Calculating dot and cross products with unit vector notation

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.130
So far, when I've told you about
the dot and the cross

00:00:03.130 --> 00:00:06.440
products, I've given you the
definition as the magnitude

00:00:06.440 --> 00:00:08.710
times either the cosine
or the sine of the

00:00:08.710 --> 00:00:09.710
angle between them.

00:00:09.710 --> 00:00:12.430
But what if you're not given
the vectors visually?

00:00:12.430 --> 00:00:14.210
And what if you're not given
the angle between them?

00:00:14.210 --> 00:00:17.240
How do you calculate the dot
and the cross products?

00:00:17.240 --> 00:00:19.160
Well, let me give you
the definition that

00:00:19.160 --> 00:00:20.000
I giving you already.

00:00:20.000 --> 00:00:26.710
So let's say I have a
dot b dot product.

00:00:26.710 --> 00:00:31.610
That's the magnitude of a times
the magnitude of b times

00:00:31.610 --> 00:00:34.200
cosine of the angle
between them.

00:00:34.200 --> 00:00:39.730
a cross b is equal to the
magnitude of a times the

00:00:39.730 --> 00:00:44.670
magnitude of b times sine of
the angle between them-- so

00:00:44.670 --> 00:00:48.360
the perpendicular projections
of them-- times the normal

00:00:48.360 --> 00:00:50.130
vector that's perpendicular
to both of them.

00:00:50.130 --> 00:00:53.750
The normal unit vector, and you
figure out which of the

00:00:53.750 --> 00:00:55.500
two perpendicular vectors
it is by using

00:00:55.500 --> 00:00:56.620
the right hand rule.

00:00:56.620 --> 00:01:00.170
But what if we don't have
the thetas; the

00:01:00.170 --> 00:01:01.320
angles between them?

00:01:01.320 --> 00:01:04.760
What if, for example, I were
to tell you that the vector

00:01:04.760 --> 00:01:09.990
a,-- if I were to give it to you
in engineering notation.

00:01:09.990 --> 00:01:12.090
In engineering notation,
you're essentially just

00:01:12.090 --> 00:01:16.270
breaking down the vector into
its x, y and z components.

00:01:16.270 --> 00:01:23.580
So let's say vector a is 5i--
i is just the unit vector in

00:01:23.580 --> 00:01:31.890
the x direction, minus
6j, plus 3k.

00:01:34.740 --> 00:01:37.790
i,j and k are just the unit
vectors of the x, y and z

00:01:37.790 --> 00:01:38.310
directions.

00:01:38.310 --> 00:01:40.700
And the 5 is how much it goes
in the x direction.

00:01:40.700 --> 00:01:43.400
The minus 6 is how much it
goes in the y direction.

00:01:43.400 --> 00:01:45.890
And the 3 is how much it goes
in the z direction.

00:01:45.890 --> 00:01:47.040
You could try to graph it.

00:01:47.040 --> 00:01:48.960
And actually, I'm trying to look
for a graphing calculator

00:01:48.960 --> 00:01:51.370
that'll do this, so I can show
you it all in videos to give

00:01:51.370 --> 00:01:52.360
you more intuition.

00:01:52.360 --> 00:01:53.830
But lets say this is
all you're given.

00:01:53.830 --> 00:02:00.100
And let's say that b-- I'm just
making these numbers up--

00:02:00.100 --> 00:02:04.170
let's say it's minus 2i-- and,
of course, we're working in

00:02:04.170 --> 00:02:14.480
three dimensions right now--
plus 7j, plus 4k.

00:02:14.480 --> 00:02:15.300
You could graph it.

00:02:15.300 --> 00:02:19.030
But obviously, if you were given
a problem, and if you

00:02:19.030 --> 00:02:22.270
were actually trying to model
vectors on a computer

00:02:22.270 --> 00:02:23.510
simulation, this is the
way you would do it.

00:02:23.510 --> 00:02:25.690
You'd break it up into the x, y,
and z components because of

00:02:25.690 --> 00:02:26.780
the add vectors.

00:02:26.780 --> 00:02:28.600
You just have to add the
respective components.

00:02:28.600 --> 00:02:31.210
But how do you multiply them
either taking the cross or the

00:02:31.210 --> 00:02:32.340
dot product?

00:02:32.340 --> 00:02:34.580
Well it actually turns out I'm
not going to prove it here but

00:02:34.580 --> 00:02:35.400
I'll just show you
how to do it.

00:02:35.400 --> 00:02:38.100
The dot product is very
easy when you have it

00:02:38.100 --> 00:02:39.330
given in this notation.

00:02:39.330 --> 00:02:40.880
And actually another way of
writing this notation,

00:02:40.880 --> 00:02:42.360
sometimes it's in bracket
notation.

00:02:42.360 --> 00:02:46.955
Sometimes they would rewrite
this as 5 minus 6, 3.

00:02:46.955 --> 00:02:49.455
Or it's just the magnitudes of
the x,y and z direction.

00:02:49.455 --> 00:02:53.170
I just want to make sure you're
comfortable with all of

00:02:53.170 --> 00:02:54.270
these various notations.

00:02:54.270 --> 00:02:57.360
You could have written b
as minus is 2, 7, 4.

00:02:57.360 --> 00:02:58.380
These are all the same things.

00:02:58.380 --> 00:03:00.360
You shouldn't get daunted if
you see one or the other.

00:03:00.360 --> 00:03:05.430
But anyway, so how do
I take a dot b?

00:03:08.110 --> 00:03:10.670
This, I think you'll find
fairly pleasant.

00:03:10.670 --> 00:03:15.410
All you do is you multiply the
i components, add that to the

00:03:15.410 --> 00:03:18.270
j components multiplied, and
then add that to the k

00:03:18.270 --> 00:03:20.210
components multiplied
together.

00:03:20.210 --> 00:03:34.350
So it would be 5 times minus 2
plus minus 6 times 7 plus 3

00:03:34.350 --> 00:03:45.260
times 4, so it equals minus
10 minus 42 plus 12.

00:03:45.260 --> 00:03:52.020
So this is minus 52 plus 12,
so it equals minus 40.

00:03:52.020 --> 00:03:52.460
That's it.

00:03:52.460 --> 00:03:54.840
It's just a number.

00:03:54.840 --> 00:03:57.090
And I'd actually be curious
to graph this on a three

00:03:57.090 --> 00:04:00.980
dimensional grapher to see
why it's minus 40.

00:04:00.980 --> 00:04:03.600
They must be going in
opposite directions.

00:04:03.600 --> 00:04:05.680
And their projections onto each
other go into opposite

00:04:05.680 --> 00:04:06.070
directions.

00:04:06.070 --> 00:04:07.770
And that's why we get
a minus number.

00:04:11.000 --> 00:04:13.030
The purpose of this-- I don't
want to get too much into the

00:04:13.030 --> 00:04:15.050
intuition-- this is just how to
calculate, but it's fairly

00:04:15.050 --> 00:04:15.900
straightforward.

00:04:15.900 --> 00:04:18.930
You just multiply the
x components.

00:04:18.930 --> 00:04:22.029
Add that to the y components
multiplied and add that to the

00:04:22.029 --> 00:04:23.450
z components multiplied.

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So whenever I am given something
in engineering or

00:04:25.710 --> 00:04:28.470
bracket notation and I have to
find the dot product, it's

00:04:28.470 --> 00:04:33.680
very, almost soothing, and
not so error prone.

00:04:33.680 --> 00:04:37.390
But, as you will see, taking the
cross product of these two

00:04:37.390 --> 00:04:40.160
vectors when given in this
notation isn't so

00:04:40.160 --> 00:04:41.490
straightforward.

00:04:41.490 --> 00:04:43.020
And I want you to keep in mind,
another way you could

00:04:43.020 --> 00:04:44.590
have done it, you could have
figured out the magnitude of

00:04:44.590 --> 00:04:49.470
each of these vectors and then
you could have used some fancy

00:04:49.470 --> 00:04:51.770
trigonometry to figure out the
thetas and then used this

00:04:51.770 --> 00:04:52.370
definition.

00:04:52.370 --> 00:04:56.230
But I think you appreciate the
fact that this is a much

00:04:56.230 --> 00:04:57.350
simpler way of doing it.

00:04:57.350 --> 00:04:59.140
So the dot product
is a lot of fun.

00:04:59.140 --> 00:05:02.570
Now let's see if we could
take the cross product.

00:05:02.570 --> 00:05:04.450
And once again, I'm not
going to prove it.

00:05:04.450 --> 00:05:06.230
I'm just going to show
you how to do it.

00:05:06.230 --> 00:05:09.370
In a future video, I'm sure I'll
get a request to do it

00:05:09.370 --> 00:05:11.710
eventually, and I'll prove it.

00:05:11.710 --> 00:05:15.270
But the cross product, this
is more involved.

00:05:15.270 --> 00:05:18.210
And I never look forward to
taking the cross product of

00:05:18.210 --> 00:05:20.290
two vectors in engineering
notation.

00:05:20.290 --> 00:05:22.700
a cross b.

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It equals.

00:05:23.760 --> 00:05:27.530
So this is an application
of matrices.

00:05:27.530 --> 00:05:31.850
So what you do is you take the
determinant-- I'll draw a big

00:05:31.850 --> 00:05:34.120
determinant line-- on the top
line of the determinant.

00:05:34.120 --> 00:05:35.190
This is really just
a way to make you

00:05:35.190 --> 00:05:37.090
memorize how to do it.

00:05:37.090 --> 00:05:39.240
It doesn't give you much
intuition, but the intuition

00:05:39.240 --> 00:05:41.690
is given by the actual
definition.

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How much of the vectors are
perpendicular to each other.

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Multiply those magnitudes.

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Right hand rule figures
out what direction

00:05:47.210 --> 00:05:48.360
you're pointing in.

00:05:48.360 --> 00:05:51.380
But the way to do it if you're
given engineering notation,

00:05:51.380 --> 00:05:55.763
you write the i, j, k unit
vectors the top row.

00:05:55.763 --> 00:06:00.080
i, j, k.

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Then you write the first vector
in the cross product,

00:06:02.230 --> 00:06:03.560
because order matters.

00:06:03.560 --> 00:06:09.550
So it's 5 minus 6, 3.

00:06:09.550 --> 00:06:12.320
Then you take the second vector
which is b, which is

00:06:12.320 --> 00:06:16.970
minus 2, 7, 4.

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So you take the determinant
of the 3 by 3 matrix,

00:06:19.880 --> 00:06:21.350
and how do I do that?

00:06:21.350 --> 00:06:25.930
Well that's equal to the
subdeterminant for i.

00:06:25.930 --> 00:06:28.460
So the subdeterminant for i, if
you get rid of this column

00:06:28.460 --> 00:06:31.920
and this row, the determinant
that's left over, so that's

00:06:31.920 --> 00:06:40.760
minus 6, 3, 7, 4 times i--
you might want to review

00:06:40.760 --> 00:06:42.430
determinants if you don't
remember how to do this, but

00:06:42.430 --> 00:06:47.770
maybe me working through it
will just jog your memory.

00:06:47.770 --> 00:06:50.590
And then remember, it's
plus, minus, plus.

00:06:50.590 --> 00:06:53.550
So then minus the subdeterminant
for j.

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What's the subdeterminant
for j?

00:06:55.500 --> 00:06:57.470
You cross out j's
row and columns.

00:06:57.470 --> 00:07:01.065
You have 5, 3, minus 2, 4.

00:07:05.030 --> 00:07:07.650
We just crossed j's
row and column.

00:07:07.650 --> 00:07:09.770
And whatever's left over, those
are the numbers in its

00:07:09.770 --> 00:07:11.470
subdeterminant.

00:07:11.470 --> 00:07:13.420
That's what I call it.

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j plus-- I want to do them all
on one line because it would

00:07:18.136 --> 00:07:19.870
have been a little bit
neater-- plus the

00:07:19.870 --> 00:07:20.840
subdeterminant for k.

00:07:20.840 --> 00:07:23.290
Cross out the row and
the column for k.

00:07:23.290 --> 00:07:35.010
We're left with 5 minus 6,
minus 2 and 7 times k.

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And now let's calculate them.

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And let me make some
space, because I've

00:07:39.440 --> 00:07:41.130
written this too big.

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I don't think we need
this anymore.

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So what do we get?

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Let's take this up here.

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So these 2 by 2 determinants
are pretty easy.

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This is minus 6 times
4 minus 7 times 3.

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I always make careless
mistakes here.

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Minus 24 minus 21 times i minus
5 times 4 is 20, minus

00:08:10.770 --> 00:08:23.270
minus 2 times 3, so minus minus
6 j, plus 5 times 7, 35

00:08:23.270 --> 00:08:25.640
minus minus 2 times minus 6.

00:08:25.640 --> 00:08:29.330
So it's minus positive 12k.

00:08:29.330 --> 00:08:34.330
We could simplify this, which
equals minus 24 minus 21.

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It is minus 35-- I didn't have
to put a parentheses-- i, and

00:08:40.830 --> 00:08:43.720
then what's 20 minus minus 6?

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Well that's 20 plus
plus 6, so 26.

00:08:46.600 --> 00:08:47.590
And then we have a
minus out here.

00:08:47.590 --> 00:08:51.640
So minus 26j.

00:08:51.640 --> 00:08:54.340
And that was 35 minus
12, that's 23.

00:08:54.340 --> 00:08:57.190
Plus 23k.

00:08:57.190 --> 00:08:58.690
So that's the cross product.

00:08:58.690 --> 00:09:01.150
And if you were to graph this in
three dimensions, you will

00:09:01.150 --> 00:09:03.710
see-- and this is what's
interesting-- you will see

00:09:03.710 --> 00:09:09.410
that vector, if my math is
correct, minus 35i, minus 26j,

00:09:09.410 --> 00:09:15.750
plus 23k, is perpendicular
to both of these vectors.

00:09:15.750 --> 00:09:19.440
I think I'll leave you there for
now, and I will see you in

00:09:19.440 --> 00:09:20.050
the next video.

00:09:20.050 --> 00:09:22.140
And hopefully, I can track down
a vector graphic program.

00:09:22.140 --> 00:09:25.880
Because I think it'll be fun to
both calculate the dot and

00:09:25.880 --> 00:09:29.130
the cross products using the
methods I just showed you and

00:09:29.130 --> 00:09:29.840
then to graph them.

00:09:29.840 --> 00:09:31.320
And to show that it
really does work.

00:09:31.320 --> 00:09:36.930
That this vector really is
perpendicular to both of these

00:09:36.930 --> 00:09:40.820
and pointing in the direction as
you would predict using the

00:09:40.820 --> 00:09:42.520
right hand rule.

00:09:42.520 --> 00:09:43.990
I'll see you in the next video.


Dot vs. cross product

WEBVTT
Kind: captions
Language: en

00:00:01.010 --> 00:00:04.070
Let's do a little compare and
contrast between the dot

00:00:04.070 --> 00:00:05.710
product and the cross product.

00:00:05.710 --> 00:00:09.670
Let me just make two vectors--
just visually draw them.

00:00:09.670 --> 00:00:15.620
And maybe if we have time,
we'll, actually figure out

00:00:15.620 --> 00:00:17.970
some dot and cross products
with real vectors.

00:00:36.170 --> 00:00:47.070
Let's call the first one--
That's the angle between them.

00:00:47.070 --> 00:00:47.550
OK.

00:00:47.550 --> 00:00:49.550
So let's just go over the
definitions and then we'll

00:00:49.550 --> 00:00:50.980
work on the intuition.

00:00:50.980 --> 00:00:53.360
And hopefully, you have a little
bit of both already.

00:00:53.360 --> 00:00:59.240
So what is a dot b?

00:00:59.240 --> 00:01:03.000
Well first of all, that's the
exact same thing as b dot a.

00:01:03.000 --> 00:01:06.590
Order does not matter when you
take the dot product because

00:01:06.590 --> 00:01:09.230
you end up with just a number.

00:01:09.230 --> 00:01:13.440
And that is equal to the
magnitude of a times the

00:01:13.440 --> 00:01:21.145
magnitude of b times cosine
of the angle between them.

00:01:21.145 --> 00:01:24.450
Let's look at the definition
of the cross product.

00:01:24.450 --> 00:01:31.810
What is a cross b?

00:01:31.810 --> 00:01:34.615
Well first of all, that does
not equal b cross a.

00:01:34.615 --> 00:01:37.650
It actually equals the opposite
direction, or you

00:01:37.650 --> 00:01:39.900
could view it as the negative
of b cross a.

00:01:39.900 --> 00:01:42.980
Because the vector that you end
up with ends up flipped,

00:01:42.980 --> 00:01:44.510
whichever order you do it in.

00:01:44.510 --> 00:01:51.600
But a cross b, that is equal to
the magnitude of vector a

00:01:51.600 --> 00:01:54.580
times the magnitude of vector
b-- so far, it looks a lot

00:01:54.580 --> 00:01:57.420
like the dot product, but this
is where the diverge is--

00:01:57.420 --> 00:02:00.690
times the sine of the
angle between them.

00:02:00.690 --> 00:02:02.030
The sine of the angle
between them.

00:02:02.030 --> 00:02:04.070
And this is where it
really diverges.

00:02:04.070 --> 00:02:05.610
When we took the dot
product, we just

00:02:05.610 --> 00:02:06.420
ended up with a number.

00:02:06.420 --> 00:02:07.520
This is just a number.

00:02:07.520 --> 00:02:08.729
There's no direction here.

00:02:08.729 --> 00:02:10.340
This is just a scalar
quantity.

00:02:10.340 --> 00:02:13.700
But the cross product, we take
the magnitude of a times the

00:02:13.700 --> 00:02:16.250
magnitude of b, times the sine
of the angle between them, and

00:02:16.250 --> 00:02:18.600
that provides a magnitude, but
it also has a direction.

00:02:18.600 --> 00:02:21.840
And that direction is provided
by this normal vector.

00:02:21.840 --> 00:02:23.550
It's a unit vector.

00:02:23.550 --> 00:02:26.820
A unit vector gets that
little hat on it.

00:02:26.820 --> 00:02:28.470
It's a unit vector, and
what direction is it?

00:02:28.470 --> 00:02:31.320
Well, that's defined by
the right hand rule.

00:02:31.320 --> 00:02:33.200
This is a vector.

00:02:33.200 --> 00:02:45.870
It's perpendicular
to both a and b.

00:02:45.870 --> 00:02:48.740
And then you might say, a and
b, the way I drew them,

00:02:48.740 --> 00:02:51.810
they're both sitting in the
plane of this video screen, or

00:02:51.810 --> 00:02:53.200
your video screen.

00:02:53.200 --> 00:02:55.670
So in order for something to
be perpendicular to both of

00:02:55.670 --> 00:02:59.550
these, it either has to pop out
of the screen or pop into

00:02:59.550 --> 00:03:00.240
the screen, right?

00:03:00.240 --> 00:03:02.110
And when you learned about the
cross product, I said there's

00:03:02.110 --> 00:03:05.110
two ways of showing a vector
popping out of the screen.

00:03:05.110 --> 00:03:07.430
It looks like that because
that's the tip of an arrow.

00:03:07.430 --> 00:03:10.110
And to show a vector going into
the screen, it's like

00:03:10.110 --> 00:03:13.040
that because that is the
back of an arrow.

00:03:13.040 --> 00:03:14.980
The rear end of an arrow.

00:03:14.980 --> 00:03:17.020
So how do you know which
of these two it is?

00:03:17.020 --> 00:03:20.350
Because both of these vectors
are perpendicular to a and b.

00:03:20.350 --> 00:03:22.480
That's where you take your right
hand and you use the

00:03:22.480 --> 00:03:23.470
right hand rule.

00:03:23.470 --> 00:03:27.120
So you take your index finger
in the direction of a, your

00:03:27.120 --> 00:03:29.730
middle finger in the direction
of b, and then your thumb

00:03:29.730 --> 00:03:32.210
points in the direction of n.

00:03:32.210 --> 00:03:33.460
So let's do that.

00:03:33.460 --> 00:03:36.695
I'm looking at my hand.

00:03:36.695 --> 00:03:40.300
It's not an easy thing to do
with your right hand, but your

00:03:40.300 --> 00:03:43.820
right hand is going to look
something like this.

00:03:43.820 --> 00:03:55.820
Your index finger will go
in the direction of a.

00:03:55.820 --> 00:03:59.200
Your middle finger goes
in the direction of b.

00:03:59.200 --> 00:04:02.040
So that's my middle finger.

00:04:02.040 --> 00:04:04.610
And then my other two fingers
just do what they need to do.

00:04:04.610 --> 00:04:06.160
I like to just bend them
out of the way.

00:04:09.160 --> 00:04:11.940
So they just curl
around my hand.

00:04:11.940 --> 00:04:14.280
And then what direction
is my thumb in?

00:04:14.280 --> 00:04:17.820
My thumb-- well, actually I drew
it at the wrong angle.

00:04:17.820 --> 00:04:21.300
My thumb is actually going
in this direction, right?

00:04:21.300 --> 00:04:22.730
Into the page.

00:04:22.730 --> 00:04:23.830
This is the top of my hand.

00:04:23.830 --> 00:04:25.550
These are like my veins.

00:04:25.550 --> 00:04:29.450
Or, if I actually drew it
correctly, where you would see

00:04:29.450 --> 00:04:32.690
your hand from side-- so it
would look like this.

00:04:32.690 --> 00:04:34.390
You would see your pinky.

00:04:34.390 --> 00:04:37.760
Your palm and your pinky
would be like that.

00:04:37.760 --> 00:04:39.630
And your other finger
like this.

00:04:39.630 --> 00:04:43.460
Your middle finger would go
in the direction of b.

00:04:43.460 --> 00:04:46.900
Your index finger goes in the
direction of a, and you

00:04:46.900 --> 00:04:49.430
wouldn't even see your thumb,
because your thumb is pointing

00:04:49.430 --> 00:04:49.960
straight down.

00:04:49.960 --> 00:04:53.520
But I think you get the point.
a cross b, this n vector is

00:04:53.520 --> 00:04:54.780
pointing straight down.

00:04:54.780 --> 00:04:55.630
It's a unit vector.

00:04:55.630 --> 00:04:56.750
And this provides
the magnitude.

00:04:56.750 --> 00:04:59.240
Unit vector just means it
has a magnitude of one.

00:04:59.240 --> 00:05:02.560
So the magnitudes of the cross
and the dot products seem

00:05:02.560 --> 00:05:03.360
pretty close.

00:05:03.360 --> 00:05:06.440
They both have the magnitude
of both vectors there.

00:05:06.440 --> 00:05:08.720
Dot product, cosine theta.

00:05:08.720 --> 00:05:09.980
Cross product sine of theta.

00:05:09.980 --> 00:05:11.710
But then, the huge difference
is that sine

00:05:11.710 --> 00:05:12.730
of theta has a direction.

00:05:12.730 --> 00:05:14.410
It is a different
vector that is

00:05:14.410 --> 00:05:16.360
perpendicular to both of these.

00:05:16.360 --> 00:05:18.340
Now, let's get the intuition.

00:05:18.340 --> 00:05:20.830
And if you've watched the videos
on the dot and the

00:05:20.830 --> 00:05:23.420
cross product, hopefully you
have a little intuition.

00:05:23.420 --> 00:05:25.740
But I review it because I think
it all fits together

00:05:25.740 --> 00:05:28.230
when you see them
with each other.

00:05:42.060 --> 00:05:45.290
First, let's study a,
b cosine of theta.

00:05:45.290 --> 00:05:49.470
If you watched the dot product
video, cosine of theta, if you

00:05:49.470 --> 00:05:51.860
took, let's say, b
cosine of theta.

00:05:51.860 --> 00:05:54.750
What is b cosine of theta?

00:05:54.750 --> 00:05:56.930
b cosine of theta-- and you
could work it out on your own

00:05:56.930 --> 00:06:03.690
time-- if you say cosine is
adjacent over hypotenuse, the

00:06:03.690 --> 00:06:06.140
magnitude of b cosine theta is
actually going to be the

00:06:06.140 --> 00:06:09.520
magnitude of, if you dropped a
perpendicular-- I'll use a

00:06:09.520 --> 00:06:14.940
different color here-- if you
dropped a perpendicular here,

00:06:14.940 --> 00:06:17.250
this length right here--
that's b cosine theta.

00:06:17.250 --> 00:06:18.680
Let me draw it separately.

00:06:18.680 --> 00:06:21.140
I don't want to mess up
this picture too much.

00:06:21.140 --> 00:06:22.390
So if that's b.

00:06:30.890 --> 00:06:40.320
If that's a-- And that's b.

00:06:40.320 --> 00:06:41.610
That's a.

00:06:41.610 --> 00:06:42.540
This is theta.

00:06:42.540 --> 00:06:47.236
b cosine theta, if you drop a
line perpendicular to a, this

00:06:47.236 --> 00:06:48.590
is a right angle.

00:06:48.590 --> 00:06:51.350
b cosine theta, adjacent
over hypotenuse is

00:06:51.350 --> 00:06:52.910
equal to cosine theta.

00:06:52.910 --> 00:06:55.210
So it would be the projection
of b going in the same

00:06:55.210 --> 00:06:57.230
direction as a.

00:06:57.230 --> 00:06:59.810
So it would be this magnitude.

00:07:02.770 --> 00:07:06.120
That is b cosine theta.

00:07:06.120 --> 00:07:13.180
So the magnitude of that vector
right there is the

00:07:13.180 --> 00:07:16.720
magnitude of b cosine
of theta.

00:07:16.720 --> 00:07:18.910
So when you're taking the dot
product, at least the example

00:07:18.910 --> 00:07:22.970
I just did, if you view it as
the magnitude of a times the

00:07:22.970 --> 00:07:26.570
magnitude of b cosine theta,
you're saying what part of b

00:07:26.570 --> 00:07:28.680
goes in the same
direction as a?

00:07:28.680 --> 00:07:30.870
And whatever that magnitude is,
let me just multiply that

00:07:30.870 --> 00:07:31.980
times the magnitude of a.

00:07:31.980 --> 00:07:34.360
And I have the dot product.

00:07:34.360 --> 00:07:35.960
Let's take the pieces that
go the same direction and

00:07:35.960 --> 00:07:36.590
multiply them.

00:07:36.590 --> 00:07:38.040
So how much do they
move together?

00:07:38.040 --> 00:07:40.310
Or do they point together?

00:07:40.310 --> 00:07:41.510
Or you could view it
the other way.

00:07:41.510 --> 00:07:44.050
You could view the dot product
as-- and I did this in the dot

00:07:44.050 --> 00:07:50.510
product video-- you could view
it as a cosine of theta, b.

00:07:50.510 --> 00:07:51.480
Because it doesn't matter.

00:07:51.480 --> 00:07:52.990
These are all scalar quantities,
so it doesn't

00:07:52.990 --> 00:07:55.560
matter what order you take
the multiplication in.

00:07:55.560 --> 00:07:57.270
And a cosine theta is
the same thing.

00:07:57.270 --> 00:08:00.350
It's the magnitude of the a
vector that's going in the

00:08:00.350 --> 00:08:01.060
same direction of b.

00:08:01.060 --> 00:08:04.270
Or the projection of a onto b.

00:08:04.270 --> 00:08:08.620
So this vector right here is a
cosine theta; the magnitude of

00:08:08.620 --> 00:08:10.940
a cosine theta.

00:08:10.940 --> 00:08:12.010
And they're actually
the same number.

00:08:12.010 --> 00:08:14.320
If you take how much of b goes
in the direction of a, and

00:08:14.320 --> 00:08:16.400
multiply that with the magnitude
of a, that gives you

00:08:16.400 --> 00:08:20.550
the same number as how much of
a goes in the direction of b,

00:08:20.550 --> 00:08:24.270
and then multiply the
two magnitudes.

00:08:24.270 --> 00:08:28.100
Now, what is a, b sine theta?

00:08:28.100 --> 00:08:29.710
a, b, sine theta.

00:08:29.710 --> 00:08:34.299
Well if this vector right here
is a cosine theta-- and you

00:08:34.299 --> 00:08:35.700
learned this when you learned
how to take the

00:08:35.700 --> 00:08:36.600
components of vectors.

00:08:36.600 --> 00:08:43.460
This vector right here is the
magnitude of a sine theta.

00:08:43.460 --> 00:08:49.330
You could rewrite this as the
magnitude of a sine theta

00:08:49.330 --> 00:08:53.750
times the magnitude of b in that
normal vector direction.

00:08:53.750 --> 00:08:56.830
So if you take a sine theta
times b, you're saying what

00:08:56.830 --> 00:08:59.460
part of a doesn't go the
same direction as b.

00:08:59.460 --> 00:09:02.290
What part of a is completely
perpendicular to b-- has

00:09:02.290 --> 00:09:03.390
nothing to do is b.

00:09:03.390 --> 00:09:05.100
They share nothing in common.

00:09:05.100 --> 00:09:06.930
It goes in a completely
different direction.

00:09:06.930 --> 00:09:08.340
That's a sine theta.

00:09:08.340 --> 00:09:12.560
And so you take the product of
this with b and then you get a

00:09:12.560 --> 00:09:13.690
third vector.

00:09:13.690 --> 00:09:15.930
And it almost says,
how different

00:09:15.930 --> 00:09:16.950
are these two vectors?

00:09:16.950 --> 00:09:18.390
And it points in a different
direction.

00:09:18.390 --> 00:09:20.810
It gives you this-- sometimes
it's called a pseudo vector,

00:09:20.810 --> 00:09:22.330
because it applies
to some concepts

00:09:22.330 --> 00:09:24.060
that are pseudo vectors.

00:09:24.060 --> 00:09:26.750
But the most important of these
concepts is torque, when

00:09:26.750 --> 00:09:30.970
we talk about the magnetic
field; the force of a magnetic

00:09:30.970 --> 00:09:32.320
field on electric charge.

00:09:32.320 --> 00:09:35.600
These are all forces, or these
are all physical phenomena,

00:09:35.600 --> 00:09:39.170
where what matters isn't the
direction of the force with

00:09:39.170 --> 00:09:41.110
another vector, it's the
direction of the force

00:09:41.110 --> 00:09:42.630
perpendicular to
another vector.

00:09:42.630 --> 00:09:45.080
And so that's where the cross
product comes in useful.

00:09:45.080 --> 00:09:48.140
Anyway, hopefully, that gave
you a little intuition.

00:09:48.140 --> 00:09:49.280
And you could have done
it the other way.

00:09:49.280 --> 00:09:51.310
You could have written
this as b sine theta.

00:09:51.310 --> 00:09:53.800
And then you would have said
that's the component of b that

00:09:53.800 --> 00:09:54.900
is perpendicular to a.

00:09:54.900 --> 00:09:58.665
So b sine theta actually would
have been this vector.

00:10:02.140 --> 00:10:03.330
Or let me draw it here.

00:10:03.330 --> 00:10:04.070
That would make more sense.

00:10:04.070 --> 00:10:08.210
This would be b sine theta.

00:10:08.210 --> 00:10:10.030
So you could switch orders.

00:10:10.030 --> 00:10:11.160
You could visualize
it either way.

00:10:11.160 --> 00:10:12.500
You could say this is the
magnitude of b that is

00:10:12.500 --> 00:10:15.550
completely perpendicular to a,
multiply the two, and use the

00:10:15.550 --> 00:10:18.000
right hand rule to get
that normal vector.

00:10:18.000 --> 00:10:22.350
And we just decided that we're
going to use the right hand

00:10:22.350 --> 00:10:23.630
rule to have a common
convention.

00:10:23.630 --> 00:10:25.470
But people could have used the
left hand rule, or they might

00:10:25.470 --> 00:10:26.730
have used it a different way.

00:10:26.730 --> 00:10:29.810
It's just a way that we have a
consistent framework, so that

00:10:29.810 --> 00:10:32.770
when we take the cross product
we all know what direction

00:10:32.770 --> 00:10:34.900
that normal vector
is pointing in.

00:10:34.900 --> 00:10:35.400
Anyway.

00:10:35.400 --> 00:10:38.300
In the next video I'll show you
how to actually compute

00:10:38.300 --> 00:10:41.430
dot and cross products when
you're given them in their

00:10:41.430 --> 00:10:43.100
component notation.

00:10:43.100 --> 00:10:44.920
See you in the next video.


The dot product

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.510
Let's learn a little bit
about the dot product.

00:00:02.510 --> 00:00:06.040
The dot product, frankly, out of
the two ways of multiplying

00:00:06.040 --> 00:00:08.410
vectors, I think is
the easier one.

00:00:08.410 --> 00:00:09.820
So what does the
dot product do?

00:00:09.820 --> 00:00:11.406
Why don't I give you the
definition, and then I'll give

00:00:11.406 --> 00:00:12.080
you an intuition.

00:00:12.080 --> 00:00:23.960
So if I have two vectors; vector
a dot vector b-- that's

00:00:23.960 --> 00:00:24.660
how I draw my arrows.

00:00:24.660 --> 00:00:26.300
I can draw my arrows
like that.

00:00:26.300 --> 00:00:31.940
That is equal to the magnitude
of vector a times the

00:00:31.940 --> 00:00:36.610
magnitude of vector b
times cosine of the

00:00:36.610 --> 00:00:38.430
angle between them.

00:00:38.430 --> 00:00:39.520
Now where does this come from?

00:00:39.520 --> 00:00:41.480
This might seem a little
arbitrary, but I think with a

00:00:41.480 --> 00:00:44.500
visual explanation, it will make
a little bit more sense.

00:00:44.500 --> 00:00:48.850
So let me draw, arbitrarily,
these two vectors.

00:00:48.850 --> 00:00:54.060
So that is my vector a-- nice
big and fat vector.

00:00:54.060 --> 00:00:56.710
It's good for showing
the point.

00:00:56.710 --> 00:00:58.810
And let me draw vector
b like that.

00:01:01.350 --> 00:01:02.380
Vector b.

00:01:02.380 --> 00:01:05.180
And then let me draw the cosine,
or let me, at least,

00:01:05.180 --> 00:01:06.650
draw the angle between them.

00:01:06.650 --> 00:01:09.090
This is theta.

00:01:09.090 --> 00:01:10.650
So there's two ways
of view this.

00:01:10.650 --> 00:01:12.600
Let me label them.

00:01:12.600 --> 00:01:14.886
This is vector a.

00:01:14.886 --> 00:01:17.010
I'm trying to be color
consistent.

00:01:17.010 --> 00:01:18.740
This is vector b.

00:01:18.740 --> 00:01:22.120
So there's two ways of
viewing this product.

00:01:22.120 --> 00:01:27.360
You could view it as vector a--
because multiplication is

00:01:27.360 --> 00:01:29.130
associative, you could
switch the order.

00:01:29.130 --> 00:01:32.400
So this could also be written
as, the magnitude of vector a

00:01:32.400 --> 00:01:39.200
times cosine of theta, times--
and I'll do it in color

00:01:39.200 --> 00:01:41.700
appropriate-- vector b.

00:01:41.700 --> 00:01:43.300
And this times, this
is the dot product.

00:01:43.300 --> 00:01:44.470
I almost don't have
to write it.

00:01:44.470 --> 00:01:45.860
This is just regular
multiplication, because these

00:01:45.860 --> 00:01:47.200
are all scalar quantities.

00:01:47.200 --> 00:01:49.300
When you see the dot between
vectors, you're talking about

00:01:49.300 --> 00:01:51.550
the vector dot product.

00:01:51.550 --> 00:01:53.640
So if we were to just rearrange
this expression this

00:01:53.640 --> 00:01:54.410
way, what does it mean?

00:01:54.410 --> 00:01:57.110
What is a cosine of theta?

00:01:57.110 --> 00:01:58.180
Let me ask you a question.

00:01:58.180 --> 00:02:01.970
If I were to drop a right
angle, right here,

00:02:01.970 --> 00:02:06.740
perpendicular to b-- so let's
just drop a right angle

00:02:06.740 --> 00:02:14.900
there-- cosine of theta
soh-coh-toa so, cah cosine--

00:02:14.900 --> 00:02:20.260
is equal to adjacent of
a hypotenuse, right?

00:02:20.260 --> 00:02:21.990
Well, what's the adjacent?

00:02:21.990 --> 00:02:24.540
It's equal to this.

00:02:24.540 --> 00:02:28.900
And the hypotenuse is equal to
the magnitude of a, right?

00:02:28.900 --> 00:02:29.740
Let me re-write that.

00:02:29.740 --> 00:02:34.230
So cosine of theta-- and this
applies to the a vector.

00:02:34.230 --> 00:02:40.270
Cosine of theta of this angle
is equal to ajacent, which

00:02:40.270 --> 00:02:42.800
is-- I don't know what you could
call this-- let's call

00:02:42.800 --> 00:02:47.130
this the projection
of a onto b.

00:02:47.130 --> 00:02:49.890
It's like if you were to shine
a light perpendicular to b--

00:02:49.890 --> 00:02:54.080
if there was a light source
here and the light was

00:02:54.080 --> 00:02:57.440
straight down, it would be
the shadow of a onto b.

00:02:57.440 --> 00:03:00.110
Or you could almost think of it
as the part of a that goes

00:03:00.110 --> 00:03:01.440
in the same direction of b.

00:03:01.440 --> 00:03:07.340
So this projection, they call
it-- at least the way I get

00:03:07.340 --> 00:03:09.050
the intuition of what a
projection is, I kind of view

00:03:09.050 --> 00:03:09.910
it as a shadow.

00:03:09.910 --> 00:03:13.480
If you had a light source that
came up perpendicular, what

00:03:13.480 --> 00:03:16.590
would be the shadow of that
vector on to this one?

00:03:16.590 --> 00:03:21.260
So if you think about it, this
shadow right here-- you could

00:03:21.260 --> 00:03:23.240
call that, the projection
of a onto b.

00:03:23.240 --> 00:03:24.180
Or, I don't know.

00:03:24.180 --> 00:03:27.200
Let's just call it, a sub b.

00:03:27.200 --> 00:03:29.240
And it's the magnitude
of it, right?

00:03:29.240 --> 00:03:33.520
It's how much of vector a goes
on vector b over-- that's the

00:03:33.520 --> 00:03:35.650
adjacent side-- over
the hypotenuse.

00:03:35.650 --> 00:03:38.790
The hypotenuse is just the
magnitude of vector a.

00:03:38.790 --> 00:03:41.810
It's just our basic calculus.

00:03:41.810 --> 00:03:44.500
Or another way you could view
it, just multiply both sides

00:03:44.500 --> 00:03:46.190
by the magnitude of vector a.

00:03:46.190 --> 00:03:51.390
You get the projection of a onto
b, which is just a fancy

00:03:51.390 --> 00:03:55.830
way of saying, this side; the
part of a that goes in the

00:03:55.830 --> 00:03:59.750
same direction as b-- is another
way to say it-- is

00:03:59.750 --> 00:04:03.590
equal to just multiplying both
sides times the magnitude of a

00:04:03.590 --> 00:04:08.240
is equal to the magnitude
of a, cosine of theta.

00:04:08.240 --> 00:04:10.430
Which is exactly what
we have up here.

00:04:10.430 --> 00:04:12.570
And the definition of
the dot product.

00:04:12.570 --> 00:04:15.860
So another way of visualizing
the dot product is, you could

00:04:15.860 --> 00:04:19.570
replace this term with the
magnitude of the projection of

00:04:19.570 --> 00:04:22.760
a onto b-- which is just
this-- times the

00:04:22.760 --> 00:04:24.870
magnitude of b.

00:04:24.870 --> 00:04:26.360
That's interesting.

00:04:26.360 --> 00:04:29.410
All the dot product of two
vectors is-- let's just take

00:04:29.410 --> 00:04:30.650
one vector.

00:04:30.650 --> 00:04:33.720
Let's figure out how much of
that vector-- what component

00:04:33.720 --> 00:04:36.650
of it's magnitude-- goes in
the same direction as the

00:04:36.650 --> 00:04:39.570
other vector, and let's
just multiply them.

00:04:39.570 --> 00:04:41.250
And where is that useful?

00:04:41.250 --> 00:04:42.040
Well, think about it.

00:04:42.040 --> 00:04:43.020
What about work?

00:04:43.020 --> 00:04:44.720
When we learned work
in physics?

00:04:44.720 --> 00:04:46.870
Work is force times distance.

00:04:46.870 --> 00:04:48.600
But it's not just
the total force

00:04:48.600 --> 00:04:49.460
times the total distance.

00:04:49.460 --> 00:04:51.110
It's the force going
in the same

00:04:51.110 --> 00:04:52.360
direction as the distance.

00:04:56.030 --> 00:04:58.320
You should review the physics
playlist if you're watching

00:04:58.320 --> 00:05:04.150
this within the calculus
playlist. Let's say I have a

00:05:04.150 --> 00:05:08.250
10 newton object.

00:05:08.250 --> 00:05:11.710
It's sitting on ice, so
there's no friction.

00:05:11.710 --> 00:05:13.920
We don't want to worry about
fiction right now.

00:05:13.920 --> 00:05:15.915
And let's say I pull on it.

00:05:15.915 --> 00:05:20.485
Let's say my force vector--
This is my force vector.

00:05:25.540 --> 00:05:34.440
Let's say my force vector
is 100 newtons.

00:05:34.440 --> 00:05:37.910
I'm making the numbers up.

00:05:37.910 --> 00:05:40.250
100 newtons.

00:05:40.250 --> 00:05:46.940
And Let's say I slide it to
the right, so my distance

00:05:46.940 --> 00:05:54.720
vector is 10 meters parallel
to the ground.

00:05:57.910 --> 00:06:02.820
And the angle between them is
equal to 60 degrees, which is

00:06:02.820 --> 00:06:05.080
the same thing is pi over 3.

00:06:05.080 --> 00:06:06.125
We'll stick to degrees.

00:06:06.125 --> 00:06:07.790
It's a little bit
more intuitive.

00:06:07.790 --> 00:06:09.660
It's 60 degrees.

00:06:09.660 --> 00:06:11.940
This distance right
here is 10 meters.

00:06:11.940 --> 00:06:16.550
So my question is, by pulling on
this rope, or whatever, at

00:06:16.550 --> 00:06:20.360
the 60 degree angle, with a
force of 100 newtons, and

00:06:20.360 --> 00:06:23.980
pulling this block to the right
for 10 meters, how much

00:06:23.980 --> 00:06:25.560
work am I doing?

00:06:25.560 --> 00:06:28.990
Well, work is force times the
distance, but not just the

00:06:28.990 --> 00:06:29.890
total force.

00:06:29.890 --> 00:06:32.880
The magnitude of the force in
the direction of the distance.

00:06:32.880 --> 00:06:35.150
So what's the magnitude
of the force in the

00:06:35.150 --> 00:06:36.590
direction of the distance?

00:06:36.590 --> 00:06:41.730
It would be the horizontal
component of this force

00:06:41.730 --> 00:06:42.690
vector, right?

00:06:42.690 --> 00:06:45.110
So it would be 100
newtons times the

00:06:45.110 --> 00:06:46.520
cosine of 60 degrees.

00:06:46.520 --> 00:06:47.620
It will tell you how
much of that 100

00:06:47.620 --> 00:06:49.180
newtons goes to the right.

00:06:49.180 --> 00:06:50.540
Or another way you could
view it if this

00:06:50.540 --> 00:06:55.390
is the force vector.

00:06:55.390 --> 00:06:59.110
And this down here is
the distance vector.

00:06:59.110 --> 00:07:03.710
You could say that the total
work you performed is equal to

00:07:03.710 --> 00:07:10.350
the force vector dot the
distance vector, using the dot

00:07:10.350 --> 00:07:11.940
product-- taking the dot
product, to the force and the

00:07:11.940 --> 00:07:12.650
distance factor.

00:07:12.650 --> 00:07:14.930
And we know that the definition
is the magnitude of

00:07:14.930 --> 00:07:18.700
the force vector, which is 100
newtons, times the magnitude

00:07:18.700 --> 00:07:22.880
of the distance vector, which is
10 meters, times the cosine

00:07:22.880 --> 00:07:23.990
of the angle between them.

00:07:23.990 --> 00:07:27.030
Cosine of the angle
is 60 degrees.

00:07:27.030 --> 00:07:32.890
So that's equal to 1,000
newton meters

00:07:32.890 --> 00:07:34.010
times cosine of 60.

00:07:34.010 --> 00:07:36.410
Cosine of 60 is what?

00:07:36.410 --> 00:07:38.510
It's square root of 3 over 2.

00:07:41.690 --> 00:07:45.040
Square root of 3 over 2, if
I remember correctly.

00:07:45.040 --> 00:07:47.750
So times the square
root of 3 over 2.

00:07:47.750 --> 00:07:50.290
So the 2 becomes 500.

00:07:50.290 --> 00:07:56.160
So it becomes 500 square roots
of 3 joules, whatever that is.

00:07:56.160 --> 00:07:57.990
I don't know 700 something,
I'm guessing.

00:07:57.990 --> 00:07:58.960
Maybe it's 800 something.

00:07:58.960 --> 00:07:59.920
I'm not quite sure.

00:07:59.920 --> 00:08:01.950
But the important thing to
realize is that the dot

00:08:01.950 --> 00:08:03.010
product is useful.

00:08:03.010 --> 00:08:04.880
It applies to work.

00:08:04.880 --> 00:08:07.230
It actually calculates what
component of what vector goes

00:08:07.230 --> 00:08:08.280
in the other direction.

00:08:08.280 --> 00:08:09.400
Now you could interpret
it the other way.

00:08:09.400 --> 00:08:12.190
You could say this is
the magnitude of a

00:08:12.190 --> 00:08:14.500
times b cosine of theta.

00:08:14.500 --> 00:08:16.110
And that's completely valid.

00:08:16.110 --> 00:08:18.370
And what's b cosine of theta?

00:08:18.370 --> 00:08:21.920
Well, if you took b cosine of
theta, and you could work this

00:08:21.920 --> 00:08:24.370
out as an exercise for yourself,
that's the amount of

00:08:24.370 --> 00:08:26.020
the magnitude of the
b vector that's

00:08:26.020 --> 00:08:27.580
going in the a direction.

00:08:27.580 --> 00:08:29.630
So it doesn't matter
what order you go.

00:08:29.630 --> 00:08:33.520
So when you take the cross
product, it matters whether

00:08:33.520 --> 00:08:35.680
you do a cross b,
or b cross a.

00:08:35.680 --> 00:08:37.980
But when you're doing the dot
product, it doesn't matter

00:08:37.980 --> 00:08:38.740
what order.

00:08:38.740 --> 00:08:42.049
So b cosine theta would be the
magnitude of vector b that

00:08:42.049 --> 00:08:43.570
goes in the direction of a.

00:08:43.570 --> 00:08:49.810
So if you were to draw a
perpendicular line here, b

00:08:49.810 --> 00:08:51.455
cosine theta would
be this vector.

00:08:54.030 --> 00:08:56.660
That would be b cosine theta.

00:08:56.660 --> 00:08:59.590
The magnitude of
b cosine theta.

00:08:59.590 --> 00:09:03.090
So you could say how much of
vector b goes in the same

00:09:03.090 --> 00:09:03.820
direction as a?

00:09:03.820 --> 00:09:05.730
Then multiply the
two magnitudes.

00:09:05.730 --> 00:09:08.460
Or you could say how much of
vector a goes in the same

00:09:08.460 --> 00:09:09.680
direction is vector b?

00:09:09.680 --> 00:09:11.970
And then multiply the
two magnitudes.

00:09:11.970 --> 00:09:14.410
And now, this is, I think, a
good time to just make sure

00:09:14.410 --> 00:09:16.590
you understand the difference
between the dot product and

00:09:16.590 --> 00:09:18.240
the cross product.

00:09:18.240 --> 00:09:20.200
The dot product ends up
with just a number.

00:09:20.200 --> 00:09:23.280
You multiply two vectors and
all you have is a number.

00:09:23.280 --> 00:09:25.990
You end up with just
a scalar quantity.

00:09:25.990 --> 00:09:27.950
And why is that interesting?

00:09:27.950 --> 00:09:32.610
Well, it tells you how much do
these-- you could almost say--

00:09:32.610 --> 00:09:34.280
these vectors reinforce
each other.

00:09:34.280 --> 00:09:37.330
Because you're taking the parts
of their magnitudes that

00:09:37.330 --> 00:09:39.305
go in the same direction
and multiplying them.

00:09:39.305 --> 00:09:42.420
The cross product is actually
almost the opposite.

00:09:42.420 --> 00:09:45.110
You're taking their orthogonal
components, right?

00:09:45.110 --> 00:09:47.390
The difference was, this
was a a sine of theta.

00:09:47.390 --> 00:09:49.110
I don't want to mess you up
this picture too much.

00:09:49.110 --> 00:09:51.460
But you should review the
cross product videos.

00:09:51.460 --> 00:09:53.020
And I'll do another video where
I actually compare and

00:09:53.020 --> 00:09:53.880
contrast them.

00:09:53.880 --> 00:09:56.240
But the cross product is, you're
saying, let's multiply

00:09:56.240 --> 00:09:59.820
the magnitudes of the vectors
that are perpendicular to each

00:09:59.820 --> 00:10:01.660
other, that aren't going in the
same direction, that are

00:10:01.660 --> 00:10:03.380
actually orthogonal
to each other.

00:10:03.380 --> 00:10:05.770
And then, you have to pick a
direction since you're not

00:10:05.770 --> 00:10:07.230
saying, well, the same
direction that

00:10:07.230 --> 00:10:08.570
they're both going in.

00:10:08.570 --> 00:10:11.060
So you're picking the direction
that's orthogonal to

00:10:11.060 --> 00:10:12.150
both vectors.

00:10:12.150 --> 00:10:14.813
And then, that's why the
orientation matters and you

00:10:14.813 --> 00:10:16.370
have to take the right hand
rule, because there's actually

00:10:16.370 --> 00:10:19.190
two vectors that are
perpendicular to any other two

00:10:19.190 --> 00:10:20.810
vectors in three dimensions.

00:10:20.810 --> 00:10:22.380
Anyway, I'm all out of time.

00:10:22.380 --> 00:10:25.750
I'll continue this, hopefully
not too confusing, discussion

00:10:25.750 --> 00:10:26.330
in the next video.

00:10:26.330 --> 00:10:28.730
I'll compare and contrast
the cross

00:10:28.730 --> 00:10:30.260
product and the dot product.

00:10:30.260 --> 00:10:32.070
See you in the next video.


Induced current in a wire

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.540
Let's say I have a
magnetic field

00:00:02.540 --> 00:00:04.220
popping out of this video.

00:00:04.220 --> 00:00:07.010
So these little brown circles
show us the tips of the

00:00:07.010 --> 00:00:10.420
vectors popping out
of our screen.

00:00:10.420 --> 00:00:14.800
And in that magnetic field I
have this wire, this off-white

00:00:14.800 --> 00:00:15.760
colored wire.

00:00:15.760 --> 00:00:19.630
And sitting on that off-white
colored wire I have a charge

00:00:19.630 --> 00:00:20.990
of charge Q.

00:00:20.990 --> 00:00:22.250
Let me write down
the other stuff.

00:00:22.250 --> 00:00:27.940
So this is the magnetic
field B coming out.

00:00:27.940 --> 00:00:30.520
Let's say I were to take
this whole wire.

00:00:30.520 --> 00:00:32.390
And let's say that the wire
overlaps with the magnetic

00:00:32.390 --> 00:00:34.820
field a distance of L.

00:00:34.820 --> 00:00:36.760
So let's say the magnetic
field stops

00:00:36.760 --> 00:00:40.800
here and stops here.

00:00:40.800 --> 00:00:43.650
And let's say that this distance
right here, that

00:00:43.650 --> 00:00:46.660
distance is L.

00:00:46.660 --> 00:00:48.300
I drew it a little bit weird
but you get the idea.

00:00:48.300 --> 00:00:49.750
From here to here is L.

00:00:49.750 --> 00:00:53.090
I have this charge sitting on
some type of conductor that we

00:00:53.090 --> 00:00:54.170
can consider a wire.

00:00:54.170 --> 00:00:57.770
And the magnetic field is
pushing out of the page.

00:00:57.770 --> 00:01:01.720
So in this current formation,
let's say I don't have any

00:01:01.720 --> 00:01:04.050
voltage across this
wire or anything.

00:01:04.050 --> 00:01:05.560
What's going to happen?

00:01:05.560 --> 00:01:08.500
Well, if I just have a
stationary charge sitting in a

00:01:08.500 --> 00:01:11.200
magnetic field, nothing really
is going to happen, right?

00:01:11.200 --> 00:01:17.920
Because we know that the force
due to a magnetic field is

00:01:17.920 --> 00:01:23.880
equal to the charge times the
cross product of the velocity

00:01:23.880 --> 00:01:27.770
of the charge and the
magnetic field.

00:01:27.770 --> 00:01:30.480
If this wires are stationary and
there's no voltage across

00:01:30.480 --> 00:01:32.620
it, et cetera, et cetera, the
velocity of this charge is

00:01:32.620 --> 00:01:33.660
going to be 0.

00:01:33.660 --> 00:01:36.950
So if the velocity is 0, we know
that the magnitude of a

00:01:36.950 --> 00:01:40.670
cross product is the
same thing as--.

00:01:40.670 --> 00:01:44.960
So Q is, that's just a scalar
quantity-- so that's just Q--

00:01:44.960 --> 00:01:49.390
times the magnitude of the
velocity times the magnitude

00:01:49.390 --> 00:01:52.140
of B times sine theta.

00:01:52.140 --> 00:01:56.430
And in these situations where
anything that's going on in

00:01:56.430 --> 00:01:58.780
this plane is going to
be perpendicular to

00:01:58.780 --> 00:01:59.720
this magnetic field--.

00:01:59.720 --> 00:02:03.190
So the angle between the
magnetic field and any

00:02:03.190 --> 00:02:05.220
velocity-- if there were any
within this plane-- would be

00:02:05.220 --> 00:02:05.710
90 degrees.

00:02:05.710 --> 00:02:07.640
So you wouldn't have to worry
about the sine theta too much.

00:02:07.640 --> 00:02:11.039
But we see if the velocity is
0 or the speed is 0, the

00:02:11.039 --> 00:02:13.850
magnitude of the velocity is 0,
that there's not going to

00:02:13.850 --> 00:02:16.570
be any net force due to the
magnetic field on this charge.

00:02:16.570 --> 00:02:18.950
And nothing interesting's
going to happen.

00:02:18.950 --> 00:02:21.650
But let's do a little
experiment.

00:02:21.650 --> 00:02:24.210
What happens if I were to move
this wire, if I were to shift

00:02:24.210 --> 00:02:27.320
it to the left, with
the velocity V?

00:02:27.320 --> 00:02:32.020
So I take this wire and I shift
it to the left with the

00:02:32.020 --> 00:02:32.935
velocity V.

00:02:32.935 --> 00:02:37.180
All right, so the whole wire
is shifting to the left.

00:02:37.180 --> 00:02:39.270
Well, if the whole wire is
shifting to the left, this

00:02:39.270 --> 00:02:43.070
charge is sitting on that wire,
so that charge is also

00:02:43.070 --> 00:02:45.730
going to move to the left
with the velocity V.

00:02:45.730 --> 00:02:47.870
And now things get
interesting.

00:02:47.870 --> 00:02:50.090
The charge is moving to the left
with the velocity V, so

00:02:50.090 --> 00:02:53.340
now we can apply the
first magnetism

00:02:53.340 --> 00:02:54.460
formula that we learned.

00:02:54.460 --> 00:02:55.630
We could apply this formula.

00:02:55.630 --> 00:02:57.490
So what's going to happen
to this charge?

00:02:57.490 --> 00:02:59.950
Well, the force of the charge
is going to be the charge

00:02:59.950 --> 00:03:04.300
times the magnitude of the
velocity cross the magnetic

00:03:04.300 --> 00:03:05.680
field vector.

00:03:05.680 --> 00:03:07.820
So we know that there's going
to be some net force.

00:03:07.820 --> 00:03:10.080
This is non-zero now.

00:03:10.080 --> 00:03:12.510
And this is non-zero,
we're assuming.

00:03:12.510 --> 00:03:13.670
And we're assuming the
charge is non-zero.

00:03:13.670 --> 00:03:16.170
So what direction is the
force going to be in?

00:03:16.170 --> 00:03:19.350
So let's do our right hand rule
on the cross product.

00:03:19.350 --> 00:03:21.420
V cross B will give
us the direction.

00:03:21.420 --> 00:03:26.740
So point your index finger in
the direction of the velocity.

00:03:26.740 --> 00:03:28.330
I have to look at my own
hand to make sure

00:03:28.330 --> 00:03:30.450
I'm doing it right.

00:03:30.450 --> 00:03:32.230
So you point your index
finger in the

00:03:32.230 --> 00:03:34.440
direction of the velocity.

00:03:34.440 --> 00:03:37.540
Point your middle finger
in the direction of

00:03:37.540 --> 00:03:38.510
the magnetic field.

00:03:38.510 --> 00:03:40.440
The magnetic field is popping
out of the page, so your

00:03:40.440 --> 00:03:41.630
middle finger is actually
going to be

00:03:41.630 --> 00:03:43.710
popping out of the page.

00:03:43.710 --> 00:03:45.430
Your next two fingers
are just going to do

00:03:45.430 --> 00:03:46.190
something like that.

00:03:46.190 --> 00:03:48.520
So you're kind of approximating
like you're

00:03:48.520 --> 00:03:49.580
shooting a gun.

00:03:49.580 --> 00:03:51.030
And then what's your
thumb going to do?

00:03:51.030 --> 00:03:55.500
Your thumb is going to
point straight up.

00:03:55.500 --> 00:03:58.660
This is the palm of--
that's your thumb.

00:03:58.660 --> 00:04:00.710
This could be your nail,
fingernail, fingernail of your

00:04:00.710 --> 00:04:03.160
thumb, fingernail of
your middle finger.

00:04:03.160 --> 00:04:04.940
This is the direction
of the velocity.

00:04:04.940 --> 00:04:06.120
Let me get a suitable color.

00:04:06.120 --> 00:04:07.960
The velocity is that way.

00:04:07.960 --> 00:04:12.180
The magnetic field is popping
out of the page.

00:04:12.180 --> 00:04:17.100
So the force on the particle--
on this charged particle or on

00:04:17.100 --> 00:04:20.480
this charge-- due to the
magnetic field is going to go

00:04:20.480 --> 00:04:22.700
in the direction
of your thumb.

00:04:22.700 --> 00:04:25.005
So the direction of the force
is in this direction.

00:04:28.300 --> 00:04:30.210
So what's going to happen?

00:04:30.210 --> 00:04:31.630
There's going to be a
net force in this

00:04:31.630 --> 00:04:33.330
direction on the charge.

00:04:33.330 --> 00:04:37.560
And the charge is going to
move upwards, right?

00:04:37.560 --> 00:04:39.830
I mean, when you start having
a moving-- you could imagine

00:04:39.830 --> 00:04:42.450
also that you had multiple
charges, right?

00:04:42.450 --> 00:04:45.040
If you had multiple charges
here and you're moving the

00:04:45.040 --> 00:04:46.670
whole wire, all of those
charges are going

00:04:46.670 --> 00:04:47.910
to be moving upwards.

00:04:47.910 --> 00:04:51.670
And what is another way to call
a bunch of moving charges

00:04:51.670 --> 00:04:53.010
along a conductor?

00:04:53.010 --> 00:04:55.015
Well, it's a current, depending
on how much charge

00:04:55.015 --> 00:04:56.380
is moving per second.

00:04:56.380 --> 00:04:59.940
So at least in very qualitative
terms, you see

00:04:59.940 --> 00:05:03.810
that when you move a wire
through a magnetic field or

00:05:03.810 --> 00:05:06.140
when you move a magnetic field
past a wire, right?

00:05:06.140 --> 00:05:08.450
Because they're kind of the same
thing, it's all about the

00:05:08.450 --> 00:05:09.530
relative motion.

00:05:09.530 --> 00:05:13.200
But if you move a wire through
a magnetic field, it is

00:05:13.200 --> 00:05:19.950
actually going to induce
a current in the wire.

00:05:19.950 --> 00:05:21.600
It's going to induce the current
in the wire, and

00:05:21.600 --> 00:05:24.990
actually this is how electric
generators are generated.

00:05:24.990 --> 00:05:28.080
And I'll do a whole series of
videos on how you-- you know,

00:05:28.080 --> 00:05:34.410
if you're using coal or steam or
hydropower, how that turns,

00:05:34.410 --> 00:05:36.700
essentially that turns these
generators around and it

00:05:36.700 --> 00:05:37.530
induces current.

00:05:37.530 --> 00:05:40.070
And that's how we get
electricity from all of these

00:05:40.070 --> 00:05:42.040
various energy sources
that essentially just

00:05:42.040 --> 00:05:44.020
make turbines turn.

00:05:44.020 --> 00:05:46.310
But anyway, let's go back
to what we were doing.

00:05:46.310 --> 00:05:47.240
So let me ask you a question.

00:05:47.240 --> 00:05:52.070
If this particle-- and this
all has a point-- if this

00:05:52.070 --> 00:05:53.760
particle starts at
the beginning.

00:05:53.760 --> 00:05:55.120
Let's say the particle
is right here.

00:05:55.120 --> 00:05:57.880
So it starts right where the
magnetic field starts

00:05:57.880 --> 00:05:59.210
affecting the wire.

00:05:59.210 --> 00:06:02.610
And how much work is going to be
done on the particle by the

00:06:02.610 --> 00:06:04.340
magnetic field?

00:06:04.340 --> 00:06:05.080
Well, what's work?

00:06:05.080 --> 00:06:13.120
Work is equal to force times
distance, where the force has

00:06:13.120 --> 00:06:15.990
to be in the same direction
as the distance, right?

00:06:15.990 --> 00:06:17.770
Force times distance, I
won't mess with the

00:06:17.770 --> 00:06:18.430
vectors right now.

00:06:18.430 --> 00:06:20.350
But they have to be in
the same direction.

00:06:20.350 --> 00:06:24.160
So how much work is going to
be done on this particle?

00:06:24.160 --> 00:06:26.950
So the work is going to be the
net force exerted on the

00:06:26.950 --> 00:06:29.030
particle times the distance.

00:06:29.030 --> 00:06:30.690
Well, this distance
is L, right?

00:06:30.690 --> 00:06:33.320
We say, once a particle gets
here there's no magnetic field

00:06:33.320 --> 00:06:36.710
up here, so the magnetic field
will stop acting on it.

00:06:36.710 --> 00:06:40.990
So the total work done: Work,
which is equal to force times

00:06:40.990 --> 00:06:45.480
distance, is equal to-- so the
net force is this up here.

00:06:45.480 --> 00:06:48.840
Q-- and I'll leave
some space-- V

00:06:48.840 --> 00:06:52.060
cross B times the distance.

00:06:52.060 --> 00:06:54.190
And the distance right here is
just a scalar quantity, so we

00:06:54.190 --> 00:06:55.830
could put it out front, right?

00:06:55.830 --> 00:07:00.010
Q times L times V
cross B, right?

00:07:00.010 --> 00:07:03.400
This is-- Q V cross B is the
force times the distance.

00:07:03.400 --> 00:07:05.360
That's just the work done.

00:07:05.360 --> 00:07:08.550
Now how much work is being
done per charge, right?

00:07:08.550 --> 00:07:11.510
This is how much work is being
done on this charge.

00:07:11.510 --> 00:07:13.430
But let's say there might have
been multiple charges, so we

00:07:13.430 --> 00:07:16.050
just want to know how much
work is done per charge.

00:07:16.050 --> 00:07:17.810
So work per charge.

00:07:17.810 --> 00:07:21.990
We could divide both
sides by charge.

00:07:21.990 --> 00:07:26.350
So work per charge is equal
to this per charge.

00:07:26.350 --> 00:07:34.390
So it is equal to the distance
times the velocity that you're

00:07:34.390 --> 00:07:41.260
pulling the wire to the left
with cross the magnetic field.

00:07:41.260 --> 00:07:43.400
This is where it gets
interesting.

00:07:43.400 --> 00:07:45.240
So what is work per charge?

00:07:45.240 --> 00:07:47.370
The units of work are
energy, right?

00:07:47.370 --> 00:07:48.230
Joules.

00:07:48.230 --> 00:07:50.240
And charge, that's
in coulombs.

00:07:50.240 --> 00:07:54.595
So what are joules
per coulomb?

00:07:59.220 --> 00:08:01.170
This is equal to volts.

00:08:01.170 --> 00:08:03.350
Volts are joules per coulomb.

00:08:03.350 --> 00:08:07.680
So this particle, or these
charges, are going to start

00:08:07.680 --> 00:08:11.260
moving in this direction as if
there is a voltage difference.

00:08:11.260 --> 00:08:14.840
As if there is a potential
difference between this point

00:08:14.840 --> 00:08:15.630
and this point.

00:08:15.630 --> 00:08:18.460
As if this is the positive
voltage terminal and this is

00:08:18.460 --> 00:08:20.230
the minus voltage terminal.

00:08:20.230 --> 00:08:22.870
So there's actually going to be
a voltage-- or a perceived

00:08:22.870 --> 00:08:24.970
voltage-- difference between
this point and this point that

00:08:24.970 --> 00:08:27.080
will start making the
current flow.

00:08:27.080 --> 00:08:28.870
Let's say you didn't even know
that there was a magnetic

00:08:28.870 --> 00:08:29.220
field here.

00:08:29.220 --> 00:08:30.310
You would just see this
current flowing.

00:08:30.310 --> 00:08:32.530
You'd be like, oh well, there
has to be a voltage difference

00:08:32.530 --> 00:08:33.789
there, right?

00:08:33.789 --> 00:08:35.940
But when we're dealing with
this-- because when we talk

00:08:35.940 --> 00:08:37.840
about voltages, that was like
a potential difference.

00:08:37.840 --> 00:08:41.900
That something-- that a particle
or a charge has a

00:08:41.900 --> 00:08:43.990
higher potential energy and
that's why it's moving.

00:08:43.990 --> 00:08:47.100
But it's hard to-- at least for
these purposes-- say, well

00:08:47.100 --> 00:08:49.670
you have a higher potential
energy here.

00:08:49.670 --> 00:08:52.440
It's really being created
by the magnetic field.

00:08:52.440 --> 00:08:55.450
So in this context, people
have said that instead of

00:08:55.450 --> 00:08:57.790
saying that this is creating a
voltage difference between

00:08:57.790 --> 00:09:00.580
this point and this point, if
the magnetic field on the

00:09:00.580 --> 00:09:03.220
moving wire is causing that,
people say that it's creating

00:09:03.220 --> 00:09:06.050
an electromotive force,
or an EMF.

00:09:06.050 --> 00:09:10.510
But EMF, the units are still
joules per coulomb or volts.

00:09:10.510 --> 00:09:13.350
And it really is-- in every way
when you're analyzing the

00:09:13.350 --> 00:09:16.240
circuits-- still the same
thing as a potential

00:09:16.240 --> 00:09:18.040
difference or as a voltage
difference.

00:09:18.040 --> 00:09:20.430
But since it seems a little bit
more proactive, it seems

00:09:20.430 --> 00:09:24.940
like this magnetic field is
actually impacting a force on

00:09:24.940 --> 00:09:27.030
this wire that is causing
the current to move.

00:09:27.030 --> 00:09:29.260
We call it EMF.

00:09:29.260 --> 00:09:34.300
So we could say that the EMF,
the electromotive force-- or

00:09:34.300 --> 00:09:37.720
the voltage across from here
to here, but they're really

00:09:37.720 --> 00:09:41.750
the same thing-- is equal to
the distance of the wire

00:09:41.750 --> 00:09:47.290
that's in the magnetic field
times the velocity-- that

00:09:47.290 --> 00:09:52.430
you're pulling the wire in--
cross the magnetic field.

00:09:52.430 --> 00:09:54.280
So let's say, I don't know,
let's just throw out a bunch

00:09:54.280 --> 00:09:54.730
of numbers.

00:09:54.730 --> 00:09:56.780
Let's say that the magnetic
field is-- I'll

00:09:56.780 --> 00:09:59.640
make it easy-- 2 teslas.

00:09:59.640 --> 00:10:05.340
My velocity to the left is
3 meters per second.

00:10:05.340 --> 00:10:08.000
And let's-- just for fun-- let's
give this a little bit

00:10:08.000 --> 00:10:11.360
of a resistance, just so we
can figure out something.

00:10:11.360 --> 00:10:14.730
So let's say this resistance is,
I don't know, let's say it

00:10:14.730 --> 00:10:19.140
is 6 ohms. There's a 6
ohm resistor here.

00:10:19.140 --> 00:10:22.000
So the resistance of the wire
from here to here is 6 ohms.

00:10:22.000 --> 00:10:23.970
All wires have some
resistance.

00:10:23.970 --> 00:10:25.920
So first of all,
what's the EMF?

00:10:25.920 --> 00:10:27.650
Oh, and let's say that
this total distance

00:10:27.650 --> 00:10:32.420
right here is 12 meters.

00:10:32.420 --> 00:10:37.370
So the EMF induced on the-- or
the electromotive force-- put

00:10:37.370 --> 00:10:40.950
on to the wire by the magnetic
field is going to equal the

00:10:40.950 --> 00:10:42.980
distance of the wire
in the magnetic

00:10:42.980 --> 00:10:47.130
field-- 12 meters-- times--.

00:10:47.130 --> 00:10:48.920
Well, when we're just taking
the cross product, we know

00:10:48.920 --> 00:10:51.660
that the velocity is
perpendicular to

00:10:51.660 --> 00:10:53.210
the magnetic field.

00:10:53.210 --> 00:10:54.930
So we don't have to worry about
sine theta because theta

00:10:54.930 --> 00:10:56.240
is already 90 degrees.

00:10:56.240 --> 00:10:57.870
So we just have to worry
about the magnitudes.

00:10:57.870 --> 00:11:01.540
So it's going to be 12 meters
times the velocity, which is 3

00:11:01.540 --> 00:11:04.660
meters per second, times the
magnetic field, or the

00:11:04.660 --> 00:11:08.140
magnitude of the magnetic
field, that's 2 teslas.

00:11:08.140 --> 00:11:10.390
And so the EMF is 12
times 3 times 2.

00:11:10.390 --> 00:11:12.160
12 times 6.

00:11:12.160 --> 00:11:14.710
Which is 72.

00:11:14.710 --> 00:11:19.770
You could say 72 volts, or
72 joules per coulomb.

00:11:19.770 --> 00:11:21.780
And now you have that potential
difference, or that

00:11:21.780 --> 00:11:26.640
EMF, across a 6 ohm
resistor, right?

00:11:26.640 --> 00:11:29.420
So that, you just go back to
voltage is equal to IR.

00:11:29.420 --> 00:11:33.640
Or you could write EMF
is equal to IR.

00:11:33.640 --> 00:11:36.300
So EMF divided by resistance.

00:11:36.300 --> 00:11:40.440
So if we take this EMF and we
divide it by the resistance--

00:11:40.440 --> 00:11:44.430
divided by 6 ohms-- we get
the current, right?

00:11:44.430 --> 00:11:48.160
EMF divided by resistance
is equal to current.

00:11:48.160 --> 00:11:52.550
So you divide 72 volts or 72
joules per coulomb divided by

00:11:52.550 --> 00:11:56.970
6 ohms. And then you get a
current going along this wire

00:11:56.970 --> 00:11:59.590
right here, due to the EMF, due
to the magnetic field-- I

00:11:59.590 --> 00:12:03.770
know it's very messy at this
point-- of 12 amperes.

00:12:03.770 --> 00:12:05.450
Anyway, I'm all out of time.

00:12:05.450 --> 00:12:07.490
I'll see you in the
next video.


Electric motors (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:02.790
So where I had left off
is we had the circuit.

00:00:02.790 --> 00:00:04.010
We had these little
leads here.

00:00:04.010 --> 00:00:05.550
This was kind of
our innovation.

00:00:05.550 --> 00:00:08.320
And this is actually called a
commutator, where this part

00:00:08.320 --> 00:00:10.800
that's connected to our rotating
piece, that's the

00:00:10.800 --> 00:00:11.080
commutator.

00:00:11.080 --> 00:00:12.300
And these are the brushes.

00:00:12.300 --> 00:00:14.570
So you could imagine, you could
design them as brushes

00:00:14.570 --> 00:00:15.950
that always stay in touch.

00:00:15.950 --> 00:00:20.900
Kind of like the brushes
on a, what was that?

00:00:20.900 --> 00:00:22.900
What are those cars at
the amusement park?

00:00:22.900 --> 00:00:24.260
Bumper cars, right?

00:00:24.260 --> 00:00:27.000
On the bumper cars you have a
pole behind your bumper car.

00:00:27.000 --> 00:00:27.960
I'll draw that for fun.

00:00:27.960 --> 00:00:30.880
So let's say this is
your bumper car.

00:00:30.880 --> 00:00:32.070
Looks like a shoe
a little bit.

00:00:32.070 --> 00:00:34.910
This is you driving
your bumper car.

00:00:34.910 --> 00:00:36.200
And they have a pole.

00:00:36.200 --> 00:00:39.350
And at the top of the pole,
you'll see these brushes that

00:00:39.350 --> 00:00:41.720
are touching the
ceiling, right?

00:00:41.720 --> 00:00:43.320
You could view that as the
same type of brush.

00:00:43.320 --> 00:00:46.260
And what it allows is a constant
electric current to

00:00:46.260 --> 00:00:48.430
flow through the ceiling.

00:00:48.430 --> 00:00:49.950
I don't know what direction
it's going in.

00:00:49.950 --> 00:00:52.580
But it allows a current to
flow through the ceiling.

00:00:52.580 --> 00:00:55.570
And maybe your car is grounded
so the current can flow down

00:00:55.570 --> 00:00:57.370
to ground, so that your car
could be powered by the

00:00:57.370 --> 00:00:59.470
ceiling and not have to carry
a battery in every car.

00:00:59.470 --> 00:01:01.890
Which would be kind of a waste
of energy and probably some

00:01:01.890 --> 00:01:03.590
type of a health hazard
and safety risk,

00:01:03.590 --> 00:01:04.540
et cetera, et cetera.

00:01:04.540 --> 00:01:07.450
So those brushes on your bumper
cars might not be all

00:01:07.450 --> 00:01:10.620
that different from the brushes
that are touching the

00:01:10.620 --> 00:01:12.420
commutators here.

00:01:12.420 --> 00:01:13.710
Just a little bit
of terminology.

00:01:13.710 --> 00:01:16.120
And it never hurts to
introduce bumper car

00:01:16.120 --> 00:01:18.672
references.

00:01:18.672 --> 00:01:20.360
I probably should have done
them earlier when we were

00:01:20.360 --> 00:01:22.860
learning about momentum
and things.

00:01:22.860 --> 00:01:24.210
But anyway.

00:01:24.210 --> 00:01:25.980
So what was happening here?

00:01:25.980 --> 00:01:27.870
So going back to our
first video.

00:01:27.870 --> 00:01:30.910
We have the current going
down like this.

00:01:30.910 --> 00:01:33.090
And then if you use your right
hand rule with the cross

00:01:33.090 --> 00:01:36.680
product, you know that the net
force from the magnetic field

00:01:36.680 --> 00:01:39.270
is going to be downwards on the
left hand side, upward on

00:01:39.270 --> 00:01:40.250
the right hand side.

00:01:40.250 --> 00:01:42.890
So you have a net torque
rotating it like that.

00:01:42.890 --> 00:01:46.050
Rotating the right out of the
page, the left into the page

00:01:46.050 --> 00:01:47.800
or into the video screen.

00:01:47.800 --> 00:01:49.900
Up to the point that you've
rotated 90 degrees and now

00:01:49.900 --> 00:01:52.880
you're looking kind of, so
this side right here.

00:01:52.880 --> 00:01:54.480
Let me do it in a different
color so you can see it.

00:01:54.480 --> 00:01:58.210
This side is this side,
right on top.

00:01:58.210 --> 00:02:00.960
And this side is on the bottom,
below the page.

00:02:00.960 --> 00:02:03.240
This side is now
above the page.

00:02:03.240 --> 00:02:05.830
If this distance is r,
this side is now r

00:02:05.830 --> 00:02:07.320
units above the page.

00:02:07.320 --> 00:02:11.180
And I said ideally maybe your
commutator loses touch with

00:02:11.180 --> 00:02:12.530
the brushes at this
point, right?

00:02:12.530 --> 00:02:13.820
Because they're popping out a
little bit, so when you're

00:02:13.820 --> 00:02:15.940
vertical, you actually lose
touch with the brushes.

00:02:15.940 --> 00:02:19.620
So you have no circuit flowing,
so you save a little

00:02:19.620 --> 00:02:20.280
battery energy.

00:02:20.280 --> 00:02:25.210
And you just let a little bit of
the angular momentum carry

00:02:25.210 --> 00:02:28.950
this whole rotating contraption
further a little

00:02:28.950 --> 00:02:31.860
bit to the point that
your configuration

00:02:31.860 --> 00:02:35.190
will look like this.

00:02:35.190 --> 00:02:40.810
So I know I keep changing
colors, but the whole

00:02:40.810 --> 00:02:44.320
contraption will now
look like this.

00:02:51.920 --> 00:02:56.420
OK, that's my positive,
negative, positive, negative,

00:02:56.420 --> 00:02:57.660
current flows like this.

00:02:57.660 --> 00:02:59.670
Now we assume that the
commutator has

00:02:59.670 --> 00:03:00.960
gotten back in touch.

00:03:00.960 --> 00:03:03.100
And let me color code this.

00:03:03.100 --> 00:03:07.610
So if this side is this
color, right?

00:03:07.610 --> 00:03:10.140
Now this is when we're looking
at top on, where it's popping

00:03:10.140 --> 00:03:12.320
out of the screen, where
it's above the screen.

00:03:12.320 --> 00:03:14.980
And now we've rotated 180
degrees and this side is on

00:03:14.980 --> 00:03:18.340
this side, right?

00:03:18.340 --> 00:03:19.860
Let me pick a suitable color.

00:03:19.860 --> 00:03:21.470
If this side was green.

00:03:21.470 --> 00:03:25.280
Now this side, we flip the whole
thing over 180 degrees.

00:03:25.280 --> 00:03:26.920
And now something interesting
happens.

00:03:26.920 --> 00:03:29.390
Remember, before we had this
commutator and everything, if

00:03:29.390 --> 00:03:33.760
we just flipped it over, the
current, because before when

00:03:33.760 --> 00:03:38.500
we didn't have the commutator,
the current here was flowing

00:03:38.500 --> 00:03:39.860
down here, up here.

00:03:39.860 --> 00:03:42.380
And before the commutator, we
had the current flowing down

00:03:42.380 --> 00:03:43.470
here and up here.

00:03:43.470 --> 00:03:45.510
And so we were switching
directions.

00:03:45.510 --> 00:03:47.540
And so you would have had this
thing that would never

00:03:47.540 --> 00:03:48.440
completely rotate.

00:03:48.440 --> 00:03:50.410
It would just keep flipping
over, right?

00:03:50.410 --> 00:03:53.640
Which may be useful for, I don't
know, if you wanted to

00:03:53.640 --> 00:03:54.600
flip things.

00:03:54.600 --> 00:03:57.120
But it's not useful
as a motor.

00:03:57.120 --> 00:03:58.390
So what happens here?

00:03:58.390 --> 00:04:01.580
Now this side, all of a sudden
instead of being connected to

00:04:01.580 --> 00:04:03.800
this lead, is now connected
to this lead.

00:04:03.800 --> 00:04:07.390
And this green side is now
connected to this lead.

00:04:07.390 --> 00:04:09.160
So something interesting
happens.

00:04:09.160 --> 00:04:12.600
Now the current on the left side
is still flowing down,

00:04:12.600 --> 00:04:14.580
right, and the current
on the right side is

00:04:14.580 --> 00:04:16.140
still flowing up.

00:04:16.140 --> 00:04:18.820
So we're back to this
configuration except that this

00:04:18.820 --> 00:04:20.820
contraption has flipped over.

00:04:20.820 --> 00:04:23.550
The brown side is now on the
left and the green side is now

00:04:23.550 --> 00:04:24.410
on the right.

00:04:24.410 --> 00:04:27.690
And what that allows is that
those net torques are still

00:04:27.690 --> 00:04:29.440
going in that same rotational
direction.

00:04:29.440 --> 00:04:30.720
Use your right hand rule.

00:04:30.720 --> 00:04:32.770
The current is flowing
down here.

00:04:32.770 --> 00:04:36.250
So if your magnetic field is
coming to the left, then the

00:04:36.250 --> 00:04:38.680
net force is going to be
down there and it's

00:04:38.680 --> 00:04:40.140
going to be up there.

00:04:40.140 --> 00:04:43.170
And so we can continue
indefinite, and we solve our

00:04:43.170 --> 00:04:44.250
other problem.

00:04:44.250 --> 00:04:47.360
That we will never keep twisting
these wires here.

00:04:47.360 --> 00:04:50.780
So now using the commutator, we
have essentially created an

00:04:50.780 --> 00:04:51.655
electric motor.

00:04:51.655 --> 00:04:53.710
And remember I drew that little
thing, that could be

00:04:53.710 --> 00:04:54.390
like the axle.

00:04:54.390 --> 00:04:57.200
Maybe that turns the wheels
or something.

00:04:57.200 --> 00:05:00.880
So if you have a constant
magnetic field and you just by

00:05:00.880 --> 00:05:04.120
using this commutator which,
as soon as you get to that

00:05:04.120 --> 00:05:07.300
kind of vertical point, it cuts
the current, and then

00:05:07.300 --> 00:05:10.060
when you go a little bit past
vertical, a little bit past 90

00:05:10.060 --> 00:05:13.190
degrees, it switches the
direction of the current.

00:05:13.190 --> 00:05:15.600
So on the left hand side you
always have the current coming

00:05:15.600 --> 00:05:17.840
down, and on the right hand
side you always have the

00:05:17.840 --> 00:05:18.720
current going up.

00:05:18.720 --> 00:05:23.450
So that the net torque is always
going to be pushing, is

00:05:23.450 --> 00:05:28.540
always going to be rotating this
contraption down on the

00:05:28.540 --> 00:05:31.990
left hand side and up on
the right hand side.

00:05:31.990 --> 00:05:34.040
Coming out of the page on the
right hand side and then down

00:05:34.040 --> 00:05:34.990
on the left hand side.

00:05:34.990 --> 00:05:37.730
And you could actually
turn a wheel now.

00:05:37.730 --> 00:05:39.400
You could create an
electric car.

00:05:39.400 --> 00:05:42.380
So that is the basics
really of how

00:05:42.380 --> 00:05:45.150
electric motors are created.

00:05:45.150 --> 00:05:46.190
Well, there's another way
you could have done it.

00:05:46.190 --> 00:05:47.320
You didn't have to use
the commutator.

00:05:47.320 --> 00:05:51.100
One methodology you could have
used is you could have had the

00:05:51.100 --> 00:05:54.650
magnetic field going until you
get to this point, and then

00:05:54.650 --> 00:05:56.770
you turn off the magnetic
field, right?

00:05:56.770 --> 00:06:02.150
And maybe you wait for this
situation to go all the way

00:06:02.150 --> 00:06:05.060
180 degrees and then you turn
the magnetic field back on

00:06:05.060 --> 00:06:06.130
again, right?

00:06:06.130 --> 00:06:07.550
That's one possibility.

00:06:07.550 --> 00:06:10.460
But that's maybe not as
efficient cause half of the

00:06:10.460 --> 00:06:11.910
cycle you're not powering it.

00:06:11.910 --> 00:06:13.325
Or maybe you switch
the direction of

00:06:13.325 --> 00:06:14.310
the magnetic field.

00:06:14.310 --> 00:06:16.240
Or another option, you don't
have to use a commutator.

00:06:16.240 --> 00:06:17.890
Maybe you use some other
contraption to switch the

00:06:17.890 --> 00:06:19.160
direction of the
magnetic field.

00:06:19.160 --> 00:06:21.030
But this is probably the
simplest way to do it.

00:06:21.030 --> 00:06:24.010
And I think it gives you a
general idea of how an

00:06:24.010 --> 00:06:25.480
electric motor can be created.

00:06:25.480 --> 00:06:28.460
And then we could play around
with the mechanics of

00:06:28.460 --> 00:06:29.230
innovations on it.

00:06:29.230 --> 00:06:32.500
But all electric motors are
essentially some variation of

00:06:32.500 --> 00:06:35.020
what you have learned
in this video.

00:06:35.020 --> 00:06:37.820
Isn't it neat to learn
something useful?

00:06:37.820 --> 00:06:39.070
See you in the next video.


Electric motors (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.010
Where I left off in the last
video we saw that if we had a

00:00:03.010 --> 00:00:06.610
magnetic field coming in from
the right and we had this loop

00:00:06.610 --> 00:00:10.900
of-- I guess we call it-- metal
or a circuit, and it's

00:00:10.900 --> 00:00:13.760
carrying a current where the
current is coming in this

00:00:13.760 --> 00:00:14.300
direction--.

00:00:14.300 --> 00:00:16.560
You can imagine positive
protons, although we know the

00:00:16.560 --> 00:00:17.810
electrons go in the
other direction.

00:00:17.810 --> 00:00:20.400
But the current is coming in
this direction and going out

00:00:20.400 --> 00:00:20.990
that direction.

00:00:20.990 --> 00:00:24.210
We figured out using the right
hand rule and just this

00:00:24.210 --> 00:00:27.890
formula, that the net force of
the magnetic field coming in

00:00:27.890 --> 00:00:31.850
this direction on this arm
of the wire or the

00:00:31.850 --> 00:00:34.520
circuit is net downwards.

00:00:34.520 --> 00:00:36.510
And on this arm, it
was net upwards.

00:00:36.510 --> 00:00:39.750
And so it provided a net
torque on this circuit.

00:00:39.750 --> 00:00:41.620
Or, as I said in the last
video, a paper clip.

00:00:41.620 --> 00:00:44.530
And where this dotted line
is the axis of rotation.

00:00:44.530 --> 00:00:46.610
And this is how I showed
you it would rotate.

00:00:46.610 --> 00:00:49.700
Where the magnetic field is
essentially pushing up on the

00:00:49.700 --> 00:00:51.840
right hand side and pushing down
on the left hand side.

00:00:51.840 --> 00:00:54.570
It has no effect over here on
the top and the bottom.

00:00:54.570 --> 00:00:58.230
So it would rotate in
this direction.

00:00:58.230 --> 00:01:01.610
And then this was kind of what
it looks like after it rotates

00:01:01.610 --> 00:01:02.040
a little bit.

00:01:02.040 --> 00:01:06.190
And the whole reason why I did
this, I said, well, this arm--

00:01:06.190 --> 00:01:08.130
which is the same as
this arm-- the net

00:01:08.130 --> 00:01:09.700
force is still upwards.

00:01:09.700 --> 00:01:11.160
Out of our screen.

00:01:11.160 --> 00:01:14.630
But that upwards direction is
now no longer going to be

00:01:14.630 --> 00:01:19.010
completely perpendicular to
the moment arm distance.

00:01:19.010 --> 00:01:20.770
That's the moment
arm distance.

00:01:20.770 --> 00:01:23.090
Now the moment arm distance is
kind of coming at an angle out

00:01:23.090 --> 00:01:23.940
of the page.

00:01:23.940 --> 00:01:27.540
So only some of this net outward
force for the magnetic

00:01:27.540 --> 00:01:30.600
field is going to
be perpendicular

00:01:30.600 --> 00:01:31.630
to the moment arm.

00:01:31.630 --> 00:01:34.300
And so the torque on it will be
less, but it's still going

00:01:34.300 --> 00:01:36.600
to be torque in that
same direction.

00:01:36.600 --> 00:01:39.090
Kind of coming out of the page
on the right and into the page

00:01:39.090 --> 00:01:39.580
on the left.

00:01:39.580 --> 00:01:41.330
And the same is true of
the left hand side.

00:01:41.330 --> 00:01:44.130
And you go all the way to the
point that the coil is

00:01:44.130 --> 00:01:45.560
actually vertical.

00:01:45.560 --> 00:01:49.580
Where this side, this side
right here, is on top.

00:01:49.580 --> 00:01:53.170
And this side is on bottom,
below the plane

00:01:53.170 --> 00:01:53.950
of your video screen.

00:01:53.950 --> 00:01:59.320
And at that point, the torque--
actually, there is no

00:01:59.320 --> 00:01:59.890
net torque.

00:01:59.890 --> 00:02:00.540
And why is that?

00:02:00.540 --> 00:02:03.810
Because on this top part, when
it's pointing straight out at

00:02:03.810 --> 00:02:09.060
you, when it's right here, the
magnetic field-- the force of

00:02:09.060 --> 00:02:13.270
it, the force that's affecting
the circuit-- is pushing

00:02:13.270 --> 00:02:14.160
straight up.

00:02:14.160 --> 00:02:17.080
So there's no longer any net
torque because the force is

00:02:17.080 --> 00:02:19.720
pushing straight up and that
moment arm distance-- this

00:02:19.720 --> 00:02:22.460
distance-- is now also
pointing straight up.

00:02:22.460 --> 00:02:25.700
And torque is also a cross
product, so you actually care

00:02:25.700 --> 00:02:28.080
about the perpendicular
forces.

00:02:28.080 --> 00:02:30.890
So there, at this vertical
point, there's no net torque.

00:02:30.890 --> 00:02:33.340
And the same is true at the
bottom of the circuit.

00:02:33.340 --> 00:02:36.160
Because at the bottom the
magnetic field force is going

00:02:36.160 --> 00:02:38.940
to be downwards, which is
parallel with the moment arm

00:02:38.940 --> 00:02:40.630
distance, so there's
no net torque.

00:02:40.630 --> 00:02:45.240
And I said, well maybe there's
a little bit of angular

00:02:45.240 --> 00:02:49.590
momentum that keeps this
object rotating.

00:02:49.590 --> 00:02:52.590
And then it will rotate to--
and this is where it gets

00:02:52.590 --> 00:02:53.360
interesting.

00:02:53.360 --> 00:02:55.810
I'll draw it neatly.

00:02:55.810 --> 00:02:58.900
Then it'll rotate
to this point.

00:02:58.900 --> 00:03:00.340
Once again I want to have
the perspective.

00:03:08.666 --> 00:03:10.200
It'll rotate here.

00:03:10.200 --> 00:03:12.820
So let me just make sure
I have all of it.

00:03:12.820 --> 00:03:14.980
So here it was rotating in this

00:03:14.980 --> 00:03:20.060
direction and in that direction.

00:03:20.060 --> 00:03:22.860
And then here maybe some--
there's no longer any torque

00:03:22.860 --> 00:03:26.070
on it, but it still might on the
top be moving to the left,

00:03:26.070 --> 00:03:28.420
and on the bottom moving
to the right.

00:03:28.420 --> 00:03:31.230
Up to a point, then it's going
to get into this configuration

00:03:31.230 --> 00:03:38.160
where soon. this side is-- so
at this point it has rotated

00:03:38.160 --> 00:03:40.310
more than 90 degrees.

00:03:40.310 --> 00:03:42.970
So this edge is now this edge.

00:03:42.970 --> 00:03:46.110
It had rotated from here all the
way-- it's still pointing

00:03:46.110 --> 00:03:47.440
out of the screen.

00:03:47.440 --> 00:03:51.130
But if this edge is the same as
this edge, now the current

00:03:51.130 --> 00:03:53.080
direction is going
to be like this.

00:03:53.080 --> 00:03:56.180
Because this edge has
rotated down.

00:03:56.180 --> 00:03:59.440
So it's rotated from that
position all the way back to

00:03:59.440 --> 00:04:00.390
this position.

00:04:00.390 --> 00:04:02.920
So the current is now coming--
let me make sure, let me draw

00:04:02.920 --> 00:04:04.040
that right.

00:04:04.040 --> 00:04:08.050
The current is coming like that,
like that, like that.

00:04:08.050 --> 00:04:13.220
Going up here, to the
right, up like that.

00:04:13.220 --> 00:04:15.610
So the current now on this left
hand side, although it

00:04:15.610 --> 00:04:17.125
was the former right
hand side.

00:04:17.125 --> 00:04:20.380
It's still going in that
upwards direction.

00:04:20.380 --> 00:04:22.800
So when you take the cross
product, what is going to be

00:04:22.800 --> 00:04:25.230
the net magnetic
field on that?

00:04:25.230 --> 00:04:27.660
Or the force of the
magnetic field?

00:04:27.660 --> 00:04:31.050
Well, you do the same
right hand rule.

00:04:31.050 --> 00:04:32.730
Point your index finger up.

00:04:32.730 --> 00:04:34.170
Put your middle finger
in the direction of

00:04:34.170 --> 00:04:35.680
the magnetic field.

00:04:35.680 --> 00:04:38.120
This is the palm, this is
your other two fingers.

00:04:38.120 --> 00:04:40.300
Let me draw the fingernails,
just so they're painted

00:04:40.300 --> 00:04:40.890
fingernails.

00:04:40.890 --> 00:04:42.460
Not that mine are.

00:04:42.460 --> 00:04:44.620
Then your thumb points
upwards.

00:04:44.620 --> 00:04:51.370
So on this side of the coil we
still have an upwards force.

00:04:51.370 --> 00:04:53.350
And if you do the cross product,
or you do the right

00:04:53.350 --> 00:04:56.790
hand rule on the bottom side,
or the behind side, if you

00:04:56.790 --> 00:04:59.450
could imagine it, you're
still going to have a

00:04:59.450 --> 00:05:00.730
net downward force.

00:05:00.730 --> 00:05:02.470
So now all of a sudden
you could imagine--

00:05:02.470 --> 00:05:04.580
the thing had rotated.

00:05:04.580 --> 00:05:09.020
So it had rotated in the way I
drew it here, where it pops

00:05:09.020 --> 00:05:12.100
out on this side and it
goes in on that side.

00:05:12.100 --> 00:05:14.760
And it had done it all the way
to the point where we had

00:05:14.760 --> 00:05:17.800
rotated more than 90 degrees,
but now all of a sudden the

00:05:17.800 --> 00:05:21.560
net force through the magnetic
field was going to reverse.

00:05:21.560 --> 00:05:24.610
Because the side that has a
current going upwards is now

00:05:24.610 --> 00:05:26.360
the left hand side.

00:05:26.360 --> 00:05:28.830
So now the force from the
magnetic field is out on this

00:05:28.830 --> 00:05:31.660
side and you're going
to want to rotate in

00:05:31.660 --> 00:05:34.160
the opposite direction.

00:05:34.160 --> 00:05:34.970
Hopefully that makes sense.

00:05:34.970 --> 00:05:35.850
Just think about what happens.

00:05:35.850 --> 00:05:38.380
Visualize this coil rotating.

00:05:38.380 --> 00:05:42.190
So what is essentially going to
happen is you're going to

00:05:42.190 --> 00:05:46.610
rotate like I did
here on the top.

00:05:46.610 --> 00:05:48.910
Maybe once you get to this level
you're going to have a

00:05:48.910 --> 00:05:52.100
little bit of angular momentum
that'll keep you rotating.

00:05:52.100 --> 00:05:55.020
Or rotational inertia that'll
keep you rotating until you're

00:05:55.020 --> 00:05:56.510
in something like this
configuration.

00:05:56.510 --> 00:06:00.220
Maybe you go all the way back
to this configuration, where

00:06:00.220 --> 00:06:04.580
it's essentially a complete
180 degree turn.

00:06:04.580 --> 00:06:08.260
And then since on this side
the current's going to be

00:06:08.260 --> 00:06:10.360
going up and on this side the
current's going down, because

00:06:10.360 --> 00:06:14.060
you've essentially flipped
this thing over, then the

00:06:14.060 --> 00:06:17.120
effect of the magnetic field is
going to say, well, upwards

00:06:17.120 --> 00:06:19.540
on the left, downwards
on the right.

00:06:19.540 --> 00:06:20.900
And so it's going to
turn the other way.

00:06:20.900 --> 00:06:23.920
So if you think about it, it's
going to keep oscillating.

00:06:23.920 --> 00:06:27.490
Let me draw it from-- well, I
don't want to draw it from

00:06:27.490 --> 00:06:28.830
that angle, because I don't
want to confuse you.

00:06:28.830 --> 00:06:30.630
So we have a problem.

00:06:30.630 --> 00:06:33.810
If we wanted to turn this into
some type of electric motor

00:06:33.810 --> 00:06:37.840
and keep it spinning, we would
either have to reverse the

00:06:37.840 --> 00:06:41.330
current once you get into this
configuration, or either turn

00:06:41.330 --> 00:06:42.380
off the magnetic field.

00:06:42.380 --> 00:06:46.630
Or maybe you could reverse the
magnetic field to get it going

00:06:46.630 --> 00:06:47.510
in the other direction.

00:06:47.510 --> 00:06:49.540
And actually you have another
problem, which is a slightly

00:06:49.540 --> 00:06:52.730
lesser problem, is if this was
a circuit and you just kept

00:06:52.730 --> 00:06:55.770
turning over and over the
circuit, the wires would get

00:06:55.770 --> 00:06:56.280
twisted here.

00:06:56.280 --> 00:06:58.430
So you couldn't do
it indefinitely.

00:06:58.430 --> 00:07:03.373
So the solution here is
something called a commutator.

00:07:03.373 --> 00:07:05.630
you So let me draw
a commutator.

00:07:10.720 --> 00:07:19.250
I have the same circuit which
I've now drawn messier.

00:07:23.140 --> 00:07:24.430
But it has these two leads.

00:07:27.120 --> 00:07:29.210
It has these leads that
essentially curve.

00:07:29.210 --> 00:07:31.960
You could imagine them curving
out of the page.

00:07:31.960 --> 00:07:35.040
And then we have a circuit.

00:07:35.040 --> 00:07:36.720
You could imagine
leads here, too.

00:07:36.720 --> 00:07:41.030
And this round thing and this
thing are touching each other

00:07:41.030 --> 00:07:43.150
the whole time, so current
could pass through it.

00:07:47.000 --> 00:07:50.800
Let me draw my battery.

00:07:50.800 --> 00:07:52.710
This is positive and
this is negative.

00:07:52.710 --> 00:07:55.140
So up here on the circuit the
current's always going to be

00:07:55.140 --> 00:07:56.440
flowing in this direction.

00:07:56.440 --> 00:07:58.600
It's always going to be flowing
in this direction,

00:07:58.600 --> 00:08:02.460
it's always going to be flowing
up and like this.

00:08:02.460 --> 00:08:04.150
Now when you're in this
configuration,

00:08:04.150 --> 00:08:05.720
what's going to happen?

00:08:05.720 --> 00:08:08.860
Well, the current is going
to flow down here.

00:08:08.860 --> 00:08:12.750
That's going to be I and
that's going to be I.

00:08:12.750 --> 00:08:15.720
And when you do your right hand
rule, we have the same

00:08:15.720 --> 00:08:16.320
magnetic field.

00:08:16.320 --> 00:08:19.280
I haven't changed the magnetic
field coming in from the left.

00:08:21.890 --> 00:08:24.700
So just like we did before I
cleared the screen, you use

00:08:24.700 --> 00:08:26.590
the right hand rule and you'll
figure out, well, the net

00:08:26.590 --> 00:08:32.090
force from the magnetic field
is going to be upwards here

00:08:32.090 --> 00:08:34.429
and downwards here.

00:08:34.429 --> 00:08:36.250
And that's what's going to
create that net torque.

00:08:36.250 --> 00:08:39.429
And you're going to
rotate this part.

00:08:39.429 --> 00:08:41.220
So this part of this

00:08:41.220 --> 00:08:43.260
contraption is going to rotate.

00:08:43.260 --> 00:08:44.510
You could imagine maybe
there's like

00:08:44.510 --> 00:08:45.870
a little pole here.

00:08:45.870 --> 00:08:48.910
Maybe it's a nonconducting pole
so that none of the-- and

00:08:48.910 --> 00:08:50.940
it's connected to an
axle somewhere.

00:08:50.940 --> 00:08:53.980
So you can rotate along
that axis, right?

00:08:53.980 --> 00:08:55.895
So the force of the
magnetic field is

00:08:55.895 --> 00:08:56.430
going to create a torque.

00:08:56.430 --> 00:09:00.930
We're going to rotate up on this
side, up out of the page

00:09:00.930 --> 00:09:03.030
on that side, and into the
page on that side.

00:09:03.030 --> 00:09:07.530
And then behind the page and
then back out of the page.

00:09:07.530 --> 00:09:09.620
That's what the net
torque would be.

00:09:09.620 --> 00:09:12.660
And then we would get it, and it
would keep doing that until

00:09:12.660 --> 00:09:14.860
you get to the vertical
configuration.

00:09:19.730 --> 00:09:22.680
So at the vertical
configuration, the circuit on

00:09:22.680 --> 00:09:24.800
the top stays exactly
the same.

00:09:29.440 --> 00:09:31.850
I'm trying my best to
draw this properly.

00:09:31.850 --> 00:09:34.190
At the vertical configuration
one of two things can happen,

00:09:34.190 --> 00:09:37.590
and probably the best thing is
that we actually lose contact

00:09:37.590 --> 00:09:38.820
with the two leads.

00:09:38.820 --> 00:09:42.600
So maybe the actual current
stops flowing when we're in

00:09:42.600 --> 00:09:43.790
the vertical configuration.

00:09:43.790 --> 00:09:44.640
I'll do it in the same color.

00:09:44.640 --> 00:09:48.790
So when we're vertical
we just see the top.

00:09:48.790 --> 00:09:50.070
We see this.

00:09:50.070 --> 00:09:51.960
And then we see it pops
out a little bit.

00:09:51.960 --> 00:09:55.030
And then we see this
arm right there.

00:09:55.030 --> 00:09:57.560
And then we see that pole that's
maybe holding it or

00:09:57.560 --> 00:10:00.600
that's helping it rotate.

00:10:00.600 --> 00:10:02.260
But we're still having
some-- you know,

00:10:02.260 --> 00:10:03.270
the current has ceased.

00:10:03.270 --> 00:10:05.580
So there's not going to be any
torque, no force through the

00:10:05.580 --> 00:10:09.070
magnetic field, because we've
lost touch at that point.

00:10:09.070 --> 00:10:10.630
Because these things
kind of point out.

00:10:10.630 --> 00:10:14.920
Hopefully you could visualize
how to build such a thing.

00:10:14.920 --> 00:10:17.920
And we're still rotating in this
direction because of some

00:10:17.920 --> 00:10:20.810
type of rotational inertia.

00:10:20.810 --> 00:10:23.350
Then this is what the
interesting part is.

00:10:23.350 --> 00:10:27.140
What happens when we rotate
more than 90 degrees?

00:10:27.140 --> 00:10:29.555
And I just realized that I'm
pushing over 10 minutes, so

00:10:29.555 --> 00:10:32.860
you can think about that a
little bit while I stop here

00:10:32.860 --> 00:10:34.610
and continue this in
the next video.

00:10:34.610 --> 00:10:35.860
See