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Thank you.
Let's continue with vectors and
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00:00:30 --> 00:00:37
operations of them.
Remember we saw the topic
9
00:00:37 --> 00:00:46
yesterday was dot product.
And remember the definition of
10
00:00:46 --> 00:00:51
dot product,
well, the dot product of two
11
00:00:51 --> 00:00:55
vectors is obtained by
multiplying the first component
12
00:00:55 --> 00:00:59
with the first component,
the second with the second and
13
00:00:59 --> 00:01:01
so on and summing these and you
get the scalar.
14
00:01:01 --> 00:01:05
And the geometric
interpretation of that is that
15
00:01:05 --> 00:01:09
you can also take the length of
A,
16
00:01:09 --> 00:01:16
take the length of B multiply
them and multiply that by the
17
00:01:16 --> 00:01:22
cosine of the angle between the
two vectors.
18
00:01:22 --> 00:01:34
We have seen several
applications of that.
19
00:01:34 --> 00:01:48
One application is to find
lengths and angles.
20
00:01:48 --> 00:01:52
For example,
you can use this relation to
21
00:01:52 --> 00:01:59
give you the cosine of the angle
between two vectors is the dot
22
00:01:59 --> 00:02:05
product divided by the product
of the lengths.
23
00:02:05 --> 00:02:14
Another application that we
have is to detect whether two
24
00:02:14 --> 00:02:21
vectors are perpendicular.
To decide if two vectors are
25
00:02:21 --> 00:02:28
perpendicular to each other,
all we have to do is compute
26
00:02:28 --> 00:02:34
our dot product and see if we
get zero.
27
00:02:34 --> 00:02:41
And one further application
that we did not have time to
28
00:02:41 --> 00:02:49
discuss yesterday that I will
mention very quickly is to find
29
00:02:49 --> 00:02:59
components of,
let's say, a vector A along a
30
00:02:59 --> 00:03:04
direction u.
So some unit vector.
31
00:03:04 --> 00:03:09
Let me explain.
Let's say that I have some
32
00:03:09 --> 00:03:11
direction.
For example,
33
00:03:11 --> 00:03:13
the horizontal axis on this
blackboard.
34
00:03:13 --> 00:03:16
But it could be any direction
in space.
35
00:03:16 --> 00:03:21
And, to describe this
direction, maybe I have a unit
36
00:03:21 --> 00:03:26
vector along this axis.
Let's say that I have any of a
37
00:03:26 --> 00:03:32
vector A and I want to find out
what is the component of A along
38
00:03:32 --> 00:03:36
u.
That means what is the length
39
00:03:36 --> 00:03:42
of this projection of A to the
given direction?
40
00:03:42 --> 00:03:55
This thing here is the
component of A along u.
41
00:03:55 --> 00:04:02
Well, how do we find that?
Well, we know that here we have
42
00:04:02 --> 00:04:07
a right angle.
So this component is just
43
00:04:07 --> 00:04:13
length A times cosine of the
angle between A and u.
44
00:04:13 --> 00:04:18
But now that means I should be
able to compute it very easily
45
00:04:18 --> 00:04:23
because that's the same as
length A times length u times
46
00:04:23 --> 00:04:27
cosine theta because u is a unit
vector.
47
00:04:27 --> 00:04:33
It is a unit vector.
That means this is equal to one.
48
00:04:33 --> 00:04:41
And so that's the same as the
dot product between A and u.
49
00:04:41 --> 00:04:43
That is very easy.
And, of course,
50
00:04:43 --> 00:04:47
the most of just cases of that
is say, for example,
51
00:04:47 --> 00:04:50
we want just to find the
component along i hat,
52
00:04:50 --> 00:04:53
the unit vector along the x
axis.
53
00:04:53 --> 00:04:57
Then you do the dot product
with i hat, which is 100.
54
00:04:57 --> 00:04:59
What you get is the first
component.
55
00:04:59 --> 00:05:01
And that is,
indeed, the x component of a
56
00:05:01 --> 00:05:04
vector.
Similarly, say you want the z
57
00:05:04 --> 00:05:08
component you do the dot product
with k that gives you the last
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00:05:08 --> 00:05:14
component of your vector.
But the same works with a unit
59
00:05:14 --> 00:05:21
vector in any direction.
So what is an application of
60
00:05:21 --> 00:05:24
that?
Well, for example,
61
00:05:24 --> 00:05:30
in physics maybe you have seen
situations where you have a
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pendulum that swings.
You have maybe some mass at the
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end of the string and that mass
swings back and forth on a
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00:05:41 --> 00:05:42
circle.
And to analyze this
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00:05:42 --> 00:05:45
mechanically you want to use,
of course,
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00:05:45 --> 00:05:50
Newton's Laws of Mechanics and
you want to use forces and so
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on,
but I claim that components of
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00:05:54 --> 00:05:59
vectors are useful here to
understand what happens
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geometrically.
What are the forces exerted on
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this pendulum?
Well, there is its weight,
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which usually points downwards,
and there is the tension of the
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00:06:21 --> 00:06:25
string.
And these two forces together
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are what explains how this
pendulum is going to move back
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00:06:30 --> 00:06:33
and forth.
Now, you could try to
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00:06:33 --> 00:06:36
understand the equations of
motion using x,
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00:06:36 --> 00:06:39
y coordinates or x,
z or whatever you want to call
77
00:06:39 --> 00:06:41
them, let's say x,
y.
78
00:06:41 --> 00:06:47
But really what causes the
pendulum to swing back and forth
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00:06:47 --> 00:06:52
and also to somehow stay a
constant distance are phenomenal
80
00:06:52 --> 00:06:56
relative to this circular
trajectory.
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00:06:56 --> 00:06:59
For example,
maybe instead of taking
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00:06:59 --> 00:07:03
components along the x and y
axis, we want to look at two
83
00:07:03 --> 00:07:09
other unit vectors.
We can look at a vector,
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00:07:09 --> 00:07:15
let's call it T,
that is tangent to the
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00:07:15 --> 00:07:18
trajectory.
Sorry. Can you read that?
86
00:07:18 --> 00:07:33
It's not very readable.
T is tangent to the trajectory.
87
00:07:33 --> 00:07:36
And, on the other hand,
we can introduce another
88
00:07:36 --> 00:07:42
vector.
Let's call that N.
89
00:07:42 --> 00:07:50
And that one is normal,
perpendicular to the
90
00:07:50 --> 00:07:55
trajectory.
And so now if you think about
91
00:07:55 --> 00:08:00
it you can look at the
components of the weight along
92
00:08:00 --> 00:08:06
the tangent direction and along
the normal direction.
93
00:08:06 --> 00:08:13
And so the component of F along
the tangent direction is what
94
00:08:13 --> 00:08:21
causes acceleration in the
direction along the trajectory.
95
00:08:21 --> 00:08:23
It is what causes the pendulum
to swing back and forth.
96
00:08:23 --> 00:08:38
97
00:08:38 --> 00:08:45
And the component along N,
on the other hand.
98
00:08:45 --> 00:08:51
That is the part of the weight
that tends to pull our mass away
99
00:08:51 --> 00:08:54
from this point.
It is what is going to be
100
00:08:54 --> 00:08:56
responsible for the tension of
the string.
101
00:08:56 --> 00:09:02
It is why the string is taut
and not actually slack and with
102
00:09:02 --> 00:09:06
things moving all over the
place.
103
00:09:06 --> 00:09:18
That one is responsible for the
tension of a string.
104
00:09:18 --> 00:09:20
And now, of course,
if you want to compute things,
105
00:09:20 --> 00:09:23
well, maybe you will call this
angle theta and then you will
106
00:09:23 --> 00:09:27
express things explicitly using
sines and cosines and you will
107
00:09:27 --> 00:09:29
solve for the equations of
motion.
108
00:09:29 --> 00:09:32
That would be a very
interesting physics problem.
109
00:09:32 --> 00:09:35
But, to save time,
we are not going to do it.
110
00:09:35 --> 00:09:40
I'm sure you've seen that in
8.01 or similar classes.
111
00:09:40 --> 00:09:48
And so to find these components
we will just do dot products.
112
00:09:48 --> 00:09:56
Any questions?
No.
113
00:09:56 --> 00:10:01
OK.
Let's move onto our next topic.
114
00:10:01 --> 00:10:06
Here we have found things about
lengths, angles and stuff like
115
00:10:06 --> 00:10:10
that.
One important concept that we
116
00:10:10 --> 00:10:17
have not understood yet in terms
of vectors is area.
117
00:10:17 --> 00:10:25
Let's say that we want to find
the area of this pentagon.
118
00:10:25 --> 00:10:28
Well, how do we compute that
using vectors?
119
00:10:28 --> 00:10:32
Can we do it using vectors?
Yes we can.
120
00:10:32 --> 00:10:36
And that is going to be the
goal.
121
00:10:36 --> 00:10:42
The first thing we should do is
probably simplify the problem.
122
00:10:42 --> 00:10:44
We don't actually need to
bother with pentagons.
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00:10:44 --> 00:10:48
All we need to know are
triangles because,
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00:10:48 --> 00:10:51
for example,
you can cut that in three
125
00:10:51 --> 00:10:56
triangles and then sum the areas
of the triangles.
126
00:10:56 --> 00:11:05
Perhaps easier,
what is the area of a triangle?
127
00:11:05 --> 00:11:12
Let's start with a triangle in
the plane.
128
00:11:12 --> 00:11:16
Well, then we need two vectors
to describe it,
129
00:11:16 --> 00:11:20
say A and B here.
How do we find the area of a
130
00:11:20 --> 00:11:23
triangle?
Well, we all know base times
131
00:11:23 --> 00:11:25
height over two.
What is the base?
132
00:11:25 --> 00:11:30
What is the height?
The area of this triangle is
133
00:11:30 --> 00:11:35
going to be one-half of the
base, which is going to be the
134
00:11:35 --> 00:11:39
length of A.
And the height,
135
00:11:39 --> 00:11:47
well, if you call theta this
angle, then this is length B
136
00:11:47 --> 00:11:51
sine theta.
Now, that looks a lot like the
137
00:11:51 --> 00:11:54
formula we had there,
except for one little catch.
138
00:11:54 --> 00:11:58
This is a sine instead of a
cosine.
139
00:11:58 --> 00:12:03
How do we deal with that?
Well, what we could do is first
140
00:12:03 --> 00:12:10
find the cosine of the angle.
We know how to find the cosine
141
00:12:10 --> 00:12:17
of the angle using dot products.
Then solve for sine using sine
142
00:12:17 --> 00:12:22
square plus cosine square equals
one.
143
00:12:22 --> 00:12:25
And then plug that back into
here.
144
00:12:25 --> 00:12:28
Well, that works but it is kind
of a very complicated way of
145
00:12:28 --> 00:12:30
doing it.
So there is an easier way.
146
00:12:30 --> 00:12:34
And that is going to be
determinants,
147
00:12:34 --> 00:12:40
but let me explain how we get
to that maybe still doing
148
00:12:40 --> 00:12:45
elementary geometry and dot
products first.
149
00:12:45 --> 00:12:53
Let's see.
What we can do is instead of
150
00:12:53 --> 00:12:55
finding the sine of theta,
well,
151
00:12:55 --> 00:12:59
we're not good at finding sines
of angles but we are very good
152
00:12:59 --> 00:13:00
now at finding cosines of
angles.
153
00:13:00 --> 00:13:05
Maybe we can find another angle
whose cosine is the same as the
154
00:13:05 --> 00:13:09
sine of theta.
Well, you have already heard
155
00:13:09 --> 00:13:14
about complimentary angles and
how I take my vector A,
156
00:13:14 --> 00:13:18
my vector B here and I have an
angle theta.
157
00:13:18 --> 00:13:24
Well, let's say that I rotate
my vector A by 90 degrees to get
158
00:13:24 --> 00:13:34
a new vector A prime.
A prime is just A rotated by 90
159
00:13:34 --> 00:13:39
degrees.
Then the angle between these
160
00:13:39 --> 00:13:45
two guys, let's say theta prime,
well, theta prime is 90 degrees
161
00:13:45 --> 00:13:49
or pi over two gradients minus
theta.
162
00:13:49 --> 00:13:56
So, in particular,
cosine of theta prime is equal
163
00:13:56 --> 00:14:01
to sine of theta.
In particular,
164
00:14:01 --> 00:14:09
that means that length A,
length B, sine theta,
165
00:14:09 --> 00:14:13
which is what we would need to
know in order to find the area
166
00:14:13 --> 00:14:17
of this triangle is equal to,
well, A and A prime have the
167
00:14:17 --> 00:14:21
same length so let me replace
that by length of A prime.
168
00:14:21 --> 00:14:28
I am not changing anything,
length B, cosine theta prime.
169
00:14:28 --> 00:14:31
And now we have something that
is much easier for us.
170
00:14:31 --> 00:14:37
Because that is just A prime
dot B.
171
00:14:37 --> 00:14:40
That looks like a very good
plan.
172
00:14:40 --> 00:14:43
There is only one small thing
which is we don't know yet how
173
00:14:43 --> 00:14:48
to find this A prime.
Well, I think it is not very
174
00:14:48 --> 00:14:52
hard.
Let's see.
175
00:14:52 --> 00:14:58
Actually, why don't you guys do
the hard work?
176
00:14:58 --> 00:15:02
Let's say that I have a plane
vector A with two components a1,
177
00:15:02 --> 00:15:05
a2.
And I want to rotate it
178
00:15:05 --> 00:15:10
counterclockwise by 90 degrees.
It looks like maybe we should
179
00:15:10 --> 00:15:14
change some signs somewhere.
Maybe we should do something
180
00:15:14 --> 00:15:24
with the components.
Can you come up with an idea of
181
00:15:24 --> 00:15:34
what it might be?
I see a lot of people answering
182
00:15:34 --> 00:15:37
three.
I see some other answers,
183
00:15:37 --> 00:15:41
but the majority vote seems to
be number three.
184
00:15:41 --> 00:15:49
Minus a2 and a1.
I think I agree, so let's see.
185
00:15:49 --> 00:16:01
Let's say that we have this
vector A with components a1.
186
00:16:01 --> 00:16:05
So a1 is here.
And a2. So a2 is here.
187
00:16:05 --> 00:16:14
Let's rotate this box by 90
degrees counterclockwise.
188
00:16:14 --> 00:16:19
This box ends up there.
It's the same box just flipped
189
00:16:19 --> 00:16:23
on its side.
This thing here becomes a1 and
190
00:16:23 --> 00:16:31
this thing here becomes a2.
And that means our new vector A
191
00:16:31 --> 00:16:37
prime is going to be -- Well,
the first component looks like
192
00:16:37 --> 00:16:40
an a2 but it is pointing to the
left when a2 is positive.
193
00:16:40 --> 00:16:47
So, actually, it is minus a2.
And the y component is going to
194
00:16:47 --> 00:16:53
be the same as this guy,
so it's going to be a1.
195
00:16:53 --> 00:16:56
If you wanted instead to rotate
clockwise then you would do the
196
00:16:56 --> 00:17:00
opposite.
You would do a2 minus a1.
197
00:17:00 --> 00:17:07
Is that reasonably clear for
everyone?
198
00:17:07 --> 00:17:14
OK.
Let's continue the calculation
199
00:17:14 --> 00:17:18
there.
A prime, we have decided,
200
00:17:18 --> 00:17:24
is minus a2,
a1 dot product with let's call
201
00:17:24 --> 00:17:33
b1 and b2, the components of B.
Then that will be minus a2,
202
00:17:33 --> 00:17:36
b1 plus a1, b2 plus a1,
b2.
203
00:17:36 --> 00:17:43
Let me write that the other way
around, a1, b2 minus a2,
204
00:17:43 --> 00:17:46
b1.
And that is a quantity that you
205
00:17:46 --> 00:17:53
may already know under the name
of determinant of vectors A and
206
00:17:53 --> 00:17:59
B, which we write symbolically
using this notation.
207
00:17:59 --> 00:18:03
We put A and B next to each
other inside a two-by-two table
208
00:18:03 --> 00:18:09
and we put these verticals bars.
And that means the determinant
209
00:18:09 --> 00:18:14
of these numbers,
this guy times this guy minus
210
00:18:14 --> 00:18:30
this guy times this guy.
That is called the determinant.
211
00:18:30 --> 00:18:34
And geometrically what it
measures is the area,
212
00:18:34 --> 00:18:38
well, not of a triangle because
we did not divide by two,
213
00:18:38 --> 00:18:42
but of a parallelogram formed
by A and B.
214
00:18:42 --> 00:18:51
It measures the area of the
parallelogram with sides A and
215
00:18:51 --> 00:18:53
B.
And, of course,
216
00:18:53 --> 00:18:56
if you want the triangle then
you will just divide by two.
217
00:18:56 --> 00:19:00
The triangle is half the
parallelogram.
218
00:19:00 --> 00:19:04
There is one small catch.
The area usually is something
219
00:19:04 --> 00:19:08
that is going to be positive.
This guy here has no reason to
220
00:19:08 --> 00:19:16
be positive or negative because,
in fact, well,
221
00:19:16 --> 00:19:20
if you compute things you will
see that where it is supposed to
222
00:19:20 --> 00:19:24
go negative it depends on
whether A and B are clockwise or
223
00:19:24 --> 00:19:26
counterclockwise from each
other.
224
00:19:26 --> 00:19:29
I mean the issue that we have
-- Well,
225
00:19:29 --> 00:19:31
when we say the area is
one-half length A,
226
00:19:31 --> 00:19:34
length B,
sine theta that was assuming
227
00:19:34 --> 00:19:37
that theta is positive,
that its sine is positive.
228
00:19:37 --> 00:19:42
Otherwise, if theta is negative
maybe we need to take the
229
00:19:42 --> 00:19:47
absolute value of this.
Just to be more truthful,
230
00:19:47 --> 00:19:56
I will say the determinant is
either plus or minus the area.
231
00:19:56 --> 00:20:13
Any questions about this?
Yes.
232
00:20:13 --> 00:20:15
Sorry.
That is not a dot product.
233
00:20:15 --> 00:20:18
That is the usual
multiplication.
234
00:20:18 --> 00:20:25
That is length A times length B
times sine theta.
235
00:20:25 --> 00:20:28
What does that equal?
And so that is equal to the
236
00:20:28 --> 00:20:31
area of a parallelogram.
Sorry.
237
00:20:31 --> 00:20:39
Let me explain that again.
If I have two vectors A and B,
238
00:20:39 --> 00:20:45
I can form a parallelogram with
them or I can form a triangle.
239
00:20:45 --> 00:20:53
And so the area of a
parallelogram is equal to length
240
00:20:53 --> 00:21:00
A, length B, sine theta,
is equal to the determinant of
241
00:21:00 --> 00:21:07
A and B.
While the area of a triangle is
242
00:21:07 --> 00:21:09
one-half of that.
243
00:21:09 --> 00:21:21
244
00:21:21 --> 00:21:25
And, again, to be truthful,
I should say these things can
245
00:21:25 --> 00:21:28
be positive or negative.
Depending on whether you count
246
00:21:28 --> 00:21:31
the angle positively or
negatively, you will get either
247
00:21:31 --> 00:21:36
the area or minus the area.
The area is actually the
248
00:21:36 --> 00:21:39
absolute value of these
quantities.
249
00:21:39 --> 00:21:49
Is that clear?
OK.
250
00:21:49 --> 00:21:57
Yes.
If you want to compute the
251
00:21:57 --> 00:21:59
area, you will just take the
absolute value of the
252
00:21:59 --> 00:22:00
determinant.
253
00:22:00 --> 00:22:15
254
00:22:15 --> 00:22:19
I should say the area of a
parallelogram so that it is
255
00:22:19 --> 00:22:32
completely clear.
Sorry. Do you have a question?
256
00:22:32 --> 00:22:34
Explain again,
sorry, was the question how a
257
00:22:34 --> 00:22:38
determinant equals the area of a
parallelogram?
258
00:22:38 --> 00:22:41
OK.
The area of a parallelogram is
259
00:22:41 --> 00:22:45
going to be the base times the
height.
260
00:22:45 --> 00:22:48
Let's take this guy to be the
base.
261
00:22:48 --> 00:22:53
The length of a base will be
length of A and the height will
262
00:22:53 --> 00:22:58
be obtained by taking B but only
looking at the vertical part.
263
00:22:58 --> 00:23:02
That will be length of B times
the sine of theta.
264
00:23:02 --> 00:23:06
That is how I got the area of a
parallelogram as length A,
265
00:23:06 --> 00:23:09
length B, sine theta.
And then I did this
266
00:23:09 --> 00:23:15
manipulation and this trick of
rotating to find a nice formula.
267
00:23:15 --> 00:23:23
Yes.
You are asking ahead of what I
268
00:23:23 --> 00:23:28
am going to do in a few minutes.
You are asking about magnitude
269
00:23:28 --> 00:23:29
of A cross B.
We are going to learn about
270
00:23:29 --> 00:23:32
cross products in a few minutes.
And the answer is yes,
271
00:23:32 --> 00:23:34
but cross product is for
vectors in space.
272
00:23:34 --> 00:23:38
Here I was simplifying things
by doing things just in the
273
00:23:38 --> 00:23:43
plane.
Just bear with me for five more
274
00:23:43 --> 00:23:48
minutes and we will do things in
space.
275
00:23:48 --> 00:23:55
Yes. That is correct.
The way you compute this in
276
00:23:55 --> 00:24:00
practice is you just do this.
That is how you compute the
277
00:24:00 --> 00:24:04
determinant.
Yes.
278
00:24:04 --> 00:24:09
What about three dimensions?
Three dimensions we are going
279
00:24:09 --> 00:24:11
to do now.
More questions?
280
00:24:11 --> 00:24:26
Should we move on?
OK. Let's move to space.
281
00:24:26 --> 00:24:32
There are two things we can do
in space.
282
00:24:32 --> 00:24:36
And you can look for the volume
of solids or you can look for
283
00:24:36 --> 00:24:39
the area of surfaces.
Let me start with the easier of
284
00:24:39 --> 00:24:42
the two.
Let me start with volumes of
285
00:24:42 --> 00:24:49
solids.
And we will go back to area,
286
00:24:49 --> 00:24:53
I promise.
I claim that there is also a
287
00:24:53 --> 00:24:59
notion of determinants in space.
And that is going to tell us
288
00:24:59 --> 00:25:08
how to find volumes.
Let's say that we have three
289
00:25:08 --> 00:25:16
vectors A, B and C.
And then the definition of
290
00:25:16 --> 00:25:23
their determinants going to be,
the notation for that in terms
291
00:25:23 --> 00:25:28
of the components is the same as
over there.
292
00:25:28 --> 00:25:35
We put the components of A,
the components of B and the
293
00:25:35 --> 00:25:40
components of C inside verticals
bars.
294
00:25:40 --> 00:25:42
And, of course,
I have to give meaning to this.
295
00:25:42 --> 00:25:45
This will be a number.
And what is that number?
296
00:25:45 --> 00:25:50
Well, the definition I will
take is that this is a1 times
297
00:25:50 --> 00:25:55
the determinant of what I get by
looking in this lower right
298
00:25:55 --> 00:26:01
corner.
The two-by-two determinant b2,
299
00:26:01 --> 00:26:08
b3, c2, c3.
Then I will subtract a2 times
300
00:26:08 --> 00:26:15
the determinant of b1,
b3, c1, c3.
301
00:26:15 --> 00:26:22
And then I will add a3 times
the determinant b1,
302
00:26:22 --> 00:26:26
b2, c1, c2.
And each of these guys means,
303
00:26:26 --> 00:26:30
again, you take b2 times c3
minus c2 times b3 and this times
304
00:26:30 --> 00:26:33
that minus this time that and so
on.
305
00:26:33 --> 00:26:35
In fact, there is a total of
six terms in here.
306
00:26:35 --> 00:26:39
And maybe some of you have
already seen a different formula
307
00:26:39 --> 00:26:42
for three-by-three determinants
where you directly have the six
308
00:26:42 --> 00:26:47
terms.
It is the same definition.
309
00:26:47 --> 00:26:50
How to remember the structure
of this formula?
310
00:26:50 --> 00:26:55
Well, this is called an
expansion according to the first
311
00:26:55 --> 00:26:57
row.
So we are going to take the
312
00:26:57 --> 00:27:02
entries in the first row,
a1, a2, a3 And for each of them
313
00:27:02 --> 00:27:05
we get the term.
Namely we multiply it by a
314
00:27:05 --> 00:27:10
two-by-two determinant that we
get by deleting the first row
315
00:27:10 --> 00:27:16
and the column where we are.
Here the coefficient next to
316
00:27:16 --> 00:27:21
a1, when we delete this column
and this row,
317
00:27:21 --> 00:27:24
we are left with b2,
b3, c2, c3.
318
00:27:24 --> 00:27:29
The next one we take a2,
we delete the row that is in it
319
00:27:29 --> 00:27:35
and the column that it is in.
And we are left with b1,
320
00:27:35 --> 00:27:38
b3, c1, c3.
And, similarly,
321
00:27:38 --> 00:27:41
with a3, we take what remains,
which is b1,
322
00:27:41 --> 00:27:45
b2, c1, c2.
Finally, last but not least,
323
00:27:45 --> 00:27:51
there is a minus sign here for
the second guy.
324
00:27:51 --> 00:28:01
It looks like a weird formula.
I mean it is a little bit weird.
325
00:28:01 --> 00:28:04
But it is a formula that you
should learn because it is
326
00:28:04 --> 00:28:06
really, really useful for a lot
of things.
327
00:28:06 --> 00:28:10
I should say if this looks very
artificial to you and you would
328
00:28:10 --> 00:28:14
like to know more there is more
in the notes,
329
00:28:14 --> 00:28:17
so read the notes.
They will tell you a bit more
330
00:28:17 --> 00:28:20
about what this means,
where it comes from and so on.
331
00:28:20 --> 00:28:23
If you want to know a lot more
then some day you should take
332
00:28:23 --> 00:28:26
18.06,
Linear Algebra where you will
333
00:28:26 --> 00:28:29
learn a lot more about
determinants in N dimensional
334
00:28:29 --> 00:28:32
space with N vectors.
And there is a generalization
335
00:28:32 --> 00:28:36
of this in arbitrary dimensions.
In this class,
336
00:28:36 --> 00:28:39
we will only deal with two or
three dimensions.
337
00:28:39 --> 00:28:44
Yes.
Why is the negative there?
338
00:28:44 --> 00:28:45
Well, that is a very good
question.
339
00:28:45 --> 00:28:49
It has to be there so that this
will actually equal,
340
00:28:49 --> 00:28:53
well, what I am going to say
right now is that this will give
341
00:28:53 --> 00:28:55
us the volume of [a box?]
with sides A,
342
00:28:55 --> 00:28:57
B, C.
And the formula just doesn't
343
00:28:57 --> 00:28:59
work if you don't put the
negative.
344
00:28:59 --> 00:29:02
There is a more fundamental
reason which has to do with
345
00:29:02 --> 00:29:06
orientation of space and the
fact that if you switch two
346
00:29:06 --> 00:29:09
coordinates in space then
basically you change what is
347
00:29:09 --> 00:29:12
called the handedness of the
coordinates.
348
00:29:12 --> 00:29:14
If you look at your right hand
and your left hand,
349
00:29:14 --> 00:29:16
they are not actually the same.
They are mirror images.
350
00:29:16 --> 00:29:18
And, if you squared two
coordinate axes,
351
00:29:18 --> 00:29:21
that is what you get.
That is the fundamental reason
352
00:29:21 --> 00:29:24
for the minus.
Again, we don't need to think
353
00:29:24 --> 00:29:33
too much about that.
All we need in this class is
354
00:29:33 --> 00:29:38
the formula.
Why do we care about this
355
00:29:38 --> 00:29:43
formula?
It is because of the theorem
356
00:29:43 --> 00:29:52
that says that geometrically the
determinant of the three vectors
357
00:29:52 --> 00:29:58
A, B, C is, again,
plus or minus.
358
00:29:58 --> 00:30:00
This determinant could be
positive or negative.
359
00:30:00 --> 00:30:03
See those minuses and all sorts
of stuff.
360
00:30:03 --> 00:30:14
Plus or minus the volume of the
parallelepiped.
361
00:30:14 --> 00:30:20
That is just a fancy name for a
box with parallelogram sides,
362
00:30:20 --> 00:30:24
in case you wonder,
with sides A,
363
00:30:24 --> 00:30:29
B and C.
You take the three vectors A,
364
00:30:29 --> 00:30:35
B and C and you form a box
whose sides are all
365
00:30:35 --> 00:30:44
parallelograms.
And when its volume is going to
366
00:30:44 --> 00:30:59
be the determinant.
Other questions?
367
00:30:59 --> 00:31:11
I'm sorry.
I cannot quite hear you.
368
00:31:11 --> 00:31:12
Yes.
We are going to see how to do
369
00:31:12 --> 00:31:14
it geometrically without a
determinant,
370
00:31:14 --> 00:31:17
but then you will see that you
actually need a determinant to
371
00:31:17 --> 00:31:21
compute it no matter what.
We are going to go back to this
372
00:31:21 --> 00:31:24
and see another formula for
volume, but you will see that
373
00:31:24 --> 00:31:26
really I am cheating.
I mean somehow computationally
374
00:31:26 --> 00:31:30
the only way to compute it is
really to use a determinant.
375
00:31:30 --> 00:31:43
376
00:31:43 --> 00:31:44
That is correct.
In general, I mean,
377
00:31:44 --> 00:31:47
actually, I could say if you
look at the two-by-two
378
00:31:47 --> 00:31:50
determinant, see,
you can also explain it in
379
00:31:50 --> 00:31:54
terms of this extension.
If you take a1 and multiply by
380
00:31:54 --> 00:31:57
this one-by-one determinant b2,
then you take a2 and you
381
00:31:57 --> 00:32:00
multiply it by this one-by-one
determinant b1 but you put a
382
00:32:00 --> 00:32:02
minus sign.
And in general,
383
00:32:02 --> 00:32:06
indeed, when you expand,
you would stop putting plus,
384
00:32:06 --> 00:32:08
minus, plus,
minus alternating.
385
00:32:08 --> 00:32:15
More about that in 18.06.
Yes.
386
00:32:15 --> 00:32:18
There is a way to do it based
on other rows as well,
387
00:32:18 --> 00:32:20
but then you have to be very
careful with the sign vectors.
388
00:32:20 --> 00:32:23
I will refer you to the notes
for that.
389
00:32:23 --> 00:32:25
I mean you could also do it
with a column,
390
00:32:25 --> 00:32:28
by the way.
I mean be careful about the
391
00:32:28 --> 00:32:30
sign rules.
Given how little we will use
392
00:32:30 --> 00:32:33
determinants in this class,
I mean we will use them in a
393
00:32:33 --> 00:32:36
way that is fundamental,
but we won't compute much.
394
00:32:36 --> 00:32:47
Let's say this is going to be
enough for us for now.
395
00:32:47 --> 00:32:50
After determinants now I can
tell you about cross product.
396
00:32:50 --> 00:32:53
And cross product is going to
be the answer to your question
397
00:32:53 --> 00:32:54
about area.
398
00:32:54 --> 00:33:32
399
00:33:32 --> 00:33:45
OK.
Let me move onto cross product.
400
00:33:45 --> 00:33:53
Cross product is something that
you can apply to two vectors in
401
00:33:53 --> 00:33:56
space.
And by that I mean really in
402
00:33:56 --> 00:33:59
three-dimensional space.
This is something that is
403
00:33:59 --> 00:34:05
specific to three dimensions.
The definition A cross B -- It
404
00:34:05 --> 00:34:11
is important to really do your
multiplication symbol well so
405
00:34:11 --> 00:34:16
that you don't mistake it with a
dot product.
406
00:34:16 --> 00:34:23
Well, that is going to be a
vector.
407
00:34:23 --> 00:34:26
That is another reason not to
confuse it with dot product.
408
00:34:26 --> 00:34:30
Dot product gives you a number.
Cross product gives you a
409
00:34:30 --> 00:34:32
vector.
They are really completely
410
00:34:32 --> 00:34:35
different operations.
They are both called product
411
00:34:35 --> 00:34:38
because someone could not come
up with a better name,
412
00:34:38 --> 00:34:42
but they are completely
different operations.
413
00:34:42 --> 00:34:45
What do we do to do the cross
product of A and B?
414
00:34:45 --> 00:34:47
Well, we do something very
strange.
415
00:34:47 --> 00:34:50
Just as I have told you that a
determinant is something where
416
00:34:50 --> 00:34:54
we put numbers and we get a
number, I am going to violate my
417
00:34:54 --> 00:34:59
own rule.
I am going to put together a
418
00:34:59 --> 00:35:06
determinant in which -- Well,
the last two rows are the
419
00:35:06 --> 00:35:11
components of the vectors A and
B but the first row strangely
420
00:35:11 --> 00:35:15
consists for unit vectors i,
j, k.
421
00:35:15 --> 00:35:19
What does that mean?
Well, that is not a determinant
422
00:35:19 --> 00:35:21
in the usual sense.
If you try to put that into
423
00:35:21 --> 00:35:24
your calculator,
it will tell you there is an
424
00:35:24 --> 00:35:26
error.
I don't know how to put vectors
425
00:35:26 --> 00:35:28
in there.
I want numbers.
426
00:35:28 --> 00:35:32
What is means is it is symbolic
notation that helps you remember
427
00:35:32 --> 00:35:35
what the formula is.
The actual formula is,
428
00:35:35 --> 00:35:40
well, you use this definition.
And, if you use that
429
00:35:40 --> 00:35:47
definition, you see that it is i
hat times some number.
430
00:35:47 --> 00:35:55
Let me write it as determinant
of a2, a3, b2,
431
00:35:55 --> 00:36:02
b3 times i hat minus
determinant a1,
432
00:36:02 --> 00:36:11
a3, b1, b3, j hat plus a1,
a2, b1, b2, k hat.
433
00:36:11 --> 00:36:15
And so that is the actual
definition in a way that makes
434
00:36:15 --> 00:36:18
complete sense,
but to remember this formula
435
00:36:18 --> 00:36:23
without too much trouble it is
much easier to think about it in
436
00:36:23 --> 00:36:27
these terms here.
That is the definition and it
437
00:36:27 --> 00:36:30
gives you a vector.
Now, as usual with definitions,
438
00:36:30 --> 00:36:32
the question is what is it good
for?
439
00:36:32 --> 00:36:36
What is the geometric meaning
of this very strange operation?
440
00:36:36 --> 00:36:48
Why do we bother to do that?
Here is what it does
441
00:36:48 --> 00:36:52
geometrically.
Remember a vector has two
442
00:36:52 --> 00:36:56
different things.
It has a length and it has a
443
00:36:56 --> 00:37:01
direction.
Let's start with the length.
444
00:37:01 --> 00:37:15
A length of a cross product is
the area of the parallelogram in
445
00:37:15 --> 00:37:24
space formed by the vectors A
and B.
446
00:37:24 --> 00:37:27
Now, if you have a
parallelogram in space,
447
00:37:27 --> 00:37:31
you can find its area just by
doing this calculation when you
448
00:37:31 --> 00:37:33
know the coordinates of the
points.
449
00:37:33 --> 00:37:35
You do this calculation and
then you take the length.
450
00:37:35 --> 00:37:40
You take this squared plus that
squared plus that squared,
451
00:37:40 --> 00:37:43
square root.
It looks like a very
452
00:37:43 --> 00:37:47
complicated formula but it works
and, actually,
453
00:37:47 --> 00:37:49
it is the simplest way to do
it.
454
00:37:49 --> 00:37:52
This time we don't actually
need to put plus or minus
455
00:37:52 --> 00:37:55
because the length of a vector
is always positive.
456
00:37:55 --> 00:38:00
We don't have to worry about
that.
457
00:38:00 --> 00:38:04
And what is even more magical
is that not only is the length
458
00:38:04 --> 00:38:07
remarkable but the direction is
also remarkable.
459
00:38:07 --> 00:38:24
The direction of A cross B is
perpendicular to the plane of a
460
00:38:24 --> 00:38:33
parallelogram.
Our two vectors A and B
461
00:38:33 --> 00:38:41
together in a plane.
What I am telling you is that
462
00:38:41 --> 00:38:51
for vector A cross B will point,
will stick straight out of that
463
00:38:51 --> 00:38:56
plane perpendicularly to it.
In fact, I would have to be
464
00:38:56 --> 00:38:58
more precise.
There are two ways that you can
465
00:38:58 --> 00:39:02
be perpendicular to this plane.
You can be perpendicular
466
00:39:02 --> 00:39:06
pointing up or pointing down.
How do I decide which?
467
00:39:06 --> 00:39:16
Well, there is something called
the right-hand rule.
468
00:39:16 --> 00:39:18
What does the right-hand rule
say?
469
00:39:18 --> 00:39:21
Well, there are various
versions for right-hand rule
470
00:39:21 --> 00:39:23
depending on which country you
learn about it.
471
00:39:23 --> 00:39:26
In France, given the culture,
you even learn about it in
472
00:39:26 --> 00:39:28
terms of a cork screw and a wine
bottle.
473
00:39:28 --> 00:39:33
I will just use the usual
version here.
474
00:39:33 --> 00:39:35
You take your right hand.
If you are left-handed,
475
00:39:35 --> 00:39:38
remember to take your right
hand and not the left one.
476
00:39:38 --> 00:39:43
The other right, OK?
Then place your hand to point
477
00:39:43 --> 00:39:46
in the direction of A.
Let's say my right hand is
478
00:39:46 --> 00:39:50
going in that direction.
Now, curl your fingers so that
479
00:39:50 --> 00:39:54
they point towards B.
Here that would be kind of into
480
00:39:54 --> 00:39:56
the blackboard.
Don't snap any bones.
481
00:39:56 --> 00:40:00
If it doesn't quite work then
rotate your arms so that you can
482
00:40:00 --> 00:40:04
actually physically do it.
Then get your thumb to stick
483
00:40:04 --> 00:40:07
straight out.
Well, here my thumb is going to
484
00:40:07 --> 00:40:11
go up.
And that tells me that A cross
485
00:40:11 --> 00:40:16
B will go up.
Let me write that down while
486
00:40:16 --> 00:40:19
you experiment with it.
Again, try not to enjoy
487
00:40:19 --> 00:40:20
yourselves.
488
00:40:20 --> 00:40:30
489
00:40:30 --> 00:40:39
First, your right hand points
parallel to vector A.
490
00:40:39 --> 00:40:47
Then your fingers point in the
direction of B.
491
00:40:47 --> 00:40:53
Then your thumb,
when you stick it out,
492
00:40:53 --> 00:41:00
is going to point in the
direction of A cross B.
493
00:41:00 --> 00:41:29
Let's do a quick example.
Where is my quick example? Here.
494
00:41:29 --> 00:41:32
Let's take i cross j.
495
00:41:32 --> 00:41:40
496
00:41:40 --> 00:41:47
I see most of you going in the
right direction.
497
00:41:47 --> 00:41:51
If you have it pointing in the
wrong direction,
498
00:41:51 --> 00:41:56
it might mean that you are
using your left hand,
499
00:41:56 --> 00:42:01
for example.
Example, I claim that i cross j
500
00:42:01 --> 00:42:07
equals k.
Let's see. I points towards us.
501
00:42:07 --> 00:42:12
J point to our right.
I guess this is your right.
502
00:42:12 --> 00:42:16
I think.
And then your thumb is going to
503
00:42:16 --> 00:42:19
point up.
That tells us it is roughly
504
00:42:19 --> 00:42:21
pointing up.
And, of course,
505
00:42:21 --> 00:42:24
the length should be one
because if you take the unit
506
00:42:24 --> 00:42:27
square in the x,
y plane, its area is one.
507
00:42:27 --> 00:42:29
And the direction should be
vertical.
508
00:42:29 --> 00:42:34
Because it should be
perpendicular to the x,
509
00:42:34 --> 00:42:37
y plane.
It looks like i cross j will be
510
00:42:37 --> 00:42:41
k.
Well, let's check with the
511
00:42:41 --> 00:42:43
definition i,
j, k.
512
00:42:43 --> 00:42:51
What is i? I is one, zero, zero.
J is zero, one, zero.
513
00:42:51 --> 00:42:58
The coefficient of i will be
zero times zero minus zero times
514
00:42:58 --> 00:43:00
one.
That is zero.
515
00:43:00 --> 00:43:04
The coefficient of j will be
one time zero minus zero times
516
00:43:04 --> 00:43:06
zero, that is a zero,
minus zero j.
517
00:43:06 --> 00:43:11
It doesn't matter.
And the coefficient of k will
518
00:43:11 --> 00:43:14
be one times one,
that is one,
519
00:43:14 --> 00:43:17
minus zero times zero,
so one k.
520
00:43:17 --> 00:43:22
So we do get i cross j equals k
both ways.
521
00:43:22 --> 00:43:24
In this case,
it is easier to do it
522
00:43:24 --> 00:43:27
geometrically.
If I give you no complicated
523
00:43:27 --> 00:43:32
vectors, probably you will
actually want to do the
524
00:43:32 --> 00:43:41
calculation.
Any questions? Yes.
525
00:43:41 --> 00:43:45
The coefficient of k,
remember I delete the first row
526
00:43:45 --> 00:43:50
and the last column so I get
this two-by-two determinant.
527
00:43:50 --> 00:43:54
And that two-by-two determinant
is one times one minus zero
528
00:43:54 --> 00:43:56
times zero so that gives me a
one.
529
00:43:56 --> 00:43:59
That is what you do with
two-by-two determinants.
530
00:43:59 --> 00:44:03
Similarly for [UNINTELLIGIBLE],
but [UNINTELLIGIBLE]
531
00:44:03 --> 00:44:11
turn out to be zero.
More questions?
532
00:44:11 --> 00:44:14
Yes.
Let me repeat how I got the one
533
00:44:14 --> 00:44:18
in front of k.
Remember the definition of a
534
00:44:18 --> 00:44:24
determinant I expand according
to the entries in the first row.
535
00:44:24 --> 00:44:28
When I get to k what I do is
delete the first row and I
536
00:44:28 --> 00:44:32
delete the last column,
the column that contains k.
537
00:44:32 --> 00:44:37
I delete these guys and these
guys and I am left with this
538
00:44:37 --> 00:44:41
two-by-two determinant.
Now, a two-by-two determinant,
539
00:44:41 --> 00:44:47
you multiply according to this
downward diagonal and then minus
540
00:44:47 --> 00:44:50
this times that.
One times one,
541
00:44:50 --> 00:44:55
let me see here,
I got one k because that is one
542
00:44:55 --> 00:45:00
times one minus zero times zero
equals one.
543
00:45:00 --> 00:45:03
Sorry.
That is really hard to read.
544
00:45:03 --> 00:45:11
Maybe it will be easier that
way.
545
00:45:11 --> 00:45:19
Yes.
Let's try.
546
00:45:19 --> 00:45:23
If I do the same for i,
I think I will also get zero.
547
00:45:23 --> 00:45:28
Let's do the same for i.
I take i, I delete the first
548
00:45:28 --> 00:45:33
row, I delete the first column,
I get this two-by-two
549
00:45:33 --> 00:45:36
determinant here and I get zero
times zero,
550
00:45:36 --> 00:45:39
that is zero,
minus zero times one.
551
00:45:39 --> 00:45:43
That is the other trick
question.
552
00:45:43 --> 00:45:49
Zero times one is zero as well.
So that zero minus zero is
553
00:45:49 --> 00:45:52
zero.
I hope on Monday you should get
554
00:45:52 --> 00:45:55
more practice in recitation
about how to compute
555
00:45:55 --> 00:45:58
determinants.
Hopefully, it will become very
556
00:45:58 --> 00:46:01
easy for you all to compute this
next.
557
00:46:01 --> 00:46:04
I know the first time it is
kind of a shock because there
558
00:46:04 --> 00:46:07
are a lot of numbers and a lot
of things to do.
559
00:46:07 --> 00:47:02
560
00:47:02 --> 00:47:08
Let me return to the question
that you asked a bit earlier
561
00:47:08 --> 00:47:13
about how do you find actually
volume if I don't want to know
562
00:47:13 --> 00:47:24
about determinants?
Well, let's have another look
563
00:47:24 --> 00:47:31
at the volume.
Let's say that I have three
564
00:47:31 --> 00:47:37
vectors.
Let me put them this way,
565
00:47:37 --> 00:47:43
A, B and C.
And let's try to see how else I
566
00:47:43 --> 00:47:49
could think about the volume of
this box.
567
00:47:49 --> 00:47:54
Probably you know that the
volume of a parallelepiped is
568
00:47:54 --> 00:47:57
the area of a base times the
height.
569
00:47:57 --> 00:48:04
Sorry.
The volume is the area of a
570
00:48:04 --> 00:48:12
base times the height.
How do we do that in practice?
571
00:48:12 --> 00:48:15
Well, what is the area of a
base?
572
00:48:15 --> 00:48:21
The base is a parallelogram in
space with sides B and C.
573
00:48:21 --> 00:48:23
How do we find the area of the
parallelogram in space?
574
00:48:23 --> 00:48:28
Well, we just discovered that.
We can do it by taking that
575
00:48:28 --> 00:48:30
cross product.
The area of a base,
576
00:48:30 --> 00:48:33
well, we take the cross product
of B and C.
577
00:48:33 --> 00:48:36
That is not quite it because
this is a vector.
578
00:48:36 --> 00:48:40
We would like a number while we
take its length.
579
00:48:40 --> 00:48:44
That is pretty good.
What about the height?
580
00:48:44 --> 00:48:48
Well, the height is going to be
the component of A in the
581
00:48:48 --> 00:48:51
direction that is perpendicular
to the base.
582
00:48:51 --> 00:48:53
Let's take a direction that is
perpendicular to the base.
583
00:48:53 --> 00:48:57
Let's call that N,
a unit vector in that
584
00:48:57 --> 00:49:00
direction.
Then we can get the height by
585
00:49:00 --> 00:49:04
taking A dot n.
That is what we saw at the
586
00:49:04 --> 00:49:10
beginning of class that A dot n
will tell me how much A goes in
587
00:49:10 --> 00:49:17
the direction of n.
Are you still with me?
588
00:49:17 --> 00:49:22
OK.
Let's keep going.
589
00:49:22 --> 00:49:24
Let's think about this vector
n.
590
00:49:24 --> 00:49:29
How do I get it?
Well, I can get it by actually
591
00:49:29 --> 00:49:34
using cross product as well.
Because I said the direction
592
00:49:34 --> 00:49:37
perpendicular to two vectors I
can get by taking that cross
593
00:49:37 --> 00:49:40
product and looking at that
direction.
594
00:49:40 --> 00:49:47
This is still B cross C length.
And this one is,
595
00:49:47 --> 00:49:56
so I claim, n can be obtained
by taking D cross C.
596
00:49:56 --> 00:49:58
Well, that comes in the right
direction but it is not a unit
597
00:49:58 --> 00:50:01
vector.
How do I get a unit vector?
598
00:50:01 --> 00:50:06
I divide by the length.
Thanks.
599
00:50:06 --> 00:50:14
I take B cross C and I divide
by length B cross C.
600
00:50:14 --> 00:50:20
Well, now I can probably
simplify between these two guys.
601
00:50:20 --> 00:50:38
And so what I will get -- What
I get out of this is that my
602
00:50:38 --> 00:50:53
volume equals A dot product with
vector B cross C.
603
00:50:53 --> 00:50:55
But, of course,
I have to be careful in which
604
00:50:55 --> 00:50:56
order I do it.
If I do it the other way
605
00:50:56 --> 00:50:58
around, A dot B,
I get a number.
606
00:50:58 --> 00:51:00
I cannot cross that.
I really have to do the cross
607
00:51:00 --> 00:51:03
product first.
I get the new vector.
608
00:51:03 --> 00:51:09
Then my dot product.
The fact is that the
609
00:51:09 --> 00:51:16
determinant of A,
B, C is equal to this so-called
610
00:51:16 --> 00:51:20
triple product.
Well, that looks good
611
00:51:20 --> 00:51:23
geometrically.
Let's try to check whether it
612
00:51:23 --> 00:51:27
makes sense with the formulas,
just one small thing.
613
00:51:27 --> 00:51:32
We saw the determinant is a1
times determinant b2,
614
00:51:32 --> 00:51:37
b3, c2, c3 minus a2 times
something plus a3 times
615
00:51:37 --> 00:51:42
something.
I will let you fill in the
616
00:51:42 --> 00:51:45
numbers.
That is this guy.
617
00:51:45 --> 00:51:48
What about this guy?
Well, dot product,
618
00:51:48 --> 00:51:50
we take the first component of
A, that is a1,
619
00:51:50 --> 00:51:53
we multiply by the first
component of B cross C.
620
00:51:53 --> 00:51:55
What is the first component of
B cross C?
621
00:51:55 --> 00:52:05
Well, it is this determinant
b2, b3, c2, c3.
622
00:52:05 --> 00:52:09
If you put B and C instead of A
and B into there you will get
623
00:52:09 --> 00:52:14
the i component is this guy plus
a2 times the second component
624
00:52:14 --> 00:52:18
which is minus some determinant
plus a3 times the third
625
00:52:18 --> 00:52:22
component which is,
again, a determinant.
626
00:52:22 --> 00:52:24
And you can check.
You get exactly the same
627
00:52:24 --> 00:52:26
expression, so everything is
fine.
628
00:52:26 --> 00:52:32
There is no contradiction in
math just yet.
629
00:52:32 --> 00:52:38
On Tuesday we will continue
with this and we will start
630
00:52:38 --> 00:52:43
going into matrices,
equations of planes and so on.
631
00:52:43 --> 00:52:46
Meanwhile, have a good weekend
and please start working on your
632
00:52:46 --> 00:52:49
Problem Sets so that you can ask
lots of questions to your TAs on
633
00:52:49 --> 00:52:51
Monday.
634
00:52:51 --> 00:52:56