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Let's see if we can get a little
bit more practice and

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intuition of what cross products
are all about.

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So in the last example,
we took a cross b.

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Let's see what happens when
we take b cross a.

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So let me erase some of this.

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I don't want to erase all of it
because it might be useful

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to give us some intuition
to compare.

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I'm going to keep that.

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Actually, I can erase
this, I think.

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So the things I have drawn
here, this was a cross b.

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Let me cordon it off so you
don't get confused.

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So that was me using the right
hand rule when I tried to do a

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cross b, and then we saw that
the magnitude of this was 25,

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and n, the direction,
pointed downwards.

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Or when I drew it here, it would
point into the page.

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So let's see what happens with
b cross a, so I'm just

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switching the order.

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b cross a.

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Well, the magnitude is going to
be the same thing, right?

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Because I'm still going to take
the magnitude of b times

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the magnitude of a times the
sine of the angle between

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them, which was pi over 6
radians and then times some

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unit vector n.

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But this is going
to be the same.

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When I multiply scalar
quantities, it doesn't matter

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what order I multiply
them in, right?

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So this is still going to be
25, whatever my units might

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have been, times
some vector n.

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And we still know that that
vector n has to be

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perpendicular to both a and b,
and now we have to figure out,

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well, is it, in being
perpendicular, it can either

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kind of point into the page here
or it could pop out of

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the page, or point
out of the page.

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So which one is it?

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And then we take our right hand
out, and we try it again.

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So what we do is we take
our right hand.

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I'm actually using my right hand
right now, although you

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can't see it, just to make sure
I draw the right thing.

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So in this example, if I take
my right hand, I take the

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index finger in the
direction of b.

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I take my middle finger in the
direction of a, so my middle

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figure is going to look
something like that, right?

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And then I have two leftover
fingers there.

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Then the thumb goes in the
direction of the cross

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product, right?

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Because your thumb has a right
angle right there.

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That's the right angle
of your thumb.

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So in this example, that's the
direction of a, this is the

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direction of b, and we're
doing b cross a.

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That's why b gets your
index finger.

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The index finger gets the first
term, your middle finger

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gets the second term, and the
thumb gets the direction of

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the cross product.

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So in this example, the
direction of the cross product

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is upwards.

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Or when we're drawing it in two
dimensions right here, the

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cross product would actually
pop out of the

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page for b cross a.

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So I'll draw it over.

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It would be the circle
with the dot.

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Or if I were to draw it
analogous to this, so this

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right here, that
was a cross b.

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And then b cross a is the exact
same magnitude, but it

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goes in the other direction.

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That's b cross a.

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It just flips in the
opposite direction.

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And that's why you have to use
your right hand, because you

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might know that, oh, something's
going to pop in or

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out of the page, et cetera, et
cetera, but you need to know

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your right hand to know
whether it goes in

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or out of the page.

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Anyway, let's see if we can
get a little bit more

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intuition of what this is all
about because this is all

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about intuition.

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And frankly, I'll tell you, the
cross product comes into

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use in a lot of concepts that
frankly we don't have a lot of

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real-life intuition, with
electrons flying through a

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magnetic field or magnetic
fields through a coil.

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A lot of things in our everyday
life experience,

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maybe if we were metal filings
living in a magnetic field--

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well, we do live in
a magnetic field.

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In a strong magnetic field,
maybe we would get an

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intuition, but it's hard to have
as deep of an intuition

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as we do for, say, falling
objects, or friction, or

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forces, or fluid dynamics even,
because we've all played

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with water.

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But anyway, let's get a little
bit more intuition.

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And let's think about why is
there that sine of theta?

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Why not just multiply the
magnitudes times each other

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and use the right hand rule and
figure out a direction?

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What is that sine of
theta all about?

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I think I need to clear this up
a little bit just so this

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could be useful.

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So why is that sine
of theta there?

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Let me redraw some vectors.

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00:05:09,780 --> 00:05:12,200
I'll draw them a
little fatter.

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So let's say that's a,
that's a, this is b.

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b doesn't always have
to be longer than a.

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So this is a and this is b.

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Now, we can think of
it a little bit.

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We could say, well, this is the
same thing as a sine theta

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times b, or we could say this
is b sine theta times a.

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I hope I'm not confusing-- all
I'm saying is you could

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interpret this as--
because these are

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just magnitudes, right?

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So it doesn't matter what order
you multiply them in.

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You could say this is a sine
theta times the magnitude of

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b, all of that in the direction
of the normal

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vector, or you could put the
sine theta the other way.

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But let's think about what
this would mean.

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a sine theta, if
this is theta.

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What is a sine theta?

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Sine is opposite over
hypotenuse, right?

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So opposite over hypotenuse.

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So this would be the
magnitude of a.

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Let me draw something.

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Let me draw a line here and
make it a real line.

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Let me draw a line there,
so I have a right angle.

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So what's a sine theta?

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This is the opposite side.

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So a sine theta is a, and sine
of theta is opposite over

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hypotenuse.

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The hypotenuse is the magnitude
of a, right?

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So sine of theta is equal to
this side, which I call o for

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opposite, over the
magnitude of a.

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So it's opposite over
the magnitude of a.

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So this term a sine theta is
actually just the magnitude of

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this line right here.

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Another way you could--
let me redraw it.

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It doesn't matter where the
vectors start from.

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All you care about is this
magnitude and direction, so

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you could shift vectors
around.

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So this vector right here, and
you could call it this

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opposite vector, that's the
same thing as this vector.

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00:07:19,880 --> 00:07:21,610
That's the same thing as this.

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I just shifted it away.

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And so another way to think
about it is, it is the

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component of vector a, right?

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00:07:30,820 --> 00:07:35,260
We're used to taking a vector
and splitting it up into x-

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and y-components, but now we're
taking a vector a, and

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00:07:38,360 --> 00:07:41,140
we're splitting it up into--
you can think of it as a

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component that's parallel to
vector b and a component that

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00:07:45,350 --> 00:07:48,430
is perpendicular to vector b.

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So a sine theta is the magnitude
of the component of

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00:07:54,460 --> 00:07:57,250
vector a that is perpendicular
to b.

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So when you're taking the cross
product of two numbers,

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you're saying, well, I don't
care about the entire

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00:08:02,500 --> 00:08:06,100
magnitude of vector a in this
example, I care about the

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00:08:06,100 --> 00:08:10,330
magnitude of vector a that is
perpendicular to vector b, and

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those are the two numbers that
I want to multiply and then

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give it that direction
as specified by

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the right hand rule.

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And I'll show you some
applications.

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This is especially important--
well, we'll use it in torque

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and we'll also use it in
magnetic fields, but it's

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important in both of those
applications to figure out the

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00:08:27,500 --> 00:08:31,610
components of the vector that
are perpendicular to either a

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00:08:31,610 --> 00:08:33,600
force or a radius in question.

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So that's why this cross product
has the sine theta

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00:08:36,659 --> 00:08:39,070
because we're taking-- so in
this, if you view it as

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00:08:39,070 --> 00:08:42,210
magnitude of a sine theta
times b, this is kind of

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00:08:42,210 --> 00:08:47,030
saying this is the magnitude
of the component of a

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00:08:47,030 --> 00:08:49,435
perpendicular to b, or
you could interpret

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00:08:49,435 --> 00:08:51,060
it the other way.

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You could interpret it as a
times b sine theta, right?

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00:08:59,120 --> 00:09:00,400
Put a parentheses here.

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00:09:00,400 --> 00:09:01,500
And then you could view
it the other way.

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00:09:01,500 --> 00:09:05,610
You could say, well, b sine
theta is the component of b

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00:09:05,610 --> 00:09:06,710
that is perpendicular to a.

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00:09:06,710 --> 00:09:11,110
Let me draw that, just to
hit the point home.

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So that's my a, that's my b.

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00:09:16,800 --> 00:09:23,260


180
00:09:23,260 --> 00:09:25,950
This is a, this is b.

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00:09:25,950 --> 00:09:29,510
So b has some component of it
that is perpendicular to a,

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00:09:29,510 --> 00:09:32,190
and that is going to look
something like-- well, I've

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00:09:32,190 --> 00:09:33,480
run out of space.

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00:09:33,480 --> 00:09:34,640
Let me draw it here.

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00:09:34,640 --> 00:09:41,210
If that's a, that's b, the
component of b that is

186
00:09:41,210 --> 00:09:42,760
perpendicular to a is going
to look like this.

187
00:09:42,760 --> 00:09:46,300


188
00:09:46,300 --> 00:09:48,370
It's going to be perpendicular
to a, and it's going to go

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00:09:48,370 --> 00:09:51,580
that far, right?

190
00:09:51,580 --> 00:09:54,230
And then you could go back to
SOH CAH TOA and you could

191
00:09:54,230 --> 00:09:58,850
prove to yourself that the
magnitude of this vector is b

192
00:09:58,850 --> 00:09:59,650
sine theta.

193
00:09:59,650 --> 00:10:01,410
So that is where the sine
theta comes from.

194
00:10:01,410 --> 00:10:03,990
It makes sure that we're not
just multiplying the vectors.

195
00:10:03,990 --> 00:10:06,320
It makes sure we're multiplying
the components of

196
00:10:06,320 --> 00:10:09,540
the vectors that are
perpendicular to each other to

197
00:10:09,540 --> 00:10:13,650
get a third vector that is
perpendicular to both of them.

198
00:10:13,650 --> 00:10:17,310
And then the people who invented
the cross product

199
00:10:17,310 --> 00:10:19,010
said, well, it's still ambiguous
because it doesn't

200
00:10:19,010 --> 00:10:21,475
tell us-- there's always two
vectors that are perpendicular

201
00:10:21,475 --> 00:10:22,300
to these two.

202
00:10:22,300 --> 00:10:24,050
One goes in, one goes out.

203
00:10:24,050 --> 00:10:25,340
They're in opposite
directions.

204
00:10:25,340 --> 00:10:26,950
And that's where the right
hand rule comes in.

205
00:10:26,950 --> 00:10:29,080
They'll say, OK, well, we're
just going to say a convention

206
00:10:29,080 --> 00:10:32,195
that you use your right hand,
point it like a gun, make all

207
00:10:32,195 --> 00:10:34,410
your fingers perpendicular,
and then you know what

208
00:10:34,410 --> 00:10:36,760
direction that vector
points in.

209
00:10:36,760 --> 00:10:38,830
Anyway, hopefully, you're
not confused.

210
00:10:38,830 --> 00:10:40,670
Now I want you to watch
the next video.

211
00:10:40,670 --> 00:10:43,890
This is actually going to be
some physics on electricity,

212
00:10:43,890 --> 00:10:46,400
magnetism and torque, and
that's essentially the

213
00:10:46,400 --> 00:10:48,250
applications of the cross
product, and it'll give you a

214
00:10:48,250 --> 00:10:49,770
little bit more intuition
of how to use it.

215
00:10:49,770 --> 00:10:51,470
See you soon.

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00:10:51,470 --> 00:00:00,000