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Thank you.
Let's continue with vectors and
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operations of them.
Remember we saw the topic
00:00:37.000 --> 00:00:46.000
yesterday was dot product.
And remember the definition of
00:00:46.000 --> 00:00:51.000
dot product,
well, the dot product of two
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vectors is obtained by
multiplying the first component
00:00:55.000 --> 00:00:59.000
with the first component,
the second with the second and
00:00:59.000 --> 00:01:01.000
so on and summing these and you
get the scalar.
00:01:01.000 --> 00:01:05.000
And the geometric
interpretation of that is that
00:01:05.000 --> 00:01:09.000
you can also take the length of
A,
00:01:09.000 --> 00:01:16.000
take the length of B multiply
them and multiply that by the
00:01:16.000 --> 00:01:22.000
cosine of the angle between the
two vectors.
00:01:22.000 --> 00:01:34.000
We have seen several
applications of that.
00:01:34.000 --> 00:01:48.000
One application is to find
lengths and angles.
00:01:48.000 --> 00:01:52.000
For example,
you can use this relation to
00:01:52.000 --> 00:01:59.000
give you the cosine of the angle
between two vectors is the dot
00:01:59.000 --> 00:02:05.000
product divided by the product
of the lengths.
00:02:05.000 --> 00:02:14.000
Another application that we
have is to detect whether two
00:02:14.000 --> 00:02:21.000
vectors are perpendicular.
To decide if two vectors are
00:02:21.000 --> 00:02:28.000
perpendicular to each other,
all we have to do is compute
00:02:28.000 --> 00:02:34.000
our dot product and see if we
get zero.
00:02:34.000 --> 00:02:41.000
And one further application
that we did not have time to
00:02:41.000 --> 00:02:49.000
discuss yesterday that I will
mention very quickly is to find
00:02:49.000 --> 00:02:59.000
components of,
let's say, a vector A along a
00:02:59.000 --> 00:03:04.000
direction u.
So some unit vector.
00:03:04.000 --> 00:03:09.000
Let me explain.
Let's say that I have some
00:03:09.000 --> 00:03:11.000
direction.
For example,
00:03:11.000 --> 00:03:13.000
the horizontal axis on this
blackboard.
00:03:13.000 --> 00:03:16.000
But it could be any direction
in space.
00:03:16.000 --> 00:03:21.000
And, to describe this
direction, maybe I have a unit
00:03:21.000 --> 00:03:26.000
vector along this axis.
Let's say that I have any of a
00:03:26.000 --> 00:03:32.000
vector A and I want to find out
what is the component of A along
00:03:32.000 --> 00:03:36.000
u.
That means what is the length
00:03:36.000 --> 00:03:42.000
of this projection of A to the
given direction?
00:03:42.000 --> 00:03:55.000
This thing here is the
component of A along u.
00:03:55.000 --> 00:04:02.000
Well, how do we find that?
Well, we know that here we have
00:04:02.000 --> 00:04:07.000
a right angle.
So this component is just
00:04:07.000 --> 00:04:13.000
length A times cosine of the
angle between A and u.
00:04:13.000 --> 00:04:18.000
But now that means I should be
able to compute it very easily
00:04:18.000 --> 00:04:23.000
because that's the same as
length A times length u times
00:04:23.000 --> 00:04:27.000
cosine theta because u is a unit
vector.
00:04:27.000 --> 00:04:33.000
It is a unit vector.
That means this is equal to one.
00:04:33.000 --> 00:04:41.000
And so that's the same as the
dot product between A and u.
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That is very easy.
And, of course,
00:04:43.000 --> 00:04:47.000
the most of just cases of that
is say, for example,
00:04:47.000 --> 00:04:50.000
we want just to find the
component along i hat,
00:04:50.000 --> 00:04:53.000
the unit vector along the x
axis.
00:04:53.000 --> 00:04:57.000
Then you do the dot product
with i hat, which is 100.
00:04:57.000 --> 00:04:59.000
What you get is the first
component.
00:04:59.000 --> 00:05:01.000
And that is,
indeed, the x component of a
00:05:01.000 --> 00:05:04.000
vector.
Similarly, say you want the z
00:05:04.000 --> 00:05:08.000
component you do the dot product
with k that gives you the last
00:05:08.000 --> 00:05:14.000
component of your vector.
But the same works with a unit
00:05:14.000 --> 00:05:21.000
vector in any direction.
So what is an application of
00:05:21.000 --> 00:05:24.000
that?
Well, for example,
00:05:24.000 --> 00:05:30.000
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
00:05:35.000 --> 00:05:41.000
end of the string and that mass
swings back and forth on a
00:05:41.000 --> 00:05:42.000
circle.
And to analyze this
00:05:42.000 --> 00:05:45.000
mechanically you want to use,
of course,
00:05:45.000 --> 00:05:50.000
Newton's Laws of Mechanics and
you want to use forces and so
00:05:50.000 --> 00:05:54.000
on,
but I claim that components of
00:05:54.000 --> 00:05:59.000
vectors are useful here to
understand what happens
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geometrically.
What are the forces exerted on
00:06:03.000 --> 00:06:10.000
this pendulum?
Well, there is its weight,
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which usually points downwards,
and there is the tension of the
00:06:21.000 --> 00:06:25.000
string.
And these two forces together
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are what explains how this
pendulum is going to move back
00:06:30.000 --> 00:06:33.000
and forth.
Now, you could try to
00:06:33.000 --> 00:06:36.000
understand the equations of
motion using x,
00:06:36.000 --> 00:06:39.000
y coordinates or x,
z or whatever you want to call
00:06:39.000 --> 00:06:41.000
them, let's say x,
y.
00:06:41.000 --> 00:06:47.000
But really what causes the
pendulum to swing back and forth
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and also to somehow stay a
constant distance are phenomenal
00:06:52.000 --> 00:06:56.000
relative to this circular
trajectory.
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For example,
maybe instead of taking
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components along the x and y
axis, we want to look at two
00:07:03.000 --> 00:07:09.000
other unit vectors.
We can look at a vector,
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let's call it T,
that is tangent to the
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trajectory.
Sorry. Can you read that?
00:07:18.000 --> 00:07:33.000
It's not very readable.
T is tangent to the trajectory.
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And, on the other hand,
we can introduce another
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vector.
Let's call that N.
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And that one is normal,
perpendicular to the
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trajectory.
And so now if you think about
00:07:55.000 --> 00:08:00.000
it you can look at the
components of the weight along
00:08:00.000 --> 00:08:06.000
the tangent direction and along
the normal direction.
00:08:06.000 --> 00:08:13.000
And so the component of F along
the tangent direction is what
00:08:13.000 --> 00:08:21.000
causes acceleration in the
direction along the trajectory.
00:08:21.000 --> 00:08:23.000
It is what causes the pendulum
to swing back and forth.
00:08:38.000 --> 00:08:45.000
And the component along N,
on the other hand.
00:08:45.000 --> 00:08:51.000
That is the part of the weight
that tends to pull our mass away
00:08:51.000 --> 00:08:54.000
from this point.
It is what is going to be
00:08:54.000 --> 00:08:56.000
responsible for the tension of
the string.
00:08:56.000 --> 00:09:02.000
It is why the string is taut
and not actually slack and with
00:09:02.000 --> 00:09:06.000
things moving all over the
place.
00:09:06.000 --> 00:09:18.000
That one is responsible for the
tension of a string.
00:09:18.000 --> 00:09:20.000
And now, of course,
if you want to compute things,
00:09:20.000 --> 00:09:23.000
well, maybe you will call this
angle theta and then you will
00:09:23.000 --> 00:09:27.000
express things explicitly using
sines and cosines and you will
00:09:27.000 --> 00:09:29.000
solve for the equations of
motion.
00:09:29.000 --> 00:09:32.000
That would be a very
interesting physics problem.
00:09:32.000 --> 00:09:35.000
But, to save time,
we are not going to do it.
00:09:35.000 --> 00:09:40.000
I'm sure you've seen that in
8.01 or similar classes.
00:09:40.000 --> 00:09:48.000
And so to find these components
we will just do dot products.
00:09:48.000 --> 00:09:56.000
Any questions?
No.
00:09:56.000 --> 00:10:01.000
OK.
Let's move onto our next topic.
00:10:01.000 --> 00:10:06.000
Here we have found things about
lengths, angles and stuff like
00:10:06.000 --> 00:10:10.000
that.
One important concept that we
00:10:10.000 --> 00:10:17.000
have not understood yet in terms
of vectors is area.
00:10:17.000 --> 00:10:25.000
Let's say that we want to find
the area of this pentagon.
00:10:25.000 --> 00:10:28.000
Well, how do we compute that
using vectors?
00:10:28.000 --> 00:10:32.000
Can we do it using vectors?
Yes we can.
00:10:32.000 --> 00:10:36.000
And that is going to be the
goal.
00:10:36.000 --> 00:10:42.000
The first thing we should do is
probably simplify the problem.
00:10:42.000 --> 00:10:44.000
We don't actually need to
bother with pentagons.
00:10:44.000 --> 00:10:48.000
All we need to know are
triangles because,
00:10:48.000 --> 00:10:51.000
for example,
you can cut that in three
00:10:51.000 --> 00:10:56.000
triangles and then sum the areas
of the triangles.
00:10:56.000 --> 00:11:05.000
Perhaps easier,
what is the area of a triangle?
00:11:05.000 --> 00:11:12.000
Let's start with a triangle in
the plane.
00:11:12.000 --> 00:11:16.000
Well, then we need two vectors
to describe it,
00:11:16.000 --> 00:11:20.000
say A and B here.
How do we find the area of a
00:11:20.000 --> 00:11:23.000
triangle?
Well, we all know base times
00:11:23.000 --> 00:11:25.000
height over two.
What is the base?
00:11:25.000 --> 00:11:30.000
What is the height?
The area of this triangle is
00:11:30.000 --> 00:11:35.000
going to be one-half of the
base, which is going to be the
00:11:35.000 --> 00:11:39.000
length of A.
And the height,
00:11:39.000 --> 00:11:47.000
well, if you call theta this
angle, then this is length B
00:11:47.000 --> 00:11:51.000
sine theta.
Now, that looks a lot like the
00:11:51.000 --> 00:11:54.000
formula we had there,
except for one little catch.
00:11:54.000 --> 00:11:58.000
This is a sine instead of a
cosine.
00:11:58.000 --> 00:12:03.000
How do we deal with that?
Well, what we could do is first
00:12:03.000 --> 00:12:10.000
find the cosine of the angle.
We know how to find the cosine
00:12:10.000 --> 00:12:17.000
of the angle using dot products.
Then solve for sine using sine
00:12:17.000 --> 00:12:22.000
square plus cosine square equals
one.
00:12:22.000 --> 00:12:25.000
And then plug that back into
here.
00:12:25.000 --> 00:12:28.000
Well, that works but it is kind
of a very complicated way of
00:12:28.000 --> 00:12:30.000
doing it.
So there is an easier way.
00:12:30.000 --> 00:12:34.000
And that is going to be
determinants,
00:12:34.000 --> 00:12:40.000
but let me explain how we get
to that maybe still doing
00:12:40.000 --> 00:12:45.000
elementary geometry and dot
products first.
00:12:45.000 --> 00:12:53.000
Let's see.
What we can do is instead of
00:12:53.000 --> 00:12:55.000
finding the sine of theta,
well,
00:12:55.000 --> 00:12:59.000
we're not good at finding sines
of angles but we are very good
00:12:59.000 --> 00:13:00.000
now at finding cosines of
angles.
00:13:00.000 --> 00:13:05.000
Maybe we can find another angle
whose cosine is the same as the
00:13:05.000 --> 00:13:09.000
sine of theta.
Well, you have already heard
00:13:09.000 --> 00:13:14.000
about complimentary angles and
how I take my vector A,
00:13:14.000 --> 00:13:18.000
my vector B here and I have an
angle theta.
00:13:18.000 --> 00:13:24.000
Well, let's say that I rotate
my vector A by 90 degrees to get
00:13:24.000 --> 00:13:34.000
a new vector A prime.
A prime is just A rotated by 90
00:13:34.000 --> 00:13:39.000
degrees.
Then the angle between these
00:13:39.000 --> 00:13:45.000
two guys, let's say theta prime,
well, theta prime is 90 degrees
00:13:45.000 --> 00:13:49.000
or pi over two gradients minus
theta.
00:13:49.000 --> 00:13:56.000
So, in particular,
cosine of theta prime is equal
00:13:56.000 --> 00:14:01.000
to sine of theta.
In particular,
00:14:01.000 --> 00:14:09.000
that means that length A,
length B, sine theta,
00:14:09.000 --> 00:14:13.000
which is what we would need to
know in order to find the area
00:14:13.000 --> 00:14:17.000
of this triangle is equal to,
well, A and A prime have the
00:14:17.000 --> 00:14:21.000
same length so let me replace
that by length of A prime.
00:14:21.000 --> 00:14:28.000
I am not changing anything,
length B, cosine theta prime.
00:14:28.000 --> 00:14:31.000
And now we have something that
is much easier for us.
00:14:31.000 --> 00:14:37.000
Because that is just A prime
dot B.
00:14:37.000 --> 00:14:40.000
That looks like a very good
plan.
00:14:40.000 --> 00:14:43.000
There is only one small thing
which is we don't know yet how
00:14:43.000 --> 00:14:48.000
to find this A prime.
Well, I think it is not very
00:14:48.000 --> 00:14:52.000
hard.
Let's see.
00:14:52.000 --> 00:14:58.000
Actually, why don't you guys do
the hard work?
00:14:58.000 --> 00:15:02.000
Let's say that I have a plane
vector A with two components a1,
00:15:02.000 --> 00:15:05.000
a2.
And I want to rotate it
00:15:05.000 --> 00:15:10.000
counterclockwise by 90 degrees.
It looks like maybe we should
00:15:10.000 --> 00:15:14.000
change some signs somewhere.
Maybe we should do something
00:15:14.000 --> 00:15:24.000
with the components.
Can you come up with an idea of
00:15:24.000 --> 00:15:34.000
what it might be?
I see a lot of people answering
00:15:34.000 --> 00:15:37.000
three.
I see some other answers,
00:15:37.000 --> 00:15:41.000
but the majority vote seems to
be number three.
00:15:41.000 --> 00:15:49.000
Minus a2 and a1.
I think I agree, so let's see.
00:15:49.000 --> 00:16:01.000
Let's say that we have this
vector A with components a1.
00:16:01.000 --> 00:16:05.000
So a1 is here.
And a2. So a2 is here.
00:16:05.000 --> 00:16:14.000
Let's rotate this box by 90
degrees counterclockwise.
00:16:14.000 --> 00:16:19.000
This box ends up there.
It's the same box just flipped
00:16:19.000 --> 00:16:23.000
on its side.
This thing here becomes a1 and
00:16:23.000 --> 00:16:31.000
this thing here becomes a2.
And that means our new vector A
00:16:31.000 --> 00:16:37.000
prime is going to be -- Well,
the first component looks like
00:16:37.000 --> 00:16:40.000
an a2 but it is pointing to the
left when a2 is positive.
00:16:40.000 --> 00:16:47.000
So, actually, it is minus a2.
And the y component is going to
00:16:47.000 --> 00:16:53.000
be the same as this guy,
so it's going to be a1.
00:16:53.000 --> 00:16:56.000
If you wanted instead to rotate
clockwise then you would do the
00:16:56.000 --> 00:17:00.000
opposite.
You would do a2 minus a1.
00:17:00.000 --> 00:17:07.000
Is that reasonably clear for
everyone?
00:17:07.000 --> 00:17:14.000
OK.
Let's continue the calculation
00:17:14.000 --> 00:17:18.000
there.
A prime, we have decided,
00:17:18.000 --> 00:17:24.000
is minus a2,
a1 dot product with let's call
00:17:24.000 --> 00:17:33.000
b1 and b2, the components of B.
Then that will be minus a2,
00:17:33.000 --> 00:17:36.000
b1 plus a1, b2 plus a1,
b2.
00:17:36.000 --> 00:17:43.000
Let me write that the other way
around, a1, b2 minus a2,
00:17:43.000 --> 00:17:46.000
b1.
And that is a quantity that you
00:17:46.000 --> 00:17:53.000
may already know under the name
of determinant of vectors A and
00:17:53.000 --> 00:17:59.000
B, which we write symbolically
using this notation.
00:17:59.000 --> 00:18:03.000
We put A and B next to each
other inside a two-by-two table
00:18:03.000 --> 00:18:09.000
and we put these verticals bars.
And that means the determinant
00:18:09.000 --> 00:18:14.000
of these numbers,
this guy times this guy minus
00:18:14.000 --> 00:18:30.000
this guy times this guy.
That is called the determinant.
00:18:30.000 --> 00:18:34.000
And geometrically what it
measures is the area,
00:18:34.000 --> 00:18:38.000
well, not of a triangle because
we did not divide by two,
00:18:38.000 --> 00:18:42.000
but of a parallelogram formed
by A and B.
00:18:42.000 --> 00:18:51.000
It measures the area of the
parallelogram with sides A and
00:18:51.000 --> 00:18:53.000
B.
And, of course,
00:18:53.000 --> 00:18:56.000
if you want the triangle then
you will just divide by two.
00:18:56.000 --> 00:19:00.000
The triangle is half the
parallelogram.
00:19:00.000 --> 00:19:04.000
There is one small catch.
The area usually is something
00:19:04.000 --> 00:19:08.000
that is going to be positive.
This guy here has no reason to
00:19:08.000 --> 00:19:16.000
be positive or negative because,
in fact, well,
00:19:16.000 --> 00:19:20.000
if you compute things you will
see that where it is supposed to
00:19:20.000 --> 00:19:24.000
go negative it depends on
whether A and B are clockwise or
00:19:24.000 --> 00:19:26.000
counterclockwise from each
other.
00:19:26.000 --> 00:19:29.000
I mean the issue that we have
-- Well,
00:19:29.000 --> 00:19:31.000
when we say the area is
one-half length A,
00:19:31.000 --> 00:19:34.000
length B,
sine theta that was assuming
00:19:34.000 --> 00:19:37.000
that theta is positive,
that its sine is positive.
00:19:37.000 --> 00:19:42.000
Otherwise, if theta is negative
maybe we need to take the
00:19:42.000 --> 00:19:47.000
absolute value of this.
Just to be more truthful,
00:19:47.000 --> 00:19:56.000
I will say the determinant is
either plus or minus the area.
00:19:56.000 --> 00:20:13.000
Any questions about this?
Yes.
00:20:13.000 --> 00:20:15.000
Sorry.
That is not a dot product.
00:20:15.000 --> 00:20:18.000
That is the usual
multiplication.
00:20:18.000 --> 00:20:25.000
That is length A times length B
times sine theta.
00:20:25.000 --> 00:20:28.000
What does that equal?
And so that is equal to the
00:20:28.000 --> 00:20:31.000
area of a parallelogram.
Sorry.
00:20:31.000 --> 00:20:39.000
Let me explain that again.
If I have two vectors A and B,
00:20:39.000 --> 00:20:45.000
I can form a parallelogram with
them or I can form a triangle.
00:20:45.000 --> 00:20:53.000
And so the area of a
parallelogram is equal to length
00:20:53.000 --> 00:21:00.000
A, length B, sine theta,
is equal to the determinant of
00:21:00.000 --> 00:21:07.000
A and B.
While the area of a triangle is
00:21:07.000 --> 00:21:09.000
one-half of that.
00:21:21.000 --> 00:21:25.000
And, again, to be truthful,
I should say these things can
00:21:25.000 --> 00:21:28.000
be positive or negative.
Depending on whether you count
00:21:28.000 --> 00:21:31.000
the angle positively or
negatively, you will get either
00:21:31.000 --> 00:21:36.000
the area or minus the area.
The area is actually the
00:21:36.000 --> 00:21:39.000
absolute value of these
quantities.
00:21:39.000 --> 00:21:49.000
Is that clear?
OK.
00:21:49.000 --> 00:21:57.000
Yes.
If you want to compute the
00:21:57.000 --> 00:21:59.000
area, you will just take the
absolute value of the
00:21:59.000 --> 00:22:00.000
determinant.
00:22:15.000 --> 00:22:19.000
I should say the area of a
parallelogram so that it is
00:22:19.000 --> 00:22:32.000
completely clear.
Sorry. Do you have a question?
00:22:32.000 --> 00:22:34.000
Explain again,
sorry, was the question how a
00:22:34.000 --> 00:22:38.000
determinant equals the area of a
parallelogram?
00:22:38.000 --> 00:22:41.000
OK.
The area of a parallelogram is
00:22:41.000 --> 00:22:45.000
going to be the base times the
height.
00:22:45.000 --> 00:22:48.000
Let's take this guy to be the
base.
00:22:48.000 --> 00:22:53.000
The length of a base will be
length of A and the height will
00:22:53.000 --> 00:22:58.000
be obtained by taking B but only
looking at the vertical part.
00:22:58.000 --> 00:23:02.000
That will be length of B times
the sine of theta.
00:23:02.000 --> 00:23:06.000
That is how I got the area of a
parallelogram as length A,
00:23:06.000 --> 00:23:09.000
length B, sine theta.
And then I did this
00:23:09.000 --> 00:23:15.000
manipulation and this trick of
rotating to find a nice formula.
00:23:15.000 --> 00:23:23.000
Yes.
You are asking ahead of what I
00:23:23.000 --> 00:23:28.000
am going to do in a few minutes.
You are asking about magnitude
00:23:28.000 --> 00:23:29.000
of A cross B.
We are going to learn about
00:23:29.000 --> 00:23:32.000
cross products in a few minutes.
And the answer is yes,
00:23:32.000 --> 00:23:34.000
but cross product is for
vectors in space.
00:23:34.000 --> 00:23:38.000
Here I was simplifying things
by doing things just in the
00:23:38.000 --> 00:23:43.000
plane.
Just bear with me for five more
00:23:43.000 --> 00:23:48.000
minutes and we will do things in
space.
00:23:48.000 --> 00:23:55.000
Yes. That is correct.
The way you compute this in
00:23:55.000 --> 00:24:00.000
practice is you just do this.
That is how you compute the
00:24:00.000 --> 00:24:04.000
determinant.
Yes.
00:24:04.000 --> 00:24:09.000
What about three dimensions?
Three dimensions we are going
00:24:09.000 --> 00:24:11.000
to do now.
More questions?
00:24:11.000 --> 00:24:26.000
Should we move on?
OK. Let's move to space.
00:24:26.000 --> 00:24:32.000
There are two things we can do
in space.
00:24:32.000 --> 00:24:36.000
And you can look for the volume
of solids or you can look for
00:24:36.000 --> 00:24:39.000
the area of surfaces.
Let me start with the easier of
00:24:39.000 --> 00:24:42.000
the two.
Let me start with volumes of
00:24:42.000 --> 00:24:49.000
solids.
And we will go back to area,
00:24:49.000 --> 00:24:53.000
I promise.
I claim that there is also a
00:24:53.000 --> 00:24:59.000
notion of determinants in space.
And that is going to tell us
00:24:59.000 --> 00:25:08.000
how to find volumes.
Let's say that we have three
00:25:08.000 --> 00:25:16.000
vectors A, B and C.
And then the definition of
00:25:16.000 --> 00:25:23.000
their determinants going to be,
the notation for that in terms
00:25:23.000 --> 00:25:28.000
of the components is the same as
over there.
00:25:28.000 --> 00:25:35.000
We put the components of A,
the components of B and the
00:25:35.000 --> 00:25:40.000
components of C inside verticals
bars.
00:25:40.000 --> 00:25:42.000
And, of course,
I have to give meaning to this.
00:25:42.000 --> 00:25:45.000
This will be a number.
And what is that number?
00:25:45.000 --> 00:25:50.000
Well, the definition I will
take is that this is a1 times
00:25:50.000 --> 00:25:55.000
the determinant of what I get by
looking in this lower right
00:25:55.000 --> 00:26:01.000
corner.
The two-by-two determinant b2,
00:26:01.000 --> 00:26:08.000
b3, c2, c3.
Then I will subtract a2 times
00:26:08.000 --> 00:26:15.000
the determinant of b1,
b3, c1, c3.
00:26:15.000 --> 00:26:22.000
And then I will add a3 times
the determinant b1,
00:26:22.000 --> 00:26:26.000
b2, c1, c2.
And each of these guys means,
00:26:26.000 --> 00:26:30.000
again, you take b2 times c3
minus c2 times b3 and this times
00:26:30.000 --> 00:26:33.000
that minus this time that and so
on.
00:26:33.000 --> 00:26:35.000
In fact, there is a total of
six terms in here.
00:26:35.000 --> 00:26:39.000
And maybe some of you have
already seen a different formula
00:26:39.000 --> 00:26:42.000
for three-by-three determinants
where you directly have the six
00:26:42.000 --> 00:26:47.000
terms.
It is the same definition.
00:26:47.000 --> 00:26:50.000
How to remember the structure
of this formula?
00:26:50.000 --> 00:26:55.000
Well, this is called an
expansion according to the first
00:26:55.000 --> 00:26:57.000
row.
So we are going to take the
00:26:57.000 --> 00:27:02.000
entries in the first row,
a1, a2, a3 And for each of them
00:27:02.000 --> 00:27:05.000
we get the term.
Namely we multiply it by a
00:27:05.000 --> 00:27:10.000
two-by-two determinant that we
get by deleting the first row
00:27:10.000 --> 00:27:16.000
and the column where we are.
Here the coefficient next to
00:27:16.000 --> 00:27:21.000
a1, when we delete this column
and this row,
00:27:21.000 --> 00:27:24.000
we are left with b2,
b3, c2, c3.
00:27:24.000 --> 00:27:29.000
The next one we take a2,
we delete the row that is in it
00:27:29.000 --> 00:27:35.000
and the column that it is in.
And we are left with b1,
00:27:35.000 --> 00:27:38.000
b3, c1, c3.
And, similarly,
00:27:38.000 --> 00:27:41.000
with a3, we take what remains,
which is b1,
00:27:41.000 --> 00:27:45.000
b2, c1, c2.
Finally, last but not least,
00:27:45.000 --> 00:27:51.000
there is a minus sign here for
the second guy.
00:27:51.000 --> 00:28:01.000
It looks like a weird formula.
I mean it is a little bit weird.
00:28:01.000 --> 00:28:04.000
But it is a formula that you
should learn because it is
00:28:04.000 --> 00:28:06.000
really, really useful for a lot
of things.
00:28:06.000 --> 00:28:10.000
I should say if this looks very
artificial to you and you would
00:28:10.000 --> 00:28:14.000
like to know more there is more
in the notes,
00:28:14.000 --> 00:28:17.000
so read the notes.
They will tell you a bit more
00:28:17.000 --> 00:28:20.000
about what this means,
where it comes from and so on.
00:28:20.000 --> 00:28:23.000
If you want to know a lot more
then some day you should take
00:28:23.000 --> 00:28:26.000
18.06,
Linear Algebra where you will
00:28:26.000 --> 00:28:29.000
learn a lot more about
determinants in N dimensional
00:28:29.000 --> 00:28:32.000
space with N vectors.
And there is a generalization
00:28:32.000 --> 00:28:36.000
of this in arbitrary dimensions.
In this class,
00:28:36.000 --> 00:28:39.000
we will only deal with two or
three dimensions.
00:28:39.000 --> 00:28:44.000
Yes.
Why is the negative there?
00:28:44.000 --> 00:28:45.000
Well, that is a very good
question.
00:28:45.000 --> 00:28:49.000
It has to be there so that this
will actually equal,
00:28:49.000 --> 00:28:53.000
well, what I am going to say
right now is that this will give
00:28:53.000 --> 00:28:55.000
us the volume of [a box?]
with sides A,
00:28:55.000 --> 00:28:57.000
B, C.
And the formula just doesn't
00:28:57.000 --> 00:28:59.000
work if you don't put the
negative.
00:28:59.000 --> 00:29:02.000
There is a more fundamental
reason which has to do with
00:29:02.000 --> 00:29:06.000
orientation of space and the
fact that if you switch two
00:29:06.000 --> 00:29:09.000
coordinates in space then
basically you change what is
00:29:09.000 --> 00:29:12.000
called the handedness of the
coordinates.
00:29:12.000 --> 00:29:14.000
If you look at your right hand
and your left hand,
00:29:14.000 --> 00:29:16.000
they are not actually the same.
They are mirror images.
00:29:16.000 --> 00:29:18.000
And, if you squared two
coordinate axes,
00:29:18.000 --> 00:29:21.000
that is what you get.
That is the fundamental reason
00:29:21.000 --> 00:29:24.000
for the minus.
Again, we don't need to think
00:29:24.000 --> 00:29:33.000
too much about that.
All we need in this class is
00:29:33.000 --> 00:29:38.000
the formula.
Why do we care about this
00:29:38.000 --> 00:29:43.000
formula?
It is because of the theorem
00:29:43.000 --> 00:29:52.000
that says that geometrically the
determinant of the three vectors
00:29:52.000 --> 00:29:58.000
A, B, C is, again,
plus or minus.
00:29:58.000 --> 00:30:00.000
This determinant could be
positive or negative.
00:30:00.000 --> 00:30:03.000
See those minuses and all sorts
of stuff.
00:30:03.000 --> 00:30:14.000
Plus or minus the volume of the
parallelepiped.
00:30:14.000 --> 00:30:20.000
That is just a fancy name for a
box with parallelogram sides,
00:30:20.000 --> 00:30:24.000
in case you wonder,
with sides A,
00:30:24.000 --> 00:30:29.000
B and C.
You take the three vectors A,
00:30:29.000 --> 00:30:35.000
B and C and you form a box
whose sides are all
00:30:35.000 --> 00:30:44.000
parallelograms.
And when its volume is going to
00:30:44.000 --> 00:30:59.000
be the determinant.
Other questions?
00:30:59.000 --> 00:31:11.000
I'm sorry.
I cannot quite hear you.
00:31:11.000 --> 00:31:12.000
Yes.
We are going to see how to do
00:31:12.000 --> 00:31:14.000
it geometrically without a
determinant,
00:31:14.000 --> 00:31:17.000
but then you will see that you
actually need a determinant to
00:31:17.000 --> 00:31:21.000
compute it no matter what.
We are going to go back to this
00:31:21.000 --> 00:31:24.000
and see another formula for
volume, but you will see that
00:31:24.000 --> 00:31:26.000
really I am cheating.
I mean somehow computationally
00:31:26.000 --> 00:31:30.000
the only way to compute it is
really to use a determinant.
00:31:43.000 --> 00:31:44.000
That is correct.
In general, I mean,
00:31:44.000 --> 00:31:47.000
actually, I could say if you
look at the two-by-two
00:31:47.000 --> 00:31:50.000
determinant, see,
you can also explain it in
00:31:50.000 --> 00:31:54.000
terms of this extension.
If you take a1 and multiply by
00:31:54.000 --> 00:31:57.000
this one-by-one determinant b2,
then you take a2 and you
00:31:57.000 --> 00:32:00.000
multiply it by this one-by-one
determinant b1 but you put a
00:32:00.000 --> 00:32:02.000
minus sign.
And in general,
00:32:02.000 --> 00:32:06.000
indeed, when you expand,
you would stop putting plus,
00:32:06.000 --> 00:32:08.000
minus, plus,
minus alternating.
00:32:08.000 --> 00:32:15.000
More about that in 18.06.
Yes.
00:32:15.000 --> 00:32:18.000
There is a way to do it based
on other rows as well,
00:32:18.000 --> 00:32:20.000
but then you have to be very
careful with the sign vectors.
00:32:20.000 --> 00:32:23.000
I will refer you to the notes
for that.
00:32:23.000 --> 00:32:25.000
I mean you could also do it
with a column,
00:32:25.000 --> 00:32:28.000
by the way.
I mean be careful about the
00:32:28.000 --> 00:32:30.000
sign rules.
Given how little we will use
00:32:30.000 --> 00:32:33.000
determinants in this class,
I mean we will use them in a
00:32:33.000 --> 00:32:36.000
way that is fundamental,
but we won't compute much.
00:32:36.000 --> 00:32:47.000
Let's say this is going to be
enough for us for now.
00:32:47.000 --> 00:32:50.000
After determinants now I can
tell you about cross product.
00:32:50.000 --> 00:32:53.000
And cross product is going to
be the answer to your question
00:32:53.000 --> 00:32:54.000
about area.
00:33:32.000 --> 00:33:45.000
OK.
Let me move onto cross product.
00:33:45.000 --> 00:33:53.000
Cross product is something that
you can apply to two vectors in
00:33:53.000 --> 00:33:56.000
space.
And by that I mean really in
00:33:56.000 --> 00:33:59.000
three-dimensional space.
This is something that is
00:33:59.000 --> 00:34:05.000
specific to three dimensions.
The definition A cross B -- It
00:34:05.000 --> 00:34:11.000
is important to really do your
multiplication symbol well so
00:34:11.000 --> 00:34:16.000
that you don't mistake it with a
dot product.
00:34:16.000 --> 00:34:23.000
Well, that is going to be a
vector.
00:34:23.000 --> 00:34:26.000
That is another reason not to
confuse it with dot product.
00:34:26.000 --> 00:34:30.000
Dot product gives you a number.
Cross product gives you a
00:34:30.000 --> 00:34:32.000
vector.
They are really completely
00:34:32.000 --> 00:34:35.000
different operations.
They are both called product
00:34:35.000 --> 00:34:38.000
because someone could not come
up with a better name,
00:34:38.000 --> 00:34:42.000
but they are completely
different operations.
00:34:42.000 --> 00:34:45.000
What do we do to do the cross
product of A and B?
00:34:45.000 --> 00:34:47.000
Well, we do something very
strange.
00:34:47.000 --> 00:34:50.000
Just as I have told you that a
determinant is something where
00:34:50.000 --> 00:34:54.000
we put numbers and we get a
number, I am going to violate my
00:34:54.000 --> 00:34:59.000
own rule.
I am going to put together a
00:34:59.000 --> 00:35:06.000
determinant in which -- Well,
the last two rows are the
00:35:06.000 --> 00:35:11.000
components of the vectors A and
B but the first row strangely
00:35:11.000 --> 00:35:15.000
consists for unit vectors i,
j, k.
00:35:15.000 --> 00:35:19.000
What does that mean?
Well, that is not a determinant
00:35:19.000 --> 00:35:21.000
in the usual sense.
If you try to put that into
00:35:21.000 --> 00:35:24.000
your calculator,
it will tell you there is an
00:35:24.000 --> 00:35:26.000
error.
I don't know how to put vectors
00:35:26.000 --> 00:35:28.000
in there.
I want numbers.
00:35:28.000 --> 00:35:32.000
What is means is it is symbolic
notation that helps you remember
00:35:32.000 --> 00:35:35.000
what the formula is.
The actual formula is,
00:35:35.000 --> 00:35:40.000
well, you use this definition.
And, if you use that
00:35:40.000 --> 00:35:47.000
definition, you see that it is i
hat times some number.
00:35:47.000 --> 00:35:55.000
Let me write it as determinant
of a2, a3, b2,
00:35:55.000 --> 00:36:02.000
b3 times i hat minus
determinant a1,
00:36:02.000 --> 00:36:11.000
a3, b1, b3, j hat plus a1,
a2, b1, b2, k hat.
00:36:11.000 --> 00:36:15.000
And so that is the actual
definition in a way that makes
00:36:15.000 --> 00:36:18.000
complete sense,
but to remember this formula
00:36:18.000 --> 00:36:23.000
without too much trouble it is
much easier to think about it in
00:36:23.000 --> 00:36:27.000
these terms here.
That is the definition and it
00:36:27.000 --> 00:36:30.000
gives you a vector.
Now, as usual with definitions,
00:36:30.000 --> 00:36:32.000
the question is what is it good
for?
00:36:32.000 --> 00:36:36.000
What is the geometric meaning
of this very strange operation?
00:36:36.000 --> 00:36:48.000
Why do we bother to do that?
Here is what it does
00:36:48.000 --> 00:36:52.000
geometrically.
Remember a vector has two
00:36:52.000 --> 00:36:56.000
different things.
It has a length and it has a
00:36:56.000 --> 00:37:01.000
direction.
Let's start with the length.
00:37:01.000 --> 00:37:15.000
A length of a cross product is
the area of the parallelogram in
00:37:15.000 --> 00:37:24.000
space formed by the vectors A
and B.
00:37:24.000 --> 00:37:27.000
Now, if you have a
parallelogram in space,
00:37:27.000 --> 00:37:31.000
you can find its area just by
doing this calculation when you
00:37:31.000 --> 00:37:33.000
know the coordinates of the
points.
00:37:33.000 --> 00:37:35.000
You do this calculation and
then you take the length.
00:37:35.000 --> 00:37:40.000
You take this squared plus that
squared plus that squared,
00:37:40.000 --> 00:37:43.000
square root.
It looks like a very
00:37:43.000 --> 00:37:47.000
complicated formula but it works
and, actually,
00:37:47.000 --> 00:37:49.000
it is the simplest way to do
it.
00:37:49.000 --> 00:37:52.000
This time we don't actually
need to put plus or minus
00:37:52.000 --> 00:37:55.000
because the length of a vector
is always positive.
00:37:55.000 --> 00:38:00.000
We don't have to worry about
that.
00:38:00.000 --> 00:38:04.000
And what is even more magical
is that not only is the length
00:38:04.000 --> 00:38:07.000
remarkable but the direction is
also remarkable.
00:38:07.000 --> 00:38:24.000
The direction of A cross B is
perpendicular to the plane of a
00:38:24.000 --> 00:38:33.000
parallelogram.
Our two vectors A and B
00:38:33.000 --> 00:38:41.000
together in a plane.
What I am telling you is that
00:38:41.000 --> 00:38:51.000
for vector A cross B will point,
will stick straight out of that
00:38:51.000 --> 00:38:56.000
plane perpendicularly to it.
In fact, I would have to be
00:38:56.000 --> 00:38:58.000
more precise.
There are two ways that you can
00:38:58.000 --> 00:39:02.000
be perpendicular to this plane.
You can be perpendicular
00:39:02.000 --> 00:39:06.000
pointing up or pointing down.
How do I decide which?
00:39:06.000 --> 00:39:16.000
Well, there is something called
the right-hand rule.
00:39:16.000 --> 00:39:18.000
What does the right-hand rule
say?
00:39:18.000 --> 00:39:21.000
Well, there are various
versions for right-hand rule
00:39:21.000 --> 00:39:23.000
depending on which country you
learn about it.
00:39:23.000 --> 00:39:26.000
In France, given the culture,
you even learn about it in
00:39:26.000 --> 00:39:28.000
terms of a cork screw and a wine
bottle.
00:39:28.000 --> 00:39:33.000
I will just use the usual
version here.
00:39:33.000 --> 00:39:35.000
You take your right hand.
If you are left-handed,
00:39:35.000 --> 00:39:38.000
remember to take your right
hand and not the left one.
00:39:38.000 --> 00:39:43.000
The other right, OK?
Then place your hand to point
00:39:43.000 --> 00:39:46.000
in the direction of A.
Let's say my right hand is
00:39:46.000 --> 00:39:50.000
going in that direction.
Now, curl your fingers so that
00:39:50.000 --> 00:39:54.000
they point towards B.
Here that would be kind of into
00:39:54.000 --> 00:39:56.000
the blackboard.
Don't snap any bones.
00:39:56.000 --> 00:40:00.000
If it doesn't quite work then
rotate your arms so that you can
00:40:00.000 --> 00:40:04.000
actually physically do it.
Then get your thumb to stick
00:40:04.000 --> 00:40:07.000
straight out.
Well, here my thumb is going to
00:40:07.000 --> 00:40:11.000
go up.
And that tells me that A cross
00:40:11.000 --> 00:40:16.000
B will go up.
Let me write that down while
00:40:16.000 --> 00:40:19.000
you experiment with it.
Again, try not to enjoy
00:40:19.000 --> 00:40:20.000
yourselves.
00:40:30.000 --> 00:40:39.000
First, your right hand points
parallel to vector A.
00:40:39.000 --> 00:40:47.000
Then your fingers point in the
direction of B.
00:40:47.000 --> 00:40:53.000
Then your thumb,
when you stick it out,
00:40:53.000 --> 00:41:00.000
is going to point in the
direction of A cross B.
00:41:00.000 --> 00:41:29.000
Let's do a quick example.
Where is my quick example? Here.
00:41:29.000 --> 00:41:32.000
Let's take i cross j.
00:41:40.000 --> 00:41:47.000
I see most of you going in the
right direction.
00:41:47.000 --> 00:41:51.000
If you have it pointing in the
wrong direction,
00:41:51.000 --> 00:41:56.000
it might mean that you are
using your left hand,
00:41:56.000 --> 00:42:01.000
for example.
Example, I claim that i cross j
00:42:01.000 --> 00:42:07.000
equals k.
Let's see. I points towards us.
00:42:07.000 --> 00:42:12.000
J point to our right.
I guess this is your right.
00:42:12.000 --> 00:42:16.000
I think.
And then your thumb is going to
00:42:16.000 --> 00:42:19.000
point up.
That tells us it is roughly
00:42:19.000 --> 00:42:21.000
pointing up.
And, of course,
00:42:21.000 --> 00:42:24.000
the length should be one
because if you take the unit
00:42:24.000 --> 00:42:27.000
square in the x,
y plane, its area is one.
00:42:27.000 --> 00:42:29.000
And the direction should be
vertical.
00:42:29.000 --> 00:42:34.000
Because it should be
perpendicular to the x,
00:42:34.000 --> 00:42:37.000
y plane.
It looks like i cross j will be
00:42:37.000 --> 00:42:41.000
k.
Well, let's check with the
00:42:41.000 --> 00:42:43.000
definition i,
j, k.
00:42:43.000 --> 00:42:51.000
What is i? I is one, zero, zero.
J is zero, one, zero.
00:42:51.000 --> 00:42:58.000
The coefficient of i will be
zero times zero minus zero times
00:42:58.000 --> 00:43:00.000
one.
That is zero.
00:43:00.000 --> 00:43:04.000
The coefficient of j will be
one time zero minus zero times
00:43:04.000 --> 00:43:06.000
zero, that is a zero,
minus zero j.
00:43:06.000 --> 00:43:11.000
It doesn't matter.
And the coefficient of k will
00:43:11.000 --> 00:43:14.000
be one times one,
that is one,
00:43:14.000 --> 00:43:17.000
minus zero times zero,
so one k.
00:43:17.000 --> 00:43:22.000
So we do get i cross j equals k
both ways.
00:43:22.000 --> 00:43:24.000
In this case,
it is easier to do it
00:43:24.000 --> 00:43:27.000
geometrically.
If I give you no complicated
00:43:27.000 --> 00:43:32.000
vectors, probably you will
actually want to do the
00:43:32.000 --> 00:43:41.000
calculation.
Any questions? Yes.
00:43:41.000 --> 00:43:45.000
The coefficient of k,
remember I delete the first row
00:43:45.000 --> 00:43:50.000
and the last column so I get
this two-by-two determinant.
00:43:50.000 --> 00:43:54.000
And that two-by-two determinant
is one times one minus zero
00:43:54.000 --> 00:43:56.000
times zero so that gives me a
one.
00:43:56.000 --> 00:43:59.000
That is what you do with
two-by-two determinants.
00:43:59.000 --> 00:44:03.000
Similarly for [UNINTELLIGIBLE],
but [UNINTELLIGIBLE]
00:44:03.000 --> 00:44:11.000
turn out to be zero.
More questions?
00:44:11.000 --> 00:44:14.000
Yes.
Let me repeat how I got the one
00:44:14.000 --> 00:44:18.000
in front of k.
Remember the definition of a
00:44:18.000 --> 00:44:24.000
determinant I expand according
to the entries in the first row.
00:44:24.000 --> 00:44:28.000
When I get to k what I do is
delete the first row and I
00:44:28.000 --> 00:44:32.000
delete the last column,
the column that contains k.
00:44:32.000 --> 00:44:37.000
I delete these guys and these
guys and I am left with this
00:44:37.000 --> 00:44:41.000
two-by-two determinant.
Now, a two-by-two determinant,
00:44:41.000 --> 00:44:47.000
you multiply according to this
downward diagonal and then minus
00:44:47.000 --> 00:44:50.000
this times that.
One times one,
00:44:50.000 --> 00:44:55.000
let me see here,
I got one k because that is one
00:44:55.000 --> 00:45:00.000
times one minus zero times zero
equals one.
00:45:00.000 --> 00:45:03.000
Sorry.
That is really hard to read.
00:45:03.000 --> 00:45:11.000
Maybe it will be easier that
way.
00:45:11.000 --> 00:45:19.000
Yes.
Let's try.
00:45:19.000 --> 00:45:23.000
If I do the same for i,
I think I will also get zero.
00:45:23.000 --> 00:45:28.000
Let's do the same for i.
I take i, I delete the first
00:45:28.000 --> 00:45:33.000
row, I delete the first column,
I get this two-by-two
00:45:33.000 --> 00:45:36.000
determinant here and I get zero
times zero,
00:45:36.000 --> 00:45:39.000
that is zero,
minus zero times one.
00:45:39.000 --> 00:45:43.000
That is the other trick
question.
00:45:43.000 --> 00:45:49.000
Zero times one is zero as well.
So that zero minus zero is
00:45:49.000 --> 00:45:52.000
zero.
I hope on Monday you should get
00:45:52.000 --> 00:45:55.000
more practice in recitation
about how to compute
00:45:55.000 --> 00:45:58.000
determinants.
Hopefully, it will become very
00:45:58.000 --> 00:46:01.000
easy for you all to compute this
next.
00:46:01.000 --> 00:46:04.000
I know the first time it is
kind of a shock because there
00:46:04.000 --> 00:46:07.000
are a lot of numbers and a lot
of things to do.
00:47:02.000 --> 00:47:08.000
Let me return to the question
that you asked a bit earlier
00:47:08.000 --> 00:47:13.000
about how do you find actually
volume if I don't want to know
00:47:13.000 --> 00:47:24.000
about determinants?
Well, let's have another look
00:47:24.000 --> 00:47:31.000
at the volume.
Let's say that I have three
00:47:31.000 --> 00:47:37.000
vectors.
Let me put them this way,
00:47:37.000 --> 00:47:43.000
A, B and C.
And let's try to see how else I
00:47:43.000 --> 00:47:49.000
could think about the volume of
this box.
00:47:49.000 --> 00:47:54.000
Probably you know that the
volume of a parallelepiped is
00:47:54.000 --> 00:47:57.000
the area of a base times the
height.
00:47:57.000 --> 00:48:04.000
Sorry.
The volume is the area of a
00:48:04.000 --> 00:48:12.000
base times the height.
How do we do that in practice?
00:48:12.000 --> 00:48:15.000
Well, what is the area of a
base?
00:48:15.000 --> 00:48:21.000
The base is a parallelogram in
space with sides B and C.
00:48:21.000 --> 00:48:23.000
How do we find the area of the
parallelogram in space?
00:48:23.000 --> 00:48:28.000
Well, we just discovered that.
We can do it by taking that
00:48:28.000 --> 00:48:30.000
cross product.
The area of a base,
00:48:30.000 --> 00:48:33.000
well, we take the cross product
of B and C.
00:48:33.000 --> 00:48:36.000
That is not quite it because
this is a vector.
00:48:36.000 --> 00:48:40.000
We would like a number while we
take its length.
00:48:40.000 --> 00:48:44.000
That is pretty good.
What about the height?
00:48:44.000 --> 00:48:48.000
Well, the height is going to be
the component of A in the
00:48:48.000 --> 00:48:51.000
direction that is perpendicular
to the base.
00:48:51.000 --> 00:48:53.000
Let's take a direction that is
perpendicular to the base.
00:48:53.000 --> 00:48:57.000
Let's call that N,
a unit vector in that
00:48:57.000 --> 00:49:00.000
direction.
Then we can get the height by
00:49:00.000 --> 00:49:04.000
taking A dot n.
That is what we saw at the
00:49:04.000 --> 00:49:10.000
beginning of class that A dot n
will tell me how much A goes in
00:49:10.000 --> 00:49:17.000
the direction of n.
Are you still with me?
00:49:17.000 --> 00:49:22.000
OK.
Let's keep going.
00:49:22.000 --> 00:49:24.000
Let's think about this vector
n.
00:49:24.000 --> 00:49:29.000
How do I get it?
Well, I can get it by actually
00:49:29.000 --> 00:49:34.000
using cross product as well.
Because I said the direction
00:49:34.000 --> 00:49:37.000
perpendicular to two vectors I
can get by taking that cross
00:49:37.000 --> 00:49:40.000
product and looking at that
direction.
00:49:40.000 --> 00:49:47.000
This is still B cross C length.
And this one is,
00:49:47.000 --> 00:49:56.000
so I claim, n can be obtained
by taking D cross C.
00:49:56.000 --> 00:49:58.000
Well, that comes in the right
direction but it is not a unit
00:49:58.000 --> 00:50:01.000
vector.
How do I get a unit vector?
00:50:01.000 --> 00:50:06.000
I divide by the length.
Thanks.
00:50:06.000 --> 00:50:14.000
I take B cross C and I divide
by length B cross C.
00:50:14.000 --> 00:50:20.000
Well, now I can probably
simplify between these two guys.
00:50:20.000 --> 00:50:38.000
And so what I will get -- What
I get out of this is that my
00:50:38.000 --> 00:50:53.000
volume equals A dot product with
vector B cross C.
00:50:53.000 --> 00:50:55.000
But, of course,
I have to be careful in which
00:50:55.000 --> 00:50:56.000
order I do it.
If I do it the other way
00:50:56.000 --> 00:50:58.000
around, A dot B,
I get a number.
00:50:58.000 --> 00:51:00.000
I cannot cross that.
I really have to do the cross
00:51:00.000 --> 00:51:03.000
product first.
I get the new vector.
00:51:03.000 --> 00:51:09.000
Then my dot product.
The fact is that the
00:51:09.000 --> 00:51:16.000
determinant of A,
B, C is equal to this so-called
00:51:16.000 --> 00:51:20.000
triple product.
Well, that looks good
00:51:20.000 --> 00:51:23.000
geometrically.
Let's try to check whether it
00:51:23.000 --> 00:51:27.000
makes sense with the formulas,
just one small thing.
00:51:27.000 --> 00:51:32.000
We saw the determinant is a1
times determinant b2,
00:51:32.000 --> 00:51:37.000
b3, c2, c3 minus a2 times
something plus a3 times
00:51:37.000 --> 00:51:42.000
something.
I will let you fill in the
00:51:42.000 --> 00:51:45.000
numbers.
That is this guy.
00:51:45.000 --> 00:51:48.000
What about this guy?
Well, dot product,
00:51:48.000 --> 00:51:50.000
we take the first component of
A, that is a1,
00:51:50.000 --> 00:51:53.000
we multiply by the first
component of B cross C.
00:51:53.000 --> 00:51:55.000
What is the first component of
B cross C?
00:51:55.000 --> 00:52:05.000
Well, it is this determinant
b2, b3, c2, c3.
00:52:05.000 --> 00:52:09.000
If you put B and C instead of A
and B into there you will get
00:52:09.000 --> 00:52:14.000
the i component is this guy plus
a2 times the second component
00:52:14.000 --> 00:52:18.000
which is minus some determinant
plus a3 times the third
00:52:18.000 --> 00:52:22.000
component which is,
again, a determinant.
00:52:22.000 --> 00:52:24.000
And you can check.
You get exactly the same
00:52:24.000 --> 00:52:26.000
expression, so everything is
fine.
00:52:26.000 --> 00:52:32.000
There is no contradiction in
math just yet.
00:52:32.000 --> 00:52:38.000
On Tuesday we will continue
with this and we will start
00:52:38.000 --> 00:52:43.000
going into matrices,
equations of planes and so on.
00:52:43.000 --> 00:52:46.000
Meanwhile, have a good weekend
and please start working on your
00:52:46.000 --> 00:52:49.000
Problem Sets so that you can ask
lots of questions to your TAs on
00:52:49.000 --> 00:52:51.000
Monday.