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2
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Let's do a little compare and
contrast between the dot

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product and the cross product.

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Let me just make two vectors--
just visually draw them.

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And maybe if we have time,
we'll, actually figure out

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00:00:15,620 --> 00:00:17,970
some dot and cross products
with real vectors.

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00:00:17,970 --> 00:00:36,170


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Let's call the first one--
That's the angle between them.

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00:00:47,070 --> 00:00:47,550
OK.

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So let's just go over the
definitions and then we'll

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work on the intuition.

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And hopefully, you have a little
bit of both already.

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So what is a dot b?

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Well first of all, that's the
exact same thing as b dot a.

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Order does not matter when you
take the dot product because

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you end up with just a number.

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And that is equal to the
magnitude of a times the

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00:01:13,440 --> 00:01:21,145
magnitude of b times cosine
of the angle between them.

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Let's look at the definition
of the cross product.

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What is a cross b?

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Well first of all, that does
not equal b cross a.

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It actually equals the opposite
direction, or you

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could view it as the negative
of b cross a.

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Because the vector that you end
up with ends up flipped,

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00:01:42,980 --> 00:01:44,510
whichever order you do it in.

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But a cross b, that is equal to
the magnitude of vector a

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times the magnitude of vector
b-- so far, it looks a lot

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like the dot product, but this
is where the diverge is--

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times the sine of the
angle between them.

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The sine of the angle
between them.

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And this is where it
really diverges.

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When we took the dot
product, we just

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ended up with a number.

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This is just a number.

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There's no direction here.

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This is just a scalar
quantity.

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But the cross product, we take
the magnitude of a times the

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magnitude of b, times the sine
of the angle between them, and

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that provides a magnitude, but
it also has a direction.

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And that direction is provided
by this normal vector.

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It's a unit vector.

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A unit vector gets that
little hat on it.

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It's a unit vector, and
what direction is it?

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Well, that's defined by
the right hand rule.

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This is a vector.

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It's perpendicular
to both a and b.

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And then you might say, a and
b, the way I drew them,

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they're both sitting in the
plane of this video screen, or

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your video screen.

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00:02:53,200 --> 00:02:55,670
So in order for something to
be perpendicular to both of

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these, it either has to pop out
of the screen or pop into

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the screen, right?

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00:03:00,240 --> 00:03:02,110
And when you learned about the
cross product, I said there's

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two ways of showing a vector
popping out of the screen.

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It looks like that because
that's the tip of an arrow.

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And to show a vector going into
the screen, it's like

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that because that is the
back of an arrow.

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The rear end of an arrow.

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00:03:14,980 --> 00:03:17,020
So how do you know which
of these two it is?

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Because both of these vectors
are perpendicular to a and b.

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That's where you take your right
hand and you use the

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right hand rule.

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So you take your index finger
in the direction of a, your

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middle finger in the direction
of b, and then your thumb

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00:03:29,730 --> 00:03:32,210
points in the direction of n.

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So let's do that.

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I'm looking at my hand.

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It's not an easy thing to do
with your right hand, but your

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right hand is going to look
something like this.

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Your index finger will go
in the direction of a.

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00:03:55,820 --> 00:03:59,200
Your middle finger goes
in the direction of b.

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00:03:59,200 --> 00:04:02,040
So that's my middle finger.

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00:04:02,040 --> 00:04:04,610
And then my other two fingers
just do what they need to do.

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00:04:04,610 --> 00:04:06,160
I like to just bend them
out of the way.

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So they just curl
around my hand.

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00:04:11,940 --> 00:04:14,280
And then what direction
is my thumb in?

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00:04:14,280 --> 00:04:17,820
My thumb-- well, actually I drew
it at the wrong angle.

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00:04:17,820 --> 00:04:21,300
My thumb is actually going
in this direction, right?

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00:04:21,300 --> 00:04:22,730
Into the page.

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00:04:22,730 --> 00:04:23,830
This is the top of my hand.

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00:04:23,830 --> 00:04:25,550
These are like my veins.

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00:04:25,550 --> 00:04:29,450
Or, if I actually drew it
correctly, where you would see

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00:04:29,450 --> 00:04:32,690
your hand from side-- so it
would look like this.

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You would see your pinky.

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Your palm and your pinky
would be like that.

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And your other finger
like this.

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00:04:39,630 --> 00:04:43,460
Your middle finger would go
in the direction of b.

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00:04:43,460 --> 00:04:46,900
Your index finger goes in the
direction of a, and you

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00:04:46,900 --> 00:04:49,430
wouldn't even see your thumb,
because your thumb is pointing

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00:04:49,430 --> 00:04:49,960
straight down.

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But I think you get the point.
a cross b, this n vector is

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pointing straight down.

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It's a unit vector.

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00:04:55,630 --> 00:04:56,750
And this provides
the magnitude.

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00:04:56,750 --> 00:04:59,240
Unit vector just means it
has a magnitude of one.

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00:04:59,240 --> 00:05:02,560
So the magnitudes of the cross
and the dot products seem

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00:05:02,560 --> 00:05:03,360
pretty close.

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00:05:03,360 --> 00:05:06,440
They both have the magnitude
of both vectors there.

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Dot product, cosine theta.

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00:05:08,720 --> 00:05:09,980
Cross product sine of theta.

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But then, the huge difference
is that sine

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00:05:11,710 --> 00:05:12,730
of theta has a direction.

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00:05:12,730 --> 00:05:14,410
It is a different
vector that is

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00:05:14,410 --> 00:05:16,360
perpendicular to both of these.

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00:05:16,360 --> 00:05:18,340
Now, let's get the intuition.

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00:05:18,340 --> 00:05:20,830
And if you've watched the videos
on the dot and the

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00:05:20,830 --> 00:05:23,420
cross product, hopefully you
have a little intuition.

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But I review it because I think
it all fits together

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when you see them
with each other.

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First, let's study a,
b cosine of theta.

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If you watched the dot product
video, cosine of theta, if you

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00:05:49,470 --> 00:05:51,860
took, let's say, b
cosine of theta.

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00:05:51,860 --> 00:05:54,750
What is b cosine of theta?

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00:05:54,750 --> 00:05:56,930
b cosine of theta-- and you
could work it out on your own

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00:05:56,930 --> 00:06:03,690
time-- if you say cosine is
adjacent over hypotenuse, the

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00:06:03,690 --> 00:06:06,140
magnitude of b cosine theta is
actually going to be the

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00:06:06,140 --> 00:06:09,520
magnitude of, if you dropped a
perpendicular-- I'll use a

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different color here-- if you
dropped a perpendicular here,

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00:06:14,940 --> 00:06:17,250
this length right here--
that's b cosine theta.

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00:06:17,250 --> 00:06:18,680
Let me draw it separately.

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I don't want to mess up
this picture too much.

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So if that's b.

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If that's a-- And that's b.

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That's a.

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This is theta.

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00:06:42,540 --> 00:06:47,236
b cosine theta, if you drop a
line perpendicular to a, this

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is a right angle.

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00:06:48,590 --> 00:06:51,350
b cosine theta, adjacent
over hypotenuse is

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equal to cosine theta.

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So it would be the projection
of b going in the same

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direction as a.

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So it would be this magnitude.

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137
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That is b cosine theta.

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So the magnitude of that vector
right there is the

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00:07:13,180 --> 00:07:16,720
magnitude of b cosine
of theta.

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00:07:16,720 --> 00:07:18,910
So when you're taking the dot
product, at least the example

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00:07:18,910 --> 00:07:22,970
I just did, if you view it as
the magnitude of a times the

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00:07:22,970 --> 00:07:26,570
magnitude of b cosine theta,
you're saying what part of b

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00:07:26,570 --> 00:07:28,680
goes in the same
direction as a?

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And whatever that magnitude is,
let me just multiply that

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00:07:30,870 --> 00:07:31,980
times the magnitude of a.

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00:07:31,980 --> 00:07:34,360
And I have the dot product.

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Let's take the pieces that
go the same direction and

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00:07:35,960 --> 00:07:36,590
multiply them.

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So how much do they
move together?

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00:07:38,040 --> 00:07:40,310
Or do they point together?

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00:07:40,310 --> 00:07:41,510
Or you could view it
the other way.

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You could view the dot product
as-- and I did this in the dot

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00:07:44,050 --> 00:07:50,510
product video-- you could view
it as a cosine of theta, b.

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00:07:50,510 --> 00:07:51,480
Because it doesn't matter.

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These are all scalar quantities,
so it doesn't

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00:07:52,990 --> 00:07:55,560
matter what order you take
the multiplication in.

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00:07:55,560 --> 00:07:57,270
And a cosine theta is
the same thing.

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00:07:57,270 --> 00:08:00,350
It's the magnitude of the a
vector that's going in the

159
00:08:00,350 --> 00:08:01,060
same direction of b.

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00:08:01,060 --> 00:08:04,270
Or the projection of a onto b.

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00:08:04,270 --> 00:08:08,620
So this vector right here is a
cosine theta; the magnitude of

162
00:08:08,620 --> 00:08:10,940
a cosine theta.

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00:08:10,940 --> 00:08:12,010
And they're actually
the same number.

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00:08:12,010 --> 00:08:14,320
If you take how much of b goes
in the direction of a, and

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00:08:14,320 --> 00:08:16,400
multiply that with the magnitude
of a, that gives you

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00:08:16,400 --> 00:08:20,550
the same number as how much of
a goes in the direction of b,

167
00:08:20,550 --> 00:08:24,270
and then multiply the
two magnitudes.

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00:08:24,270 --> 00:08:28,100
Now, what is a, b sine theta?

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00:08:28,100 --> 00:08:29,710
a, b, sine theta.

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00:08:29,710 --> 00:08:34,299
Well if this vector right here
is a cosine theta-- and you

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00:08:34,299 --> 00:08:35,700
learned this when you learned
how to take the

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00:08:35,700 --> 00:08:36,600
components of vectors.

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00:08:36,600 --> 00:08:43,460
This vector right here is the
magnitude of a sine theta.

174
00:08:43,460 --> 00:08:49,330
You could rewrite this as the
magnitude of a sine theta

175
00:08:49,330 --> 00:08:53,750
times the magnitude of b in that
normal vector direction.

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00:08:53,750 --> 00:08:56,830
So if you take a sine theta
times b, you're saying what

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00:08:56,830 --> 00:08:59,460
part of a doesn't go the
same direction as b.

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00:08:59,460 --> 00:09:02,290
What part of a is completely
perpendicular to b-- has

179
00:09:02,290 --> 00:09:03,390
nothing to do is b.

180
00:09:03,390 --> 00:09:05,100
They share nothing in common.

181
00:09:05,100 --> 00:09:06,930
It goes in a completely
different direction.

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00:09:06,930 --> 00:09:08,340
That's a sine theta.

183
00:09:08,340 --> 00:09:12,560
And so you take the product of
this with b and then you get a

184
00:09:12,560 --> 00:09:13,690
third vector.

185
00:09:13,690 --> 00:09:15,930
And it almost says,
how different

186
00:09:15,930 --> 00:09:16,950
are these two vectors?

187
00:09:16,950 --> 00:09:18,390
And it points in a different
direction.

188
00:09:18,390 --> 00:09:20,810
It gives you this-- sometimes
it's called a pseudo vector,

189
00:09:20,810 --> 00:09:22,330
because it applies
to some concepts

190
00:09:22,330 --> 00:09:24,060
that are pseudo vectors.

191
00:09:24,060 --> 00:09:26,750
But the most important of these
concepts is torque, when

192
00:09:26,750 --> 00:09:30,970
we talk about the magnetic
field; the force of a magnetic

193
00:09:30,970 --> 00:09:32,320
field on electric charge.

194
00:09:32,320 --> 00:09:35,600
These are all forces, or these
are all physical phenomena,

195
00:09:35,600 --> 00:09:39,170
where what matters isn't the
direction of the force with

196
00:09:39,170 --> 00:09:41,110
another vector, it's the
direction of the force

197
00:09:41,110 --> 00:09:42,630
perpendicular to
another vector.

198
00:09:42,630 --> 00:09:45,080
And so that's where the cross
product comes in useful.

199
00:09:45,080 --> 00:09:48,140
Anyway, hopefully, that gave
you a little intuition.

200
00:09:48,140 --> 00:09:49,280
And you could have done
it the other way.

201
00:09:49,280 --> 00:09:51,310
You could have written
this as b sine theta.

202
00:09:51,310 --> 00:09:53,800
And then you would have said
that's the component of b that

203
00:09:53,800 --> 00:09:54,900
is perpendicular to a.

204
00:09:54,900 --> 00:09:58,665
So b sine theta actually would
have been this vector.

205
00:09:58,665 --> 00:10:02,140


206
00:10:02,140 --> 00:10:03,330
Or let me draw it here.

207
00:10:03,330 --> 00:10:04,070
That would make more sense.

208
00:10:04,070 --> 00:10:08,210
This would be b sine theta.

209
00:10:08,210 --> 00:10:10,030
So you could switch orders.

210
00:10:10,030 --> 00:10:11,160
You could visualize
it either way.

211
00:10:11,160 --> 00:10:12,500
You could say this is the
magnitude of b that is

212
00:10:12,500 --> 00:10:15,550
completely perpendicular to a,
multiply the two, and use the

213
00:10:15,550 --> 00:10:18,000
right hand rule to get
that normal vector.

214
00:10:18,000 --> 00:10:22,350
And we just decided that we're
going to use the right hand

215
00:10:22,350 --> 00:10:23,630
rule to have a common
convention.

216
00:10:23,630 --> 00:10:25,470
But people could have used the
left hand rule, or they might

217
00:10:25,470 --> 00:10:26,730
have used it a different way.

218
00:10:26,730 --> 00:10:29,810
It's just a way that we have a
consistent framework, so that

219
00:10:29,810 --> 00:10:32,770
when we take the cross product
we all know what direction

220
00:10:32,770 --> 00:10:34,900
that normal vector
is pointing in.

221
00:10:34,900 --> 00:10:35,400
Anyway.

222
00:10:35,400 --> 00:10:38,300
In the next video I'll show you
how to actually compute

223
00:10:38,300 --> 00:10:41,430
dot and cross products when
you're given them in their

224
00:10:41,430 --> 00:10:43,100
component notation.

225
00:10:43,100 --> 00:00:00,000
See you in the next video.