first component looks like
191
00:16:37,000 --> 00:16:40,000
an a2 but it is pointing to the
left when a2 is positive.
192
00:16:40,000 --> 00:16:47,000
So, actually, it is minus a2.
And the y component is going to
193
00:16:47,000 --> 00:16:53,000
be the same as this guy,
so it's going to be a1.
194
00:16:53,000 --> 00:16:56,000
If you wanted instead to rotate
clockwise then you would do the
195
00:16:56,000 --> 00:17:00,000
opposite.
You would do a2 minus a1.
196
00:17:00,000 --> 00:17:07,000
Is that reasonably clear for
everyone?
197
00:17:07,000 --> 00:17:14,000
OK.
Let's continue the calculation
198
00:17:14,000 --> 00:17:18,000
there.
A prime, we have decided,
199
00:17:18,000 --> 00:17:24,000
is minus a2,
a1 dot product with let's call
200
00:17:24,000 --> 00:17:33,000
b1 and b2, the components of B.
Then that will be minus a2,
201
00:17:33,000 --> 00:17:36,000
b1 plus a1, b2 plus a1,
b2.
202
00:17:36,000 --> 00:17:43,000
Let me write that the other way
around, a1, b2 minus a2,
203
00:17:43,000 --> 00:17:46,000
b1.
And that is a quantity that you
204
00:17:46,000 --> 00:17:53,000
may already know under the name
of determinant of vectors A and
205
00:17:53,000 --> 00:17:59,000
B, which we write symbolically
using this notation.
206
00:17:59,000 --> 00:18:03,000
We put A and B next to each
other inside a two-by-two table
207
00:18:03,000 --> 00:18:09,000
and we put these verticals bars.
And that means the determinant
208
00:18:09,000 --> 00:18:14,000
of these numbers,
this guy times this guy minus
209
00:18:14,000 --> 00:18:30,000
this guy times this guy.
That is called the determinant.
210
00:18:30,000 --> 00:18:34,000
And geometrically what it
measures is the area,
211
00:18:34,000 --> 00:18:38,000
well, not of a triangle because
we did not divide by two,
212
00:18:38,000 --> 00:18:42,000
but of a parallelogram formed
by A and B.
213
00:18:42,000 --> 00:18:51,000
It measures the area of the
parallelogram with sides A and
214
00:18:51,000 --> 00:18:53,000
B.
And, of course,
215
00:18:53,000 --> 00:18:56,000
if you want the triangle then
you will just divide by two.
216
00:18:56,000 --> 00:19:00,000
The triangle is half the
parallelogram.
217
00:19:00,000 --> 00:19:04,000
There is one small catch.
The area usually is something
218
00:19:04,000 --> 00:19:08,000
that is going to be positive.
This guy here has no reason to
219
00:19:08,000 --> 00:19:16,000
be positive or negative because,
in fact, well,
220
00:19:16,000 --> 00:19:20,000
if you compute things you will
see that where it is supposed to
221
00:19:20,000 --> 00:19:24,000
go negative it depends on
whether A and B are clockwise or
222
00:19:24,000 --> 00:19:26,000
counterclockwise from each
other.
223
00:19:26,000 --> 00:19:29,000
I mean the issue that we have
-- Well,
224
00:19:29,000 --> 00:19:31,000
when we say the area is
one-half length A,
225
00:19:31,000 --> 00:19:34,000
length B,
sine theta that was assuming
226
00:19:34,000 --> 00:19:37,000
that theta is positive,
that its sine is positive.
227
00:19:37,000 --> 00:19:42,000
Otherwise, if theta is negative
maybe we need to take the
228
00:19:42,000 --> 00:19:47,000
absolute value of this.
Just to be more truthful,
229
00:19:47,000 --> 00:19:56,000
I will say the determinant is
either plus or minus the area.
230
00:19:56,000 --> 00:20:13,000
Any questions about this?
Yes.
231
00:20:13,000 --> 00:20:15,000
Sorry.
That is not a dot product.
232
00:20:15,000 --> 00:20:18,000
That is the usual
multiplication.
233
00:20:18,000 --> 00:20:25,000
That is length A times length B
times sine theta.
234
00:20:25,000 --> 00:20:28,000
What does that equal?
And so that is equal to the
235
00:20:28,000 --> 00:20:31,000
area of a parallelogram.
Sorry.
236
00:20:31,000 --> 00:20:39,000
Let me explain that again.
If I have two vectors A and B,
237
00:20:39,000 --> 00:20:45,000
I can form a parallelogram with
them or I can form a triangle.
238
00:20:45,000 --> 00:20:53,000
And so the area of a
parallelogram is equal to length
239
00:20:53,000 --> 00:21:00,000
A, length B, sine theta,
is equal to the determinant of
240
00:21:00,000 --> 00:21:07,000
A and B.
While the area of a triangle is
241
00:21:07,000 --> 00:21:09,000
one-half of that.
242
00:21:21,000 --> 00:21:25,000
And, again, to be truthful,
I should say these things can
243
00:21:25,000 --> 00:21:28,000
be positive or negative.
Depending on whether you count
244
00:21:28,000 --> 00:21:31,000
the angle positively or
negatively, you will get either
245
00:21:31,000 --> 00:21:36,000
the area or minus the area.
The area is actually the
246
00:21:36,000 --> 00:21:39,000
absolute value of these
quantities.
247
00:21:39,000 --> 00:21:49,000
Is that clear?
OK.
248
00:21:49,000 --> 00:21:57,000
Yes.
If you want to compute the
249
00:21:57,000 --> 00:21:59,000
area, you will just take the
absolute value of the
250
00:21:59,000 --> 00:22:00,000
determinant.
251
00:22:15,000 --> 00:22:19,000
I should say the area of a
parallelogram so that it is
252
00:22:19,000 --> 00:22:32,000
completely clear.
Sorry. Do you have a question?
253
00:22:32,000 --> 00:22:34,000
Explain again,
sorry, was the question how a
254
00:22:34,000 --> 00:22:38,000
determinant equals the area of a
parallelogram?
255
00:22:38,000 --> 00:22:41,000
OK.
The area of a parallelogram is
256
00:22:41,000 --> 00:22:45,000
going to be the base times the
height.
257
00:22:45,000 --> 00:22:48,000
Let's take this guy to be the
base.
258
00:22:48,000 --> 00:22:53,000
The length of a base will be
length of A and the height will
259
00:22:53,000 --> 00:22:58,000
be obtained by taking B but only
looking at the vertical part.
260
00:22:58,000 --> 00:23:02,000
That will be length of B times
the sine of theta.
261
00:23:02,000 --> 00:23:06,000
That is how I got the area of a
parallelogram as length A,
262
00:23:06,000 --> 00:23:09,000
length B, sine theta.
And then I did this
263
00:23:09,000 --> 00:23:15,000
manipulation and this trick of
rotating to find a nice formula.
264
00:23:15,000 --> 00:23:23,000
Yes.
You are asking ahead of what I
265
00:23:23,000 --> 00:23:28,000
am going to do in a few minutes.
You are asking about magnitude
266
00:23:28,000 --> 00:23:29,000
of A cross B.
We are going to learn about
267
00:23:29,000 --> 00:23:32,000
cross products in a few minutes.
And the answer is yes,
268
00:23:32,000 --> 00:23:34,000
but cross product is for
vectors in space.
269
00:23:34,000 --> 00:23:38,000
Here I was simplifying things
by doing things just in the
270
00:23:38,000 --> 00:23:43,000
plane.
Just bear with me for five more
271
00:23:43,000 --> 00:23:48,000
minutes and we will do things in
space.
272
00:23:48,000 --> 00:23:55,000
Yes. That is correct.
The way you compute this in
273
00:23:55,000 --> 00:24:00,000
practice is you just do this.
That is how you compute the
274
00:24:00,000 --> 00:24:04,000
determinant.
Yes.
275
00:24:04,000 --> 00:24:09,000
What about three dimensions?
Three dimensions we are going
276
00:24:09,000 --> 00:24:11,000
to do now.
More questions?
277
00:24:11,000 --> 00:24:26,000
Should we move on?
OK. Let's move to space.
278
00:24:26,000 --> 00:24:32,000
There are two things we can do
in space.
279
00:24:32,000 --> 00:24:36,000
And you can look for the volume
of solids or you can look for
280
00:24:36,000 --> 00:24:39,000
the area of surfaces.
Let me start with the easier of
281
00:24:39,000 --> 00:24:42,000
the two.
Let me start with volumes of
282
00:24:42,000 --> 00:24:49,000
solids.
And we will go back to area,
283
00:24:49,000 --> 00:24:53,000
I promise.
I claim that there is also a
284
00:24:53,000 --> 00:24:59,000
notion of determinants in space.
And that is going to tell us
285
00:24:59,000 --> 00:25:08,000
how to find volumes.
Let's say that we have three
286
00:25:08,000 --> 00:25:16,000
vectors A, B and C.
And then the definition of
287
00:25:16,000 --> 00:25:23,000
their determinants going to be,
the notation for that in terms
288
00:25:23,000 --> 00:25:28,000
of the components is the same as
over there.
289
00:25:28,000 --> 00:25:35,000
We put the components of A,
the components of B and the
290
00:25:35,000 --> 00:25:40,000
components of C inside verticals
bars.
291
00:25:40,000 --> 00:25:42,000
And, of course,
I have to give meaning to this.
292
00:25:42,000 --> 00:25:45,000
This will be a number.
And what is that number?
293
00:25:45,000 --> 00:25:50,000
Well, the definition I will
take is that this is a1 times
294
00:25:50,000 --> 00:25:55,000
the determinant of what I get by
looking in this lower right
295
00:25:55,000 --> 00:26:01,000
corner.
The two-by-two determinant b2,
296
00:26:01,000 --> 00:26:08,000
b3, c2, c3.
Then I will subtract a2 times
297
00:26:08,000 --> 00:26:15,000
the determinant of b1,
b3, c1, c3.
298
00:26:15,000 --> 00:26:22,000
And then I will add a3 times
the determinant b1,
299
00:26:22,000 --> 00:26:26,000
b2, c1, c2.
And each of these guys means,
300
00:26:26,000 --> 00:26:30,000
again, you take b2 times c3
minus c2 times b3 and this times
301
00:26:30,000 --> 00:26:33,000
that minus this time that and so
on.
302
00:26:33,000 --> 00:26:35,000
In fact, there is a total of
six terms in here.
303
00:26:35,000 --> 00:26:39,000
And maybe some of you have
already seen a different formula
304
00:26:39,000 --> 00:26:42,000
for three-by-three determinants
where you directly have the six
305
00:26:42,000 --> 00:26:47,000
terms.
It is the same definition.
306
00:26:47,000 --> 00:26:50,000
How to remember the structure
of this formula?
307
00:26:50,000 --> 00:26:55,000
Well, this is called an
expansion according to the first
308
00:26:55,000 --> 00:26:57,000
row.
So we are going to take the
309
00:26:57,000 --> 00:27:02,000
entries in the first row,
a1, a2, a3 And for each of them
310
00:27:02,000 --> 00:27:05,000
we get the term.
Namely we multiply it by a
311
00:27:05,000 --> 00:27:10,000
two-by-two determinant that we
get by deleting the first row
312
00:27:10,000 --> 00:27:16,000
and the column where we are.
Here the coefficient next to
313
00:27:16,000 --> 00:27:21,000
a1, when we delete this column
and this row,
314
00:27:21,000 --> 00:27:24,000
we are left with b2,
b3, c2, c3.
315
00:27:24,000 --> 00:27:29,000
The next one we take a2,
we delete the row that is in it
316
00:27:29,000 --> 00:27:35,000
and the column that it is in.
And we are left with b1,
317
00:27:35,000 --> 00:27:38,000
b3, c1, c3.
And, similarly,
318
00:27:38,000 --> 00:27:41,000
with a3, we take what remains,
which is b1,
319
00:27:41,000 --> 00:27:45,000
b2, c1, c2.
Finally, last but not least,
320
00:27:45,000 --> 00:27:51,000
there is a minus sign here for
the second guy.
321
00:27:51,000 --> 00:28:01,000
It looks like a weird formula.
I mean it is a little bit weird.
322
00:28:01,000 --> 00:28:04,000
But it is a formula that you
should learn because it is
323
00:28:04,000 --> 00:28:06,000
really, really useful for a lot
of things.
324
00:28:06,000 --> 00:28:10,000
I should say if this looks very
artificial to you and you would
325
00:28:10,000 --> 00:28:14,000
like to know more there is more
in the notes,
326
00:28:14,000 --> 00:28:17,000
so read the notes.
They will tell you a bit more
327
00:28:17,000 --> 00:28:20,000
about what this means,
where it comes from and so on.
328
00:28:20,000 --> 00:28:23,000
If you want to know a lot more
then some day you should take
329
00:28:23,000 --> 00:28:26,000
18.06,
Linear Algebra where you will
330
00:28:26,000 --> 00:28:29,000
learn a lot more about
determinants in N dimensional
331
00:28:29,000 --> 00:28:32,000
space with N vectors.
And there is a generalization
332
00:28:32,000 --> 00:28:36,000
of this in arbitrary dimensions.
In this class,
333
00:28:36,000 --> 00:28:39,000
we will only deal with two or
three dimensions.
334
00:28:39,000 --> 00:28:44,000
Yes.
Why is the negative there?
335
00:28:44,000 --> 00:28:45,000
Well, that is a very good
question.
336
00:28:45,000 --> 00:28:49,000
It has to be there so that this
will actually equal,
337
00:28:49,000 --> 00:28:53,000
well, what I am going to say
right now is that this will give
338
00:28:53,000 --> 00:28:55,000
us the volume of [a box?]
with sides A,
339
00:28:55,000 --> 00:28:57,000
B, C.
And the formula just doesn't
340
00:28:57,000 --> 00:28:59,000
work if you don't put the
negative.
341
00:28:59,000 --> 00:29:02,000
There is a more fundamental
reason which has to do with
342
00:29:02,000 --> 00:29:06,000
orientation of space and the
fact that if you switch two
343
00:29:06,000 --> 00:29:09,000
coordinates in space then
basically you change what is
344
00:29:09,000 --> 00:29:12,000
called the handedness of the
coordinates.
345
00:29:12,000 --> 00:29:14,000
If you look at your right hand
and your left hand,
346
00:29:14,000 --> 00:29:16,000
they are not actually the same.
They are mirror images.
347
00:29:16,000 --> 00:29:18,000
And, if you squared two
coordinate axes,
348
00:29:18,000 --> 00:29:21,000
that is what you get.
That is the fundamental reason
349
00:29:21,000 --> 00:29:24,000
for the minus.
Again, we don't need to think
350
00:29:24,000 --> 00:29:33,000
too much about that.
All we need in this class is
351
00:29:33,000 --> 00:29:38,000
the formula.
Why do we care about this
352
00:29:38,000 --> 00:29:43,000
formula?
It is because of the theorem
353
00:29:43,000 --> 00:29:52,000
that says that geometrically the
determinant of the three vectors
354
00:29:52,000 --> 00:29:58,000
A, B, C is, again,
plus or minus.
355
00:29:58,000 --> 00:30:00,000
This determinant could be
positive or negative.
356
00:30:00,000 --> 00:30:03,000
See those minuses and all sorts
of stuff.
357
00:30:03,000 --> 00:30:14,000
Plus or minus the volume of the
parallelepiped.
358
00:30:14,000 --> 00:30:20,000
That is just a fancy name for a
box with parallelogram sides,
359
00:30:20,000 --> 00:30:24,000
in case you wonder,
with sides A,
360
00:30:24,000 --> 00:30:29,000
B and C.
You take the three vectors A,
361
00:30:29,000 --> 00:30:35,000
B and C and you form a box
whose sides are all
362
00:30:35,000 --> 00:30:44,000
parallelograms.
And when its volume is going to
363
00:30:44,000 --> 00:30:59,000
be the determinant.
Other questions?
364
00:30:59,000 --> 00:31:11,000
I'm sorry.
I cannot quite hear you.
365
00:31:11,000 --> 00:31:12,000
Yes.
We are going to see how to do
366
00:31:12,000 --> 00:31:14,000
it geometrically without a
determinant,
367
00:31:14,000 --> 00:31:17,000
but then you will see that you
actually need a determinant to
368
00:31:17,000 --> 00:31:21,000
compute it no matter what.
We are going to go back to this
369
00:31:21,000 --> 00:31:24,000
and see another formula for
volume, but you will see that
370
00:31:24,000 --> 00:31:26,000
really I am cheating.
I mean somehow computationally
371
00:31:26,000 --> 00:31:30,000
the only way to compute it is
really to use a determinant.
372
00:31:43,000 --> 00:31:44,000
That is correct.
In general, I mean,
373
00:31:44,000 --> 00:31:47,000
actually, I could say if you
look at the two-by-two
374
00:31:47,000 --> 00:31:50,000
determinant, see,
you can also explain it in
375
00:31:50,000 --> 00:31:54,000
terms of this extension.
If you take a1 and multiply by
376
00:31:54,000 --> 00:31:57,000
this one-by-one determinant b2,
then you take a2 and you
377
00:31:57,000 --> 00:32:00,000
multiply it by this one-by-one
determinant b1 but you put a
378
00:32:00,000 --> 00:32:02,000
minus sign.
And in general,
379
00:32:02,000 --> 00:32:06,000
indeed, when you expand,
you would stop putting plus,
380
00:32:06,000 --> 00:32:08,000
minus, plus,
minus alternating.
381
00:32:08,000 --> 00:32:15,000
More about that in 18.06.
Yes.
382
00:32:15,000 --> 00:32:18,000
There is a way to do it based
on other rows as well,
383
00:32:18,000 --> 00:32:20,000
but then you have to be very
careful with the sign vectors.
384
00:32:20,000 --> 00:32:23,000
I will refer you to the notes
for that.
385
00:32:23,000 --> 00:32:25,000
I mean you could also do it
with a column,
386
00:32:25,000 --> 00:32:28,000
by the way.
I mean be careful about the
387
00:32:28,000 --> 00:32:30,000
sign rules.
Given how little we will use
388
00:32:30,000 --> 00:32:33,000
determinants in this class,
I mean we will use them in a
389
00:32:33,000 --> 00:32:36,000
way that is fundamental,
but we won't compute much.
390
00:32:36,000 --> 00:32:47,000
Let's say this is going to be
enough for us for now.
391
00:32:47,000 --> 00:32:50,000
After determinants now I can
tell you about cross product.
392
00:32:50,000 --> 00:32:53,000
And cross product is going to
be the answer to your question
393
00:32:53,000 --> 00:32:54,000
about area.
394
00:33:32,000 --> 00:33:45,000
OK.
Let me move onto cross product.
395
00:33:45,000 --> 00:33:53,000
Cross product is something that
you can apply to two vectors in
396
00:33:53,000 --> 00:33:56,000
space.
And by that I mean really in
397
00:33:56,000 --> 00:33:59,000
three-dimensional space.
This is something that is
398
00:33:59,000 --> 00:34:05,000
specific to three dimensions.
The definition A cross B -- It
399
00:34:05,000 --> 00:34:11,000
is important to really do your
multiplication symbol well so
400
00:34:11,000 --> 00:34:16,000
that you don't mistake it with a
dot product.
401
00:34:16,000 --> 00:34:23,000
Well, that is going to be a
vector.
402
00:34:23,000 --> 00:34:26,000
That is another reason not to
confuse it with dot product.
403
00:34:26,000 --> 00:34:30,000
Dot product gives you a number.
Cross product gives you a
404
00:34:30,000 --> 00:34:32,000
vector.
They are really completely
405
00:34:32,000 --> 00:34:35,000
different operations.
They are both called product
406
00:34:35,000 --> 00:34:38,000
because someone could not come
up with a better name,
407
00:34:38,000 --> 00:34:42,000
but they are completely
different operations.
408
00:34:42,000 --> 00:34:45,000
What do we do to do the cross
product of A and B?
409
00:34:45,000 --> 00:34:47,000
Well, we do something very
strange.
410
00:34:47,000 --> 00:34:50,000
Just as I have told you that a
determinant is something where
411
00:34:50,000 --> 00:34:54,000
we put numbers and we get a
number, I am going to violate my
412
00:34:54,000 --> 00:34:59,000
own rule.
I am going to put together a
413
00:34:59,000 --> 00:35:06,000
determinant in which -- Well,
the last two rows are the
414
00:35:06,000 --> 00:35:11,000
components of the vectors A and
B but the first row strangely
415
00:35:11,000 --> 00:35:15,000
consists for unit vectors i,
j, k.
416
00:35:15,000 --> 00:35:19,000
What does that mean?
Well, that is not a determinant
417
00:35:19,000 --> 00:35:21,000
in the usual sense.
If you try to put that into
418
00:35:21,000 --> 00:35:24,000
your calculator,
it will tell you there is an
419
00:35:24,000 --> 00:35:26,000
error.
I don't know how to put vectors
420
00:35:26,000 --> 00:35:28,000
in there.
I want numbers.
421
00:35:28,000 --> 00:35:32,000
What is means is it is symbolic
notation that helps you remember
422
00:35:32,000 --> 00:35:35,000
what the formula is.
The actual formula is,
423
00:35:35,000 --> 00:35:40,000
well, you use this definition.
And, if you use that
424
00:35:40,000 --> 00:35:47,000
definition, you see that it is i
hat times some number.
425
00:35:47,000 --> 00:35:55,000
Let me write it as determinant
of a2, a3, b2,
426
00:35:55,000 --> 00:36:02,000
b3 times i hat minus
determinant a1,
427
00:36:02,000 --> 00:36:11,000
a3, b1, b3, j hat plus a1,
a2, b1, b2, k hat.
428
00:36:11,000 --> 00:36:15,000
And so that is the actual
definition in a way that makes
429
00:36:15,000 --> 00:36:18,000
complete sense,
but to remember this formula
430
00:36:18,000 --> 00:36:23,000
without too much trouble it is
much easier to think about it in
431
00:36:23,000 --> 00:36:27,000
these terms here.
That is the definition and it
432
00:36:27,000 --> 00:36:30,000
gives you a vector.
Now, as usual with definitions,
433
00:36:30,000 --> 00:36:32,000
the question is what is it good
for?
434
00:36:32,000 --> 00:36:36,000
What is the geometric meaning
of this very strange operation?
435
00:36:36,000 --> 00:36:48,000
Why do we bother to do that?
Here is what it does
436
00:36:48,000 --> 00:36:52,000
geometrically.
Remember a vector has two
437
00:36:52,000 --> 00:36:56,000
different things.
It has a length and it has a
438
00:36:56,000 --> 00:37:01,000
direction.
Let's start with the length.
439
00:37:01,000 --> 00:37:15,000
A length of a cross product is
the area of the parallelogram in
440
00:37:15,000 --> 00:37:24,000
space formed by the vectors A
and B.
441
00:37:24,000 --> 00:37:27,000
Now, if you have a
parallelogram in space,
442
00:37:27,000 --> 00:37:31,000
you can find its area just by
doing this calculation when you
443
00:37:31,000 --> 00:37:33,000
know the coordinates of the
points.
444
00:37:33,000 --> 00:37:35,000
You do this calculation and
then you take the length.
445
00:37:35,000 --> 00:37:40,000
You take this squared plus that
squared plus that squared,
446
00:37:40,000 --> 00:37:43,000
square root.
It looks like a very
447
00:37:43,000 --> 00:37:47,000
complicated formula but it works
and, actually,
448
00:37:47,000 --> 00:37:49,000
it is the simplest way to do
it.
449
00:37:49,000 --> 00:37:52,000
This time we don't actually
need to put plus or minus
450
00:37:52,000 --> 00:37:55,000
because the length of a vector
is always positive.
451
00:37:55,000 --> 00:38:00,000
We don't have to worry about
that.
452
00:38:00,000 --> 00:38:04,000
And what is even more magical
is that not only is the length
453
00:38:04,000 --> 00:38:07,000
remarkable but the direction is
also remarkable.
454
00:38:07,000 --> 00:38:24,000
The direction of A cross B is
perpendicular to the plane of a
455
00:38:24,000 --> 00:38:33,000
parallelogram.
Our two vectors A and B
456
00:38:33,000 --> 00:38:41,000
together in a plane.
What I am telling you is that
457
00:38:41,000 --> 00:38:51,000
for vector A cross B will point,
will stick straight out of that
458
00:38:51,000 --> 00:38:56,000
plane perpendicularly to it.
In fact, I would have to be
459
00:38:56,000 --> 00:38:58,000
more precise.
There are two ways that you can
460
00:38:58,000 --> 00:39:02,000
be perpendicular to this plane.
You can be perpendicular
461
00:39:02,000 --> 00:39:06,000
pointing up or pointing down.
How do I decide which?
462
00:39:06,000 --> 00:39:16,000
Well, there is something called
the right-hand rule.
463
00:39:16,000 --> 00:39:18,000
What does the right-hand rule
say?
464
00:39:18,000 --> 00:39:21,000
Well, there are various
versions for right-hand rule
465
00:39:21,000 --> 00:39:23,000
depending on which country you
learn about it.
466
00:39:23,000 --> 00:39:26,000
In France, given the culture,
you even learn about it in
467
00:39:26,000 --> 00:39:28,000
terms of a cork screw and a wine
bottle.
468
00:39:28,000 --> 00:39:33,000
I will just use the usual
version here.
469
00:39:33,000 --> 00:39:35,000
You take your right hand.
If you are left-handed,
470
00:39:35,000 --> 00:39:38,000
remember to take your right
hand and not the left one.
471
00:39:38,000 --> 00:39:43,000
The other right, OK?
Then place your hand to point
472
00:39:43,000 --> 00:39:46,000
in the direction of A.
Let's say my right hand is
473
00:39:46,000 --> 00:39:50,000
going in that direction.
Now, curl your fingers so that
474
00:39:50,000 --> 00:39:54,000
they point towards B.
Here that would be kind of into
475
00:39:54,000 --> 00:39:56,000
the blackboard.
Don't snap any bones.
476
00:39:56,000 --> 00:40:00,000
If it doesn't quite work then
rotate your arms so that you can
477
00:40:00,000 --> 00:40:04,000
actually physically do it.
Then get your thumb to stick
478
00:40:04,000 --> 00:40:07,000
straight out.
Well, here my thumb is going to
479
00:40:07,000 --> 00:40:11,000
go up.
And that tells me that A cross
480
00:40:11,000 --> 00:40:16,000
B will go up.
Let me write that down while
481
00:40:16,000 --> 00:40:19,000
you experiment with it.
Again, try not to enjoy
482
00:40:19,000 --> 00:40:20,000
yourselves.
483
00:40:30,000 --> 00:40:39,000
First, your right hand points
parallel to vector A.
484
00:40:39,000 --> 00:40:47,000
Then your fingers point in the
direction of B.
485
00:40:47,000 --> 00:40:53,000
Then your thumb,
when you stick it out,
486
00:40:53,000 --> 00:41:00,000
is going to point in the
direction of A cross B.
487
00:41:00,000 --> 00:41:29,000
Let's do a quick example.
Where is my quick example? Here.
488
00:41:29,000 --> 00:41:32,000
Let's take i cross j.
489
00:41:40,000 --> 00:41:47,000
I see most of you going in the
right direction.
490
00:41:47,000 --> 00:41:51,000
If you have it pointing in the
wrong direction,
491
00:41:51,000 --> 00:41:56,000
it might mean that you are
using your left hand,
492
00:41:56,000 --> 00:42:01,000
for example.
Example, I claim that i cross j
493
00:42:01,000 --> 00:42:07,000
equals k.
Let's see. I points towards us.
494
00:42:07,000 --> 00:42:12,000
J point to our right.
I guess this is your right.
495
00:42:12,000 --> 00:42:16,000
I think.
And then your thumb is going to
496
00:42:16,000 --> 00:42:19,000
point up.
That tells us it is roughly
497
00:42:19,000 --> 00:42:21,000
pointing up.
And, of course,
498
00:42:21,000 --> 00:42:24,000
the length should be one
because if you take the unit
499
00:42:24,000 --> 00:42:27,000
square in the x,
y plane, its area is one.
500
00:42:27,000 --> 00:42:29,000
And the direction should be
vertical.
501
00:42:29,000 --> 00:42:34,000
Because it should be
perpendicular to the x,
502
00:42:34,000 --> 00:42:37,000
y plane.
It looks like i cross j will be
503
00:42:37,000 --> 00:42:41,000
k.
Well, let's check with the
504
00:42:41,000 --> 00:42:43,000
definition i,
j, k.
505
00:42:43,000 --> 00:42:51,000
What is i? I is one, zero, zero.
J is zero, one, zero.
506
00:42:51,000 --> 00:42:58,000
The coefficient of i will be
zero times zero minus zero times
507
00:42:58,000 --> 00:43:00,000
one.
That is zero.
508
00:43:00,000 --> 00:43:04,000
The coefficient of j will be
one time zero minus zero times
509
00:43:04,000 --> 00:43:06,000
zero, that is a zero,
minus zero j.
510
00:43:06,000 --> 00:43:11,000
It doesn't matter.
And the coefficient of k will
511
00:43:11,000 --> 00:43:14,000
be one times one,
that is one,
512
00:43:14,000 --> 00:43:17,000
minus zero times zero,
so one k.
513
00:43:17,000 --> 00:43:22,000
So we do get i cross j equals k
both ways.
514
00:43:22,000 --> 00:43:24,000
In this case,
it is easier to do it
515
00:43:24,000 --> 00:43:27,000
geometrically.
If I give you no complicated
516
00:43:27,000 --> 00:43:32,000
vectors, probably you will
actually want to do the
517
00:43:32,000 --> 00:43:41,000
calculation.
Any questions? Yes.
518
00:43:41,000 --> 00:43:45,000
The coefficient of k,
remember I delete the first row
519
00:43:45,000 --> 00:43:50,000
and the last column so I get
this two-by-two determinant.
520
00:43:50,000 --> 00:43:54,000
And that two-by-two determinant
is one times one minus zero
521
00:43:54,000 --> 00:43:56,000
times zero so that gives me a
one.
522
00:43:56,000 --> 00:43:59,000
That is what you do with
two-by-two determinants.
523
00:43:59,000 --> 00:44:03,000
Similarly for [UNINTELLIGIBLE],
but [UNINTELLIGIBLE]
524
00:44:03,000 --> 00:44:11,000
turn out to be zero.
More questions?
525
00:44:11,000 --> 00:44:14,000
Yes.
Let me repeat how I got the one
526
00:44:14,000 --> 00:44:18,000
in front of k.
Remember the definition of a
527
00:44:18,000 --> 00:44:24,000
determinant I expand according
to the entries in the first row.
528
00:44:24,000 --> 00:44:28,000
When I get to k what I do is
delete the first row and I
529
00:44:28,000 --> 00:44:32,000
delete the last column,
the column that contains k.
530
00:44:32,000 --> 00:44:37,000
I delete these guys and these
guys and I am left with this
531
00:44:37,000 --> 00:44:41,000
two-by-two determinant.
Now, a two-by-two determinant,
532
00:44:41,000 --> 00:44:47,000
you multiply according to this
downward diagonal and then minus
533
00:44:47,000 --> 00:44:50,000
this times that.
One times one,
534
00:44:50,000 --> 00:44:55,000
let me see here,
I got one k because that is one
535
00:44:55,000 --> 00:45:00,000
times one minus zero times zero
equals one.
536
00:45:00,000 --> 00:45:03,000
Sorry.
That is really hard to read.
537
00:45:03,000 --> 00:45:11,000
Maybe it will be easier that
way.
538
00:45:11,000 --> 00:45:19,000
Yes.
Let's try.
539
00:45:19,000 --> 00:45:23,000
If I do the same for i,
I think I will also get zero.
540
00:45:23,000 --> 00:45:28,000
Let's do the same for i.
I take i, I delete the first
541
00:45:28,000 --> 00:45:33,000
row, I delete the first column,
I get this two-by-two
542
00:45:33,000 --> 00:45:36,000
determinant here and I get zero
times zero,
543
00:45:36,000 --> 00:45:39,000
that is zero,
minus zero times one.
544
00:45:39,000 --> 00:45:43,000
That is the other trick
question.
545
00:45:43,000 --> 00:45:49,000
Zero times one is zero as well.
So that zero minus zero is
546
00:45:49,000 --> 00:45:52,000
zero.
I hope on Monday you should get
547
00:45:52,000 --> 00:45:55,000
more practice in recitation
about how to compute
548
00:45:55,000 --> 00:45:58,000
determinants.
Hopefully, it will become very
549
00:45:58,000 --> 00:46:01,000
easy for you all to compute this
next.
550
00:46:01,000 --> 00:46:04,000
I know the first time it is
kind of a shock because there
551
00:46:04,000 --> 00:46:07,000
are a lot of numbers and a lot
of things to do.
552
00:47:02,000 --> 00:47:08,000
Let me return to the question
that you asked a bit earlier
553
00:47:08,000 --> 00:47:13,000
about how do you find actually
volume if I don't want to know
554
00:47:13,000 --> 00:47:24,000
about determinants?
Well, let's have another look
555
00:47:24,000 --> 00:47:31,000
at the volume.
Let's say that I have three
556
00:47:31,000 --> 00:47:37,000
vectors.
Let me put them this way,
557
00:47:37,000 --> 00:47:43,000
A, B and C.
And let's try to see how else I
558
00:47:43,000 --> 00:47:49,000
could think about the volume of
this box.
559
00:47:49,000 --> 00:47:54,000
Probably you know that the
volume of a parallelepiped is
560
00:47:54,000 --> 00:47:57,000
the area of a base times the
height.
561
00:47:57,000 --> 00:48:04,000
Sorry.
The volume is the area of a
562
00:48:04,000 --> 00:48:12,000
base times the height.
How do we do that in practice?
563
00:48:12,000 --> 00:48:15,000
Well, what is the area of a
base?
564
00:48:15,000 --> 00:48:21,000
The base is a parallelogram in
space with sides B and C.
565
00:48:21,000 --> 00:48:23,000
How do we find the area of the
parallelogram in space?
566
00:48:23,000 --> 00:48:28,000
Well, we just discovered that.
We can do it by taking that
567
00:48:28,000 --> 00:48:30,000
cross product.
The area of a base,
568
00:48:30,000 --> 00:48:33,000
well, we take the cross product
of B and C.
569
00:48:33,000 --> 00:48:36,000
That is not quite it because
this is a vector.
570
00:48:36,000 --> 00:48:40,000
We would like a number while we
take its length.
571
00:48:40,000 --> 00:48:44,000
That is pretty good.
What about the height?
572
00:48:44,000 --> 00:48:48,000
Well, the height is going to be
the component of A in the
573
00:48:48,000 --> 00:48:51,000
direction that is perpendicular
to the base.
574
00:48:51,000 --> 00:48:53,000
Let's take a direction that is
perpendicular to the base.
575
00:48:53,000 --> 00:48:57,000
Let's call that N,
a unit vector in that
576
00:48:57,000 --> 00:49:00,000
direction.
Then we can get the height by
577
00:49:00,000 --> 00:49:04,000
taking A dot n.
That is what we saw at the
578
00:49:04,000 --> 00:49:10,000
beginning of class that A dot n
will tell me how much A goes in
579
00:49:10,000 --> 00:49:17,000
the direction of n.
Are you still with me?
580
00:49:17,000 --> 00:49:22,000
OK.
Let's keep going.
581
00:49:22,000 --> 00:49:24,000
Let's think about this vector
n.
582
00:49:24,000 --> 00:49:29,000
How do I get it?
Well, I can get it by actually
583
00:49:29,000 --> 00:49:34,000
using cross product as well.
Because I said the direction
584
00:49:34,000 --> 00:49:37,000
perpendicular to two vectors I
can get by taking that cross
585
00:49:37,000 --> 00:49:40,000
product and looking at that
direction.
586
00:49:40,000 --> 00:49:47,000
This is still B cross C length.
And this one is,
587
00:49:47,000 --> 00:49:56,000
so I claim, n can be obtained
by taking D cross C.
588
00:49:56,000 --> 00:49:58,000
Well, that comes in the right
direction but it is not a unit
589
00:49:58,000 --> 00:50:01,000
vector.
How do I get a unit vector?
590
00:50:01,000 --> 00:50:06,000
I divide by the length.
Thanks.
591
00:50:06,000 --> 00:50:14,000
I take B cross C and I divide
by length B cross C.
592
00:50:14,000 --> 00:50:20,000
Well, now I can probably
simplify between these two guys.
593
00:50:20,000 --> 00:50:38,000
And so what I will get -- What
I get out of this is that my
594
00:50:38,000 --> 00:50:53,000
volume equals A dot product with
vector B cross C.
595
00:50:53,000 --> 00:50:55,000
But, of course,
I have to be careful in which
596
00:50:55,000 --> 00:50:56,000
order I do it.
If I do it the other way
597
00:50:56,000 --> 00:50:58,000
around, A dot B,
I get a number.
598
00:50:58,000 --> 00:51:00,000
I cannot cross that.
I really have to do the cross
599
00:51:00,000 --> 00:51:03,000
product first.
I get the new vector.
600
00:51:03,000 --> 00:51:09,000
Then my dot product.
The fact is that the
601
00:51:09,000 --> 00:51:16,000
determinant of A,
B, C is equal to this so-called
602
00:51:16,000 --> 00:51:20,000
triple product.
Well, that looks good
603
00:51:20,000 --> 00:51:23,000
geometrically.
Let's try to check whether it
604
00:51:23,000 --> 00:51:27,000
makes sense with the formulas,
just one small thing.
605
00:51:27,000 --> 00:51:32,000
We saw the determinant is a1
times determinant b2,
606
00:51:32,000 --> 00:51:37,000
b3, c2, c3 minus a2 times
something plus a3 times
607
00:51:37,000 --> 00:51:42,000
something.
I will let you fill in the
608
00:51:42,000 --> 00:51:45,000
numbers.
That is this guy.
609
00:51:45,000 --> 00:51:48,000
What about this guy?
Well, dot product,
610
00:51:48,000 --> 00:51:50,000
we take the first component of
A, that is a1,
611
00:51:50,000 --> 00:51:53,000
we multiply by the first
component of B cross C.
612
00:51:53,000 --> 00:51:55,000
What is the first component of
B cross C?
613
00:51:55,000 --> 00:52:05,000
Well, it is this determinant
b2, b3, c2, c3.
614
00:52:05,000 --> 00:52:09,000
If you put B and C instead of A
and B into there you will get
615
00:52:09,000 --> 00:52:14,000
the i component is this guy plus
a2 times the second component
616
00:52:14,000 --> 00:52:18,000
which is minus some determinant
plus a3 times the third
617
00:52:18,000 --> 00:52:22,000
component which is,
again, a determinant.
618
00:52:22,000 --> 00:52:24,000
And you can check.
You get exactly the same
619
00:52:24,000 --> 00:52:26,000
expression, so everything is
fine.
620
00:52:26,000 --> 00:52:32,000
There is no contradiction in
math just yet.
621
00:52:32,000 --> 00:52:38,000
On Tuesday we will continue
with this and we will start
622
00:52:38,000 --> 00:52:43,000
going into matrices,
equations of planes and so on.
623
00:52:43,000 --> 00:52:46,000
Meanwhile, have a good weekend
and please start working on your
624
00:52:46,000 --> 00:52:49,000
Problem Sets so that you can ask
lots of questions to your TAs on
625
00:52:49,000 --> 00:52:51,000
Monday.