Magnetic force on a proton example (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:04.370
In the last video we figured
out that if we had a proton

00:00:04.370 --> 00:00:07.990
coming into the right at a
velocity of 6 times 10 to the

00:00:07.990 --> 00:00:09.470
seventh meters per second.

00:00:09.470 --> 00:00:11.360
So the magnitude of the
velocity is 1/5

00:00:11.360 --> 00:00:12.490
the speed of light.

00:00:12.490 --> 00:00:16.740
And if it were to cross this
magnetic field, we used this

00:00:16.740 --> 00:00:21.560
formula to figure out that the
magnitude of the force on this

00:00:21.560 --> 00:00:26.650
proton would be 4.8 times 10
to the negative 12 newtons.

00:00:26.650 --> 00:00:28.600
And then the direction, we
used our right hand rule

00:00:28.600 --> 00:00:29.990
because it's a cross product.

00:00:29.990 --> 00:00:33.180
And we figured out that it would
be perpendicular-- well,

00:00:33.180 --> 00:00:34.820
it has to be perpendicular to
both, because we're taking the

00:00:34.820 --> 00:00:38.470
cross product-- and right when
it enters, the net force will

00:00:38.470 --> 00:00:39.500
be downwards.

00:00:39.500 --> 00:00:41.020
But then think about
what happens.

00:00:41.020 --> 00:00:43.830
If you have a downward force
right there, then the particle

00:00:43.830 --> 00:00:46.950
will be deflected downward a
little bit, so its velocity

00:00:46.950 --> 00:00:48.900
vector will then look
something like that.

00:00:48.900 --> 00:00:50.800
But it's still in the magnetic
field, right?

00:00:50.800 --> 00:00:53.090
And not only is it still in the
magnetic field, but since

00:00:53.090 --> 00:00:57.470
the particle is still moving
within the plane of your video

00:00:57.470 --> 00:01:00.145
screen, it's still completely
perpendicular to

00:01:00.145 --> 00:01:01.130
the magnetic field.

00:01:01.130 --> 00:01:06.660
And so the magnitude of the
force on the moving particle

00:01:06.660 --> 00:01:08.980
won't change, just the
direction will.

00:01:08.980 --> 00:01:11.430
Because if we do the right hand
rule here, but if we just

00:01:11.430 --> 00:01:14.260
move our hand down a little bit,
if we tilt it down, then

00:01:14.260 --> 00:01:17.080
our thumb's going to be pointing
in this direction.

00:01:17.080 --> 00:01:18.200
And that just keeps happening.

00:01:18.200 --> 00:01:19.590
It gets deflected that
way a little bit.

00:01:19.590 --> 00:01:22.560
So the magnitude of the velocity
doesn't ever change.

00:01:22.560 --> 00:01:26.670
It always stays perpendicular to
the magnetic field because

00:01:26.670 --> 00:01:28.230
it's always staying
in this plane.

00:01:28.230 --> 00:01:30.510
But the orientation does change
within the plane.

00:01:30.510 --> 00:01:32.280
And because of that, because
the orientation of the

00:01:32.280 --> 00:01:34.130
velocity changes,
the orientation

00:01:34.130 --> 00:01:35.400
of the force changes.

00:01:35.400 --> 00:01:37.550
So when the velocity is here,
the force is perpendicular.

00:01:37.550 --> 00:01:41.100
So it acts as kind of a
centripetal force, and so the

00:01:41.100 --> 00:01:44.340
particle will start moving
in a circle.

00:01:44.340 --> 00:01:47.290
So let's see if we can break
out our toolkit from what

00:01:47.290 --> 00:01:49.740
we've learned before in
classical mechanics, and

00:01:49.740 --> 00:01:53.940
figure out what the radius
of that circle is.

00:01:53.940 --> 00:01:56.110
And that might seem more
daunting than it really is.

00:01:56.110 --> 00:01:58.530
Well, what do we know about
centripetal forces and

00:01:58.530 --> 00:02:00.140
radiuses of circles,
et cetera?

00:02:19.880 --> 00:02:22.250
So, what is the formula
for centripetal force?

00:02:22.250 --> 00:02:24.750
And we proved it many, many
videos ago, early in the

00:02:24.750 --> 00:02:29.090
physics playlist. Well,
centripetal acceleration is

00:02:29.090 --> 00:02:34.100
the magnitude of the velocity
vector squared over the radius

00:02:34.100 --> 00:02:35.630
of the circle.

00:02:35.630 --> 00:02:37.330
And since this is acceleration,
if we want to

00:02:37.330 --> 00:02:39.920
know the centripetal force,
it's just the mass times

00:02:39.920 --> 00:02:40.590
acceleration.

00:02:40.590 --> 00:02:43.250
So it's the mass of the
particle, or the object in

00:02:43.250 --> 00:02:46.270
question, times the magnitude
of its velocity squared

00:02:46.270 --> 00:02:47.740
divided by the radius
of the circle.

00:02:47.740 --> 00:02:50.452
In this case, this is the
radius of the circle.

00:02:50.452 --> 00:02:53.270
And that's what we're going
to try to solve for.

00:02:53.270 --> 00:02:54.940
And what do we know about
the centripetal force?

00:02:54.940 --> 00:02:56.550
What is causing the
centripetal force?

00:02:56.550 --> 00:02:59.930
Well, it's the magnetic field
and we've figured that out.

00:02:59.930 --> 00:03:03.460
This is going to be equal to
this, which we figured out is

00:03:03.460 --> 00:03:06.770
going to be equal to-- at least
the magnitudes-- the

00:03:06.770 --> 00:03:09.060
magnitude of this is equal
to the magnitude of this.

00:03:09.060 --> 00:03:15.790
And that magnitude is
4.8 times 10 to

00:03:15.790 --> 00:03:20.150
the minus 12 newtons.

00:03:20.150 --> 00:03:25.290
And so the radius is going to
be-- let's see, if we flip

00:03:25.290 --> 00:03:33.510
both sides of this equation,
we get radius over mass

00:03:33.510 --> 00:03:38.580
velocity squared is equal
to 1 over 4.8 times 10

00:03:38.580 --> 00:03:40.460
to the minus 12.

00:03:40.460 --> 00:03:42.210
I could just figure out what
that number is, but I won't

00:03:42.210 --> 00:03:42.830
worry about that now.

00:03:42.830 --> 00:03:49.330
Then we can multiply both sides
times this mv squared.

00:03:49.330 --> 00:03:52.840
And we get that the radius of
the circle is going to be

00:03:52.840 --> 00:03:58.030
equal to the mass of the proton
times the magnitude of

00:03:58.030 --> 00:04:06.910
its velocity squared divided
by the force from

00:04:06.910 --> 00:04:07.930
the magnetic field.

00:04:07.930 --> 00:04:09.610
The centripetal force.

00:04:09.610 --> 00:04:13.830
4.8 times 10 to the
minus 12 newtons.

00:04:13.830 --> 00:04:15.830
And the radius should be in
meters, since everything is

00:04:15.830 --> 00:04:17.300
kind of in the standard
SI units.

00:04:17.300 --> 00:04:21.149
And let's see if we can
figure this out.

00:04:21.149 --> 00:04:22.530
Get our calculator.

00:04:22.530 --> 00:04:25.470
And this is where that constant
function is useful

00:04:25.470 --> 00:04:28.090
again, because what is
the mass of a proton?

00:04:28.090 --> 00:04:29.210
Well, that's something that I

00:04:29.210 --> 00:04:31.020
personally don't have memorized.

00:04:31.020 --> 00:04:33.090
But if we go into the built-in
constants on the

00:04:33.090 --> 00:04:36.850
TI-85-- let's see more.

00:04:36.850 --> 00:04:37.800
Mass of a proton.

00:04:37.800 --> 00:04:41.920
This is mass of an electron.

00:04:41.920 --> 00:04:43.170
This is mass of a proton.

00:04:43.170 --> 00:04:47.420
So mass of a proton-- that's
what we care about-- times the

00:04:47.420 --> 00:04:48.740
magnitude of the velocity
squared.

00:04:48.740 --> 00:04:49.500
What was the velocity?

00:04:49.500 --> 00:04:53.440
It was 6 times 10 to the seventh
meters per second.

00:04:53.440 --> 00:05:00.220
So times 6 times 10
to the seventh

00:05:00.220 --> 00:05:03.050
meters per second squared.

00:05:06.120 --> 00:05:12.380
And all of that divided
by the magnitude of

00:05:12.380 --> 00:05:13.750
the centripetal force.

00:05:13.750 --> 00:05:16.040
Which is the force that's
being generated by the

00:05:16.040 --> 00:05:16.520
magnetic field.

00:05:16.520 --> 00:05:19.590
That's 4.8 times 10 to
the negative 12.

00:05:19.590 --> 00:05:26.380
Divided by 4.8 E minus 12.

00:05:26.380 --> 00:05:26.940
Let's see.

00:05:26.940 --> 00:05:29.340
Hopefully we don't get
something funky.

00:05:29.340 --> 00:05:30.330
There we go.

00:05:30.330 --> 00:05:31.900
That's actually a pretty
neat number.

00:05:31.900 --> 00:05:34.510
1.25 meters.

00:05:34.510 --> 00:05:38.630
That's actually a number
that we can imagine.

00:05:38.630 --> 00:05:46.520
So if you have a proton going
in this direction at 1/5 the

00:05:46.520 --> 00:05:49.140
speed of light through a--
what'd I say it was?

00:05:49.140 --> 00:05:52.830
It was a 0.5 tesla magnetic
field, where the vectors are

00:05:52.830 --> 00:05:55.220
pointing out of the video.

00:05:55.220 --> 00:05:59.510
We have just shown that this
proton will go in a circle of

00:05:59.510 --> 00:06:03.210
radius 1.25 meters.

00:06:03.210 --> 00:06:04.820
Which is neat because
it's a number that I

00:06:04.820 --> 00:06:06.520
can actually visualize.

00:06:06.520 --> 00:06:09.940
And so this whole business of
magnetic fields making charged

00:06:09.940 --> 00:06:12.820
particles go into circles, this
is one of the few times

00:06:12.820 --> 00:06:16.100
that I can actually say has a
direct application into things

00:06:16.100 --> 00:06:17.040
that you've seen.

00:06:17.040 --> 00:06:18.510
Namely, your TV.

00:06:18.510 --> 00:06:22.790
Or at least the old-school TVs.
The non-plasma or LCD

00:06:22.790 --> 00:06:26.410
TVs, your cathode ray TVs,
take advantage of this.

00:06:26.410 --> 00:06:27.760
Where you essentially
have a beam of

00:06:27.760 --> 00:06:30.015
not protons but electrons.

00:06:30.015 --> 00:06:32.570
And a magnet-- if you take
apart a TV, which I don't

00:06:32.570 --> 00:06:34.170
think you should do, because
you're more likely to hurt

00:06:34.170 --> 00:06:35.730
yourself because there's a
vacuum in there that can

00:06:35.730 --> 00:06:38.280
implode, and all that-- but
essentially, you have a magnet

00:06:38.280 --> 00:06:42.700
that deflects this electron beam
and does it really fast

00:06:42.700 --> 00:06:45.930
so it scans your entire screen
of different intensities, and

00:06:45.930 --> 00:06:46.980
that's what forms the image.

00:06:46.980 --> 00:06:48.000
I won't go into that detail.

00:06:48.000 --> 00:06:51.060
Maybe one day I'll do a whole
video on how TVs work.

00:06:51.060 --> 00:06:55.090
So that's one application of a
magnetic field causing a beam

00:06:55.090 --> 00:06:57.210
of charged particles to curve.

00:06:57.210 --> 00:06:58.670
And then the other application,
and this is

00:06:58.670 --> 00:07:00.600
actually one where it's actually
useful to make the

00:07:00.600 --> 00:07:03.550
particle go in a circle, is
these cyclotrons that you read

00:07:03.550 --> 00:07:06.200
about, where they take these
protons and they make them go

00:07:06.200 --> 00:07:07.700
in circles really, really
fast, and then

00:07:07.700 --> 00:07:08.470
they smash them together.

00:07:08.470 --> 00:07:10.990
Well, have you ever wondered,
how do they even make a proton

00:07:10.990 --> 00:07:12.090
go in a circle?

00:07:12.090 --> 00:07:14.440
It's not like you could
hold it and guide

00:07:14.440 --> 00:07:15.300
it around in a circle.

00:07:15.300 --> 00:07:15.770
Well, that's what they do.

00:07:15.770 --> 00:07:21.120
They pass it through an
appropriate strength magnetic

00:07:21.120 --> 00:07:23.620
field, and it curves the path
of the proton so that it can

00:07:23.620 --> 00:07:26.610
keep going through the same
field over and over again.

00:07:26.610 --> 00:07:31.670
And then they can actually use
those electric fields.

00:07:31.670 --> 00:07:33.610
I don't claim to have any
expertise in this, but then

00:07:33.610 --> 00:07:35.620
they can keep speeding it up
using the same devices,

00:07:35.620 --> 00:07:36.750
because it keeps passing
through the

00:07:36.750 --> 00:07:41.440
same part of the collider.

00:07:41.440 --> 00:07:42.180
And then once it collides,
you've

00:07:42.180 --> 00:07:43.430
probably seen those pictures.

00:07:48.090 --> 00:07:49.860
You know, that you spend
billions of dollars on

00:07:49.860 --> 00:07:52.020
supercolliders, and you end
up with these pictures.

00:07:52.020 --> 00:07:54.620
And somehow these physicists
are able to take these

00:07:54.620 --> 00:07:57.470
pictures and say, oh, this is
some new particle because of

00:07:57.470 --> 00:07:58.580
the way it moved.

00:07:58.580 --> 00:08:00.980
Well, what they're actually
talking about is these are

00:08:00.980 --> 00:08:02.700
moving at relativistic speeds.

00:08:02.700 --> 00:08:05.580
And since they're at
relativistic speeds as they

00:08:05.580 --> 00:08:08.430
move at different velocities,
their mass is

00:08:08.430 --> 00:08:09.050
changing, and all that.

00:08:09.050 --> 00:08:12.770
But the basic idea is what
we just learned.

00:08:12.770 --> 00:08:14.110
They move in circles.

00:08:14.110 --> 00:08:15.680
They move in circles because
they're going through a

00:08:15.680 --> 00:08:16.540
magnetic field.

00:08:16.540 --> 00:08:19.180
But their radiuses are different
because their

00:08:19.180 --> 00:08:23.010
charges and their velocities
are going to be different.

00:08:23.010 --> 00:08:24.917
And actually some will move
to the left and some

00:08:24.917 --> 00:08:25.640
will move to the right.

00:08:25.640 --> 00:08:28.480
And that might be because
they're positive or negative,

00:08:28.480 --> 00:08:30.570
and then the radius will be
dependent on their masses.

00:08:30.570 --> 00:08:31.810
Anyway, I don't want
to confuse you.

00:08:31.810 --> 00:08:33.559
But I just wanted to show you
that we actually are touching

00:08:33.559 --> 00:08:36.650
on some physics that
a physicist would

00:08:36.650 --> 00:08:38.309
actually care about.

00:08:38.309 --> 00:08:42.377
Now with that said, what would
have happened if this wasn't a

00:08:42.377 --> 00:08:45.980
proton but if this was an
electron moving at this

00:08:45.980 --> 00:08:49.360
velocity at 6 times 10 to the
seventh meters per second

00:08:49.360 --> 00:08:53.260
through a 0.5 tesla
magnetic field

00:08:53.260 --> 00:08:54.240
popping out of this video.

00:08:54.240 --> 00:08:56.260
What would have happened?

00:08:56.260 --> 00:08:58.360
Well, this formula would
have still been safe.

00:08:58.360 --> 00:09:01.335
The magnitude of the force is
the charge-- but it wouldn't

00:09:01.335 --> 00:09:02.630
have been the charge of a
proton, it would have been the

00:09:02.630 --> 00:09:06.040
charge of an electron, times 6
times 10 to the seventh meters

00:09:06.040 --> 00:09:08.490
per second times 0.5 teslas.

00:09:08.490 --> 00:09:10.210
So what's the difference between
the charge of a proton

00:09:10.210 --> 00:09:11.820
and the charge of an electron?

00:09:11.820 --> 00:09:13.840
Well, the charge of an
electron is negative.

00:09:13.840 --> 00:09:17.040
So if this was an electron,
then the net force would

00:09:17.040 --> 00:09:19.510
actually end up being
a negative number.

00:09:19.510 --> 00:09:21.030
So what does that mean?

00:09:21.030 --> 00:09:23.260
Well, when we used the right
hand rule with the proton

00:09:23.260 --> 00:09:26.800
example, we said that the-- at
least when the proton is

00:09:26.800 --> 00:09:28.110
moving in this direction--
that the net

00:09:28.110 --> 00:09:30.690
force would be downwards.

00:09:30.690 --> 00:09:33.880
But now, all of a sudden, if we
reverse the charge, if we

00:09:33.880 --> 00:09:35.850
say we have a negative charge--
the same magnitude

00:09:35.850 --> 00:09:36.980
but it's negative,
because it's an

00:09:36.980 --> 00:09:39.410
electron-- what happens?

00:09:39.410 --> 00:09:41.910
The force is now in this
direction, using the right

00:09:41.910 --> 00:09:43.880
hand rule, but it
is a negative.

00:09:43.880 --> 00:09:46.530
So really it's going to be a
positive force of the same

00:09:46.530 --> 00:09:48.640
magnitude in this direction.

00:09:48.640 --> 00:09:50.700
So if we have a proton,
it'll go in a

00:09:50.700 --> 00:09:51.810
circle in this direction.

00:09:51.810 --> 00:09:53.610
It'll go like this.

00:09:53.610 --> 00:09:55.820
But if we have an electron,
it'll go in a circle of the

00:09:55.820 --> 00:09:57.810
other direction.

00:09:57.810 --> 00:09:59.110
Now let me ask a question.

00:09:59.110 --> 00:10:01.520
Is that circle going
to be a tighter

00:10:01.520 --> 00:10:05.870
circle or a wider circle?

00:10:05.870 --> 00:10:08.843
Well, the mass of an electron is
a lot smaller than the mass

00:10:08.843 --> 00:10:09.830
of a proton.

00:10:09.830 --> 00:10:11.680
And we had the radius is equal
to the mass times the velocity

00:10:11.680 --> 00:10:13.990
squared divided by the
centripetal force.

00:10:13.990 --> 00:10:15.625
So this mass is smaller
and the radius

00:10:15.625 --> 00:10:16.460
is going to be smaller.

00:10:16.460 --> 00:10:21.370
So the electron's path would
actually move up and it would

00:10:21.370 --> 00:10:22.990
be a smaller radius.

00:10:22.990 --> 00:10:27.570
Actually proportional to the
difference in their radiuses

00:10:27.570 --> 00:10:29.960
is the difference in their
masses, actually.

00:10:29.960 --> 00:10:31.650
But that would be the path
of the electron.

00:10:31.650 --> 00:10:34.920
Anyway, I thought you'd be
interested in that, as well.

00:10:34.920 --> 00:10:36.370
I have run out of time.

00:10:36.370 --> 00:10:38.800
I will see you in
the next video.


Introduction to magnetism

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:02.860
We've learned a little
bit about gravity.

00:00:02.860 --> 00:00:05.120
We've learned a little bit
about electrostatic.

00:00:05.120 --> 00:00:09.280
So, time to learn about
a new fundamental

00:00:09.280 --> 00:00:10.300
force of the universe.

00:00:10.300 --> 00:00:13.180
And this one is probably second
most familiar to us,

00:00:13.180 --> 00:00:13.890
next to gravity.

00:00:13.890 --> 00:00:15.140
And that's magnetism.

00:00:17.310 --> 00:00:18.240
Where does the word come from?

00:00:18.240 --> 00:00:21.810
Well, I think several
civilizations-- I'm no

00:00:21.810 --> 00:00:29.810
historian-- found these
lodestones, these objects that

00:00:29.810 --> 00:00:33.260
would attract other objects
like it, other magnets.

00:00:33.260 --> 00:00:37.720
Or would even attract metallic
objects like iron.

00:00:37.720 --> 00:00:39.260
Ferrous objects.

00:00:39.260 --> 00:00:41.320
And they're called lodestones.

00:00:41.320 --> 00:00:43.150
That's, I guess, the Western
term for it.

00:00:43.150 --> 00:00:45.800
And the reason why they're
called magnets is because

00:00:45.800 --> 00:00:50.590
they're named after lodestones
that were found near the Greek

00:00:50.590 --> 00:00:51.840
province of Magnesia.

00:00:56.800 --> 00:00:58.330
And I actually think the people
who lived there were

00:00:58.330 --> 00:01:00.160
called Magnetes.

00:01:00.160 --> 00:01:03.790
But anyway, you could Wikipedia
that and learn more

00:01:03.790 --> 00:01:05.459
about it than I know.

00:01:05.459 --> 00:01:08.090
But anyway let's focus
on what magnetism is.

00:01:08.090 --> 00:01:12.010
And I think most of us have at
least a working knowledge of

00:01:12.010 --> 00:01:15.490
what it is; we've all played
with magnets and we've dealt

00:01:15.490 --> 00:01:16.470
with compasses.

00:01:16.470 --> 00:01:20.910
But I'll tell you this right
now, what it really is, is

00:01:20.910 --> 00:01:22.250
pretty deep.

00:01:22.250 --> 00:01:25.740
And I think it's fairly-- I
don't think anyone has-- we

00:01:25.740 --> 00:01:28.940
can mathematically understand
it and manipulate it and see

00:01:28.940 --> 00:01:31.410
how it relates to electricity.

00:01:31.410 --> 00:01:33.890
We actually will show you the
electrostatic force and the

00:01:33.890 --> 00:01:37.480
magnetic force are actually the
same thing, just viewed

00:01:37.480 --> 00:01:39.090
from different frames
of reference.

00:01:39.090 --> 00:01:40.420
I know that all of
that sounds very

00:01:40.420 --> 00:01:41.560
complicated and all of that.

00:01:41.560 --> 00:01:45.370
But in our classical Newtonian
world we treat them as two

00:01:45.370 --> 00:01:46.040
different forces.

00:01:46.040 --> 00:01:48.830
But what I'm saying is although
we're kind of used to

00:01:48.830 --> 00:01:52.480
a magnet just like we're used to
gravity, just like gravity

00:01:52.480 --> 00:01:54.620
is also fairly mysterious when
you really think about what it

00:01:54.620 --> 00:01:56.530
is, so is magnetism.

00:01:56.530 --> 00:01:59.290
So with that said, let's at
least try to get some working

00:01:59.290 --> 00:02:02.550
knowledge of how we can
deal with magnetism.

00:02:02.550 --> 00:02:07.650
So we're all familiar
with a magnet.

00:02:07.650 --> 00:02:10.610
I didn't want it to be yellow.

00:02:10.610 --> 00:02:12.060
I could make the boundary
yellow.

00:02:12.060 --> 00:02:16.100
No, I didn't want it to
be like that either.

00:02:16.100 --> 00:02:21.360
So if this is a magnet,
we know that a magnet

00:02:21.360 --> 00:02:24.120
always has two poles.

00:02:24.120 --> 00:02:28.330
It has a north pole
and a south pole.

00:02:28.330 --> 00:02:30.450
And these were just labeled
by convention.

00:02:30.450 --> 00:02:33.630
Because when people first
discovered these lodestones,

00:02:33.630 --> 00:02:37.300
or they took a lodestone and
they magnetized a needle with

00:02:37.300 --> 00:02:40.190
that lodestone, and then that
needle they put on a cork in a

00:02:40.190 --> 00:02:44.840
bucket of water, and that needle
would point to the

00:02:44.840 --> 00:02:46.000
Earth's north pole.

00:02:46.000 --> 00:02:48.250
They said, oh, well the side of
the needle that is pointing

00:02:48.250 --> 00:02:50.820
to the Earth's north, let's
call that the north pole.

00:02:50.820 --> 00:02:52.860
And the point of the needle
that's pointing to the south

00:02:52.860 --> 00:02:57.300
pole-- sorry, the point of the
needle that's pointing to the

00:02:57.300 --> 00:02:59.210
Earth's geographic
south, we'll call

00:02:59.210 --> 00:03:00.230
that the south pole.

00:03:00.230 --> 00:03:03.050
Or another way to put it,
if we have a magnet, the

00:03:03.050 --> 00:03:05.790
direction of the magnet or the
side of the magnet that

00:03:05.790 --> 00:03:07.980
orients itself-- if it's allowed
to orient freely

00:03:07.980 --> 00:03:11.330
without friction-- towards our
geographic north, we call that

00:03:11.330 --> 00:03:12.280
the north pole.

00:03:12.280 --> 00:03:14.820
And the other side is
the south pole.

00:03:14.820 --> 00:03:23.710
And this is actually a little
bit-- obviously we call the

00:03:23.710 --> 00:03:26.930
top of the Earth
the north pole.

00:03:26.930 --> 00:03:30.600
You know, this is
the north pole.

00:03:30.600 --> 00:03:34.530
And we call this
the south pole.

00:03:34.530 --> 00:03:38.840
And there's another notion
of magnetic north.

00:03:38.840 --> 00:03:42.380
And that's where-- I guess, you
could kind of say-- that

00:03:42.380 --> 00:03:47.270
is where a compass, the
north point of a

00:03:47.270 --> 00:03:48.690
compass, will point to.

00:03:48.690 --> 00:03:51.730
And actually, magnetic north
moves around because we have

00:03:51.730 --> 00:03:54.150
all of this moving fluid
inside of the earth.

00:03:54.150 --> 00:03:55.215
And a bunch of other
interactions.

00:03:55.215 --> 00:03:56.690
It's a very complex
interaction.

00:03:56.690 --> 00:04:01.130
But magnetic north is actually
roughly in northern Canada.

00:04:01.130 --> 00:04:04.240
So magnetic north
might be here.

00:04:04.240 --> 00:04:05.510
So that might be
magnetic north.

00:04:05.510 --> 00:04:07.420
And magnetic south, I don't know
exactly where that is.

00:04:07.420 --> 00:04:09.340
But it can kind of move
around a little bit.

00:04:09.340 --> 00:04:10.360
It's not in the same place.

00:04:10.360 --> 00:04:14.860
So it's a little bit off the
axis of the geographic north

00:04:14.860 --> 00:04:16.180
pole and the south pole.

00:04:16.180 --> 00:04:19.540
And this is another slightly
confusing thing.

00:04:19.540 --> 00:04:23.720
Magnetic north is the geographic
location, where the

00:04:23.720 --> 00:04:26.970
north pole of a magnet
will point to.

00:04:26.970 --> 00:04:31.730
But that would actually be the
south pole, if you viewed the

00:04:31.730 --> 00:04:32.980
Earth as a magnet.

00:04:35.240 --> 00:04:38.800
So if the Earth was a big
magnet, you would actually

00:04:38.800 --> 00:04:40.820
view that as a south
pole of the magnet.

00:04:40.820 --> 00:04:43.780
And the geographic
south pole is the

00:04:43.780 --> 00:04:44.640
north pole of the magnet.

00:04:44.640 --> 00:04:46.390
You could read more about that
on Wikipedia, I know it's a

00:04:46.390 --> 00:04:47.390
little bit confusing.

00:04:47.390 --> 00:04:49.520
But in general, when most
people refer to magnetic

00:04:49.520 --> 00:04:51.920
north, or the north pole,
they're talking about the

00:04:51.920 --> 00:04:53.930
geographic north area.

00:04:53.930 --> 00:04:56.940
And the south pole is the
geographic south area.

00:04:56.940 --> 00:05:00.170
But the reason why I make this
distinction is because we know

00:05:00.170 --> 00:05:03.520
when we deal with magnets,
just like electricity, or

00:05:03.520 --> 00:05:06.590
electrostatics-- but I'll show
a key difference very

00:05:06.590 --> 00:05:10.760
shortly-- is that opposite
poles attract.

00:05:10.760 --> 00:05:14.770
So if this side of my magnet is
attracted to Earth's north

00:05:14.770 --> 00:05:19.730
pole then Earth's north pole--
or Earth's magnetic north--

00:05:19.730 --> 00:05:22.690
actually must be the south
pole of that magnet.

00:05:22.690 --> 00:05:23.370
And vice versa.

00:05:23.370 --> 00:05:26.110
The south pole of my magnet here
is going to be attracted

00:05:26.110 --> 00:05:28.090
to Earth's magnetic south.

00:05:28.090 --> 00:05:29.770
Which is actually the
north pole of the

00:05:29.770 --> 00:05:31.660
magnet we call Earth.

00:05:31.660 --> 00:05:34.220
Anyway, I'll take Earth out of
the equation because it gets a

00:05:34.220 --> 00:05:35.070
little bit confusing.

00:05:35.070 --> 00:05:37.650
And we'll just stick to bars
because that tends to be a

00:05:37.650 --> 00:05:40.930
little bit more consistent.

00:05:40.930 --> 00:05:44.220
Let me erase this.

00:05:44.220 --> 00:05:45.500
There you go.

00:05:45.500 --> 00:05:46.750
I'll erase my Magnesia.

00:05:49.680 --> 00:05:52.200
I wonder if the element
magnesium was first discovered

00:05:52.200 --> 00:05:53.370
in Magnesia, as well.

00:05:53.370 --> 00:05:55.870
Probably.

00:05:55.870 --> 00:05:57.200
And I actually looked
up Milk of

00:05:57.200 --> 00:05:58.920
Magnesia, which is a laxative.

00:05:58.920 --> 00:06:02.780
And it was not discovered
in Magnesia, but it has

00:06:02.780 --> 00:06:04.780
magnesium in it.

00:06:04.780 --> 00:06:08.060
So I guess its roots could be
in Magnesia if magnesium was

00:06:08.060 --> 00:06:09.410
discovered in Magnesia.

00:06:09.410 --> 00:06:11.530
Anyway, enough about Magnesia.

00:06:11.530 --> 00:06:13.870
Back to the magnets.

00:06:13.870 --> 00:06:18.350
So if this is a magnet, and let
me draw another magnet.

00:06:18.350 --> 00:06:20.700
Actually, let me erase
all of this.

00:06:23.990 --> 00:06:24.935
All right.

00:06:24.935 --> 00:06:26.730
So let me draw two
more magnets.

00:06:32.840 --> 00:06:36.700
We know from experimentation
when we were all kids, this is

00:06:36.700 --> 00:06:38.810
the north pole, this
is the south pole.

00:06:38.810 --> 00:06:43.210
That the north pole is going to
be attracted to the south

00:06:43.210 --> 00:06:45.700
pole of another magnet.

00:06:45.700 --> 00:06:48.490
And that if I were to flip this
magnet around, it would

00:06:48.490 --> 00:06:52.480
actually repel north-- two north
facing magnets would

00:06:52.480 --> 00:06:54.330
repel each other.

00:06:54.330 --> 00:06:56.930
And so we have this notion,
just like we had in

00:06:56.930 --> 00:07:01.120
electrostatics, that a magnet
generates a field.

00:07:01.120 --> 00:07:05.520
It generates these vectors
around it, that if you put

00:07:05.520 --> 00:07:09.840
something in that field that can
be affected by it, it'll

00:07:09.840 --> 00:07:15.630
be some net force
acting on it.

00:07:15.630 --> 00:07:17.850
So actually, before I go into
magnetic field, I actually

00:07:17.850 --> 00:07:24.030
want to make one huge
distinction between magnetism

00:07:24.030 --> 00:07:25.360
and electrostatics.

00:07:25.360 --> 00:07:30.140
Magnetism always comes in
the form of a dipole.

00:07:30.140 --> 00:07:30.910
What does a dipole mean?

00:07:30.910 --> 00:07:32.700
It means that we
have two poles.

00:07:32.700 --> 00:07:34.520
A north and a south.

00:07:34.520 --> 00:07:40.430
In electrostatics, you
do have two charges.

00:07:40.430 --> 00:07:43.480
You have a positive charge
and a negative charge.

00:07:43.480 --> 00:07:44.630
So you do have two charges.

00:07:44.630 --> 00:07:46.530
But they could be
by themselves.

00:07:46.530 --> 00:07:48.230
You could just have a proton.

00:07:48.230 --> 00:07:49.600
You don't have to have
an electron there

00:07:49.600 --> 00:07:50.490
right next to it.

00:07:50.490 --> 00:07:54.620
You could just have a proton and
it would create a positive

00:07:54.620 --> 00:07:56.140
electrostatic field.

00:07:56.140 --> 00:07:57.950
And our field lines are
what a positive point

00:07:57.950 --> 00:07:59.000
charge would do.

00:07:59.000 --> 00:08:00.260
And it would be repelled.

00:08:00.260 --> 00:08:05.340
So you don't always have to have
a negative charge there.

00:08:05.340 --> 00:08:07.250
Similarly you could just
have an electron.

00:08:07.250 --> 00:08:08.690
And you don't have to
have a proton there.

00:08:08.690 --> 00:08:10.460
So you could have monopoles.

00:08:10.460 --> 00:08:12.740
These are called monopoles, when
you just have one charge

00:08:12.740 --> 00:08:14.770
when you're talking about
electrostatics.

00:08:14.770 --> 00:08:17.420
But with magnetism you
always have a dipole.

00:08:17.420 --> 00:08:20.200
If I were to take this magnet,
this one right here, and if I

00:08:20.200 --> 00:08:28.510
were to cut it in half, somehow
miraculously each of

00:08:28.510 --> 00:08:30.150
those halves of that
magnet will turn

00:08:30.150 --> 00:08:31.560
into two more magnets.

00:08:31.560 --> 00:08:34.260
Where this will be the south,
this'll be the north, this'll

00:08:34.260 --> 00:08:36.679
be the south, this will
be the north.

00:08:36.679 --> 00:08:41.200
And actually, theoretically,
I've read-- my own abilities

00:08:41.200 --> 00:08:44.110
don't go this far-- there could
be such a thing as a

00:08:44.110 --> 00:08:46.200
magnetic monopole, although
it has not been

00:08:46.200 --> 00:08:48.320
observed yet in nature.

00:08:48.320 --> 00:08:51.130
So everything we've seen in
nature has been a dipole.

00:08:51.130 --> 00:08:54.260
So you could just keep cutting
this up, all the way down to

00:08:54.260 --> 00:08:56.000
if it's just one
electron left.

00:08:56.000 --> 00:08:59.230
And it actually turns out that
even one electron is still a

00:08:59.230 --> 00:09:00.400
magnetic dipole.

00:09:00.400 --> 00:09:04.400
It still is generating, it still
has a north pole and a

00:09:04.400 --> 00:09:05.310
south pole.

00:09:05.310 --> 00:09:08.830
And actually it turns out, all
magnets, the magnetic field is

00:09:08.830 --> 00:09:12.380
actually generated by the
electrons within it.

00:09:12.380 --> 00:09:14.710
By the spin of electrons and
that-- you know, when we talk

00:09:14.710 --> 00:09:16.970
about electron spin we
imagine some little

00:09:16.970 --> 00:09:18.430
ball of charge spinning.

00:09:18.430 --> 00:09:20.910
But electrons are-- you
know, it's hard to--

00:09:20.910 --> 00:09:21.900
they do have mass.

00:09:21.900 --> 00:09:23.590
But it starts to get
fuzzy whether they

00:09:23.590 --> 00:09:24.720
are energy or mass.

00:09:24.720 --> 00:09:26.990
And then how does a ball
of energy spin?

00:09:26.990 --> 00:09:27.510
Et cetera, et cetera.

00:09:27.510 --> 00:09:31.900
So it gets very almost
metaphysical.

00:09:31.900 --> 00:09:33.220
So I don't want to go
too far into it.

00:09:33.220 --> 00:09:35.710
And frankly, I don't think you
really can get an intuition.

00:09:35.710 --> 00:09:38.080
It is almost-- it is a
realm that we don't

00:09:38.080 --> 00:09:39.790
normally operate in.

00:09:39.790 --> 00:09:43.340
But even these large magnets
you deal with, the magnetic

00:09:43.340 --> 00:09:47.120
field is generated by the
electron spins inside of it

00:09:47.120 --> 00:09:51.040
and by the actual magnetic
fields generated by the

00:09:51.040 --> 00:09:53.620
electron motion around
the protons.

00:09:53.620 --> 00:09:56.350
Well, I hope I'm not
overwhelming you.

00:09:56.350 --> 00:09:58.370
And you might say, well, how
come sometimes a metal bar can

00:09:58.370 --> 00:10:00.700
be magnetized and sometimes
it won't be?

00:10:00.700 --> 00:10:02.940
Well, when all of the electrons
are doing random

00:10:02.940 --> 00:10:06.220
different things in a metal bar,
then it's not magnetized.

00:10:06.220 --> 00:10:10.590
Because the magnetic spins, or
the magnetism created by the

00:10:10.590 --> 00:10:11.920
electrons are all canceling
each other out,

00:10:11.920 --> 00:10:12.790
because it's random.

00:10:12.790 --> 00:10:16.100
But if you align the spins of
the electrons, and if you

00:10:16.100 --> 00:10:20.130
align their rotations, then you
will have a magnetically

00:10:20.130 --> 00:10:21.060
charged bar.

00:10:21.060 --> 00:10:23.620
But anyway, I'm past the
ten-minute mark, but hopefully

00:10:23.620 --> 00:10:26.140
that gives you a little bit
of a working knowledge of

00:10:26.140 --> 00:10:27.190
what a magnet is.

00:10:27.190 --> 00:10:31.170
And in the next video, I will
show what the effect is.

00:10:31.170 --> 00:10:34.490
Well, one, I'll explain how we
think about a magnetic field.

00:10:34.490 --> 00:10:36.100
And then what the effect
of a magnetic

00:10:36.100 --> 00:10:38.870
field is on an electron.

00:10:38.870 --> 00:10:41.140
Or not an electron, on
a moving charge.

00:10:41.140 --> 00:10:43.010
See you in the next video.


Cross product and torque

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.400
In all of the torque problems
I've done so far in the

00:00:03.400 --> 00:00:05.910
physics playlist, we only just
figured out the magnitude of

00:00:05.910 --> 00:00:09.610
torque, frankly because that's
what normally matters, but

00:00:09.610 --> 00:00:13.500
torque is actually
a vector and it's

00:00:13.500 --> 00:00:15.380
direction can be found.

00:00:15.380 --> 00:00:21.160
And that is because torque is
defined as the cross product

00:00:21.160 --> 00:00:26.330
between the radial distance from
your axis of rotation and

00:00:26.330 --> 00:00:29.370
the rotational force
being applied.

00:00:29.370 --> 00:00:30.810
So these are both vectors.

00:00:30.810 --> 00:00:33.350
So let's take a look at how I
taught you vectors the first

00:00:33.350 --> 00:00:36.300
time, and then I'll show you
how that's really the same

00:00:36.300 --> 00:00:38.860
thing as what we're doing here
with the cross product.

00:00:38.860 --> 00:00:41.570
Except now with the cross
product, besides just the

00:00:41.570 --> 00:00:43.590
magnitude for torque, we're also
getting the direction.

00:00:43.590 --> 00:00:45.850
But then we'll also see that
direction is a little bit--

00:00:45.850 --> 00:00:49.080
it's just the definition of
the direction of torque.

00:00:49.080 --> 00:00:51.300
I don't know how intuitive
it really is.

00:00:51.300 --> 00:00:52.950
But what did I teach you
before about torque?

00:00:52.950 --> 00:01:02.410
Well, let's say I had some arm,
and let's say this could

00:01:02.410 --> 00:01:05.269
be the hand of a clock or it's
pinned down to the wall there.

00:01:05.269 --> 00:01:07.690
So it would rotate around
this object.

00:01:14.060 --> 00:01:21.070
Let's say it's some distance,
r, from the pivot.

00:01:21.070 --> 00:01:25.350
Let's say that distance is 10.

00:01:25.350 --> 00:01:29.670
This is the same thing as r,
and the magnitude of r is

00:01:29.670 --> 00:01:30.870
equal to 10.

00:01:30.870 --> 00:01:35.290
At some distance 10 from the
pivot, I apply some force F,

00:01:35.290 --> 00:01:39.260
and F I will do in yellow.

00:01:39.260 --> 00:01:42.370
I apply some force F.

00:01:42.370 --> 00:01:45.190
Let me draw it straight.

00:01:45.190 --> 00:01:48.360
I apply some force
F at some angle.

00:01:50.960 --> 00:01:52.530
That's my force F.

00:01:52.530 --> 00:01:53.390
It's also a vector.

00:01:53.390 --> 00:01:57.610
It has magnitude
and direction.

00:01:57.610 --> 00:02:01.280
Let's say that this is 10
meters, and let's say that I

00:02:01.280 --> 00:02:07.120
apply a force of 7 newtons.

00:02:07.120 --> 00:02:08.669
Let me make it more
interesting.

00:02:08.669 --> 00:02:13.120
Let's say I apply a force of
square root of 3 newtons.

00:02:13.120 --> 00:02:14.840
And I just threw that out there
because I think the

00:02:14.840 --> 00:02:16.450
numbers will all work out.

00:02:16.450 --> 00:02:19.790
And let's say that the angle
between my force and the lever

00:02:19.790 --> 00:02:22.550
arm, or the arm that's
rotating-- let's stick to

00:02:22.550 --> 00:02:25.720
radians this time.

00:02:25.720 --> 00:02:28.950
Let's say it's pi over 3, but if
you need to visualize that,

00:02:28.950 --> 00:02:30.480
that's 60 degrees.

00:02:30.480 --> 00:02:35.920
pi over 3 radians is
equal to theta.

00:02:35.920 --> 00:02:39.870
And so just based on what we
already know about moments or

00:02:39.870 --> 00:02:45.120
torque, what is the torque
around this pivot?

00:02:45.120 --> 00:02:47.600
Or how much torque is being
applied by this force?

00:02:47.600 --> 00:02:50.420
And when we learn torque or we
learn moments, we realize

00:02:50.420 --> 00:02:54.180
really the only hard part about
these problems is that

00:02:54.180 --> 00:02:58.410
you don't just multiply the
entire rotational force times

00:02:58.410 --> 00:03:00.910
the distance from the
axis of rotation.

00:03:00.910 --> 00:03:03.230
You have to multiply the
component of that force that

00:03:03.230 --> 00:03:05.660
is actually doing the rotation,
or the component of

00:03:05.660 --> 00:03:09.860
the force that is perpendicular
to this rotating

00:03:09.860 --> 00:03:13.090
arm, or perpendicular
to this moment arm.

00:03:13.090 --> 00:03:14.260
So how do we figure that out?

00:03:14.260 --> 00:03:18.480
Well, the component of this
force that is perpendicular to

00:03:18.480 --> 00:03:21.465
this arm-- I can visually
draw it here.

00:03:21.465 --> 00:03:24.850
Let's see, it would look
something like this.

00:03:29.330 --> 00:03:30.160
I could draw it there.

00:03:30.160 --> 00:03:32.150
I could also draw
it here, right?

00:03:32.150 --> 00:03:34.310
This would be the component, or
this would be the component

00:03:34.310 --> 00:03:38.620
that is perpendicular to this
rotating arm, and the

00:03:38.620 --> 00:03:40.570
component that is parallel would
be this, but we don't

00:03:40.570 --> 00:03:41.210
care about that.

00:03:41.210 --> 00:03:43.970
That's not contributing
to the rotation.

00:03:43.970 --> 00:03:45.350
The only thing that is
contributing to the rotation

00:03:45.350 --> 00:03:46.890
is this component
of the force.

00:03:49.600 --> 00:03:53.880
And what is the magnitude of
this vector right here?

00:03:53.880 --> 00:03:59.200
The component of vector F that
is perpendicular to this arm.

00:03:59.200 --> 00:04:00.560
Well, if this angle--
let me draw a little

00:04:00.560 --> 00:04:01.810
triangle down here.

00:04:04.610 --> 00:04:10.140
If this is square root of 3,
this is pi over 3 radians, or

00:04:10.140 --> 00:04:15.770
60 degrees, and this is a right
angle, it's pi over 3.

00:04:15.770 --> 00:04:17.250
I know it's hard to read.

00:04:17.250 --> 00:04:18.740
What is this length
right here?

00:04:18.740 --> 00:04:21.970
Well, it's a 30-60-90 triangle,
and we know that

00:04:21.970 --> 00:04:24.510
this length here-- I mean,
there's a couple of ways you

00:04:24.510 --> 00:04:25.410
can think about it.

00:04:25.410 --> 00:04:28.420
Now that we know trigonometry,
we know that this is just the

00:04:28.420 --> 00:04:33.170
square root of 3 times the sine
of pi over 3, or the sine

00:04:33.170 --> 00:04:36.020
of 60 degrees, and so that
equals the square root of 3.

00:04:36.020 --> 00:04:38.790
Sine of pi over 3, or sine
of 60 degrees, is square

00:04:38.790 --> 00:04:40.820
root of 3 over 2.

00:04:40.820 --> 00:04:44.190
So the square root of 3 times
the square root of 3 is just

00:04:44.190 --> 00:04:48.110
3, so that equals 3/2.

00:04:48.110 --> 00:04:54.190
So the magnitude of this force
vector that is perpendicular,

00:04:54.190 --> 00:04:57.660
the component that is
perpendicular to the arm, is

00:04:57.660 --> 00:05:01.580
3/2 newtons, and now we can
figure out the magnitude of

00:05:01.580 --> 00:05:02.135
the torque.

00:05:02.135 --> 00:05:05.700
It's 3/2 newtons times
10 meters.

00:05:05.700 --> 00:05:08.970
So we know the magnitude of the
torque, and I'm being a

00:05:08.970 --> 00:05:11.200
little bit more careful with
my notation right now to

00:05:11.200 --> 00:05:13.680
remind you that torque actually
is a vector, or you

00:05:13.680 --> 00:05:16.520
can almost view it as, they use
this term pseudovector,

00:05:16.520 --> 00:05:21.330
because it's kind of a-- well,
anyway, I won't go into that.

00:05:21.330 --> 00:05:24.060
So what is the magnitude
of the torque vector?

00:05:24.060 --> 00:05:31.930
Well, it's 3/2 newtons times the
distance, and remember, I

00:05:31.930 --> 00:05:33.830
just drew this vector
here just to

00:05:33.830 --> 00:05:34.450
show you the component.

00:05:34.450 --> 00:05:36.400
I could just shift the vector
here because this is actually

00:05:36.400 --> 00:05:37.550
where the force is
being applied.

00:05:37.550 --> 00:05:39.810
You could draw that same vector
here because you can

00:05:39.810 --> 00:05:43.730
shift vectors around, so this is
also 3/2 newtons and maybe

00:05:43.730 --> 00:05:44.950
that makes it a little
bit clearer.

00:05:44.950 --> 00:05:49.600
So it's 3/2 newtons times the
distance that you are from

00:05:49.600 --> 00:05:54.960
your pivot arm, so times
10 meters, and so

00:05:54.960 --> 00:05:56.770
that is equal to what?

00:05:56.770 --> 00:06:00.590
15 newton meters.

00:06:00.590 --> 00:06:04.940
So the magnitude of the torque
is 15 newton meters.

00:06:04.940 --> 00:06:07.720
But all we did now-- and
hopefully this looks a little

00:06:07.720 --> 00:06:08.180
bit familiar.

00:06:08.180 --> 00:06:10.530
This is what we learned when we
learned moments and torque,

00:06:10.530 --> 00:06:13.690
but all we did now is we figured
out the magnitude of

00:06:13.690 --> 00:06:14.010
the torque.

00:06:14.010 --> 00:06:16.470
But what if we wanted to
know the direction?

00:06:16.470 --> 00:06:20.300
And that's where the cross
product comes in.

00:06:20.300 --> 00:06:22.080
So what was the definition
of the cross product?

00:06:22.080 --> 00:06:29.610
Cross product: r cross F, that
is equal to magnitude of r

00:06:29.610 --> 00:06:35.910
times the magnitude of F times
sine of the smallest angle

00:06:35.910 --> 00:06:41.400
between them times some vector
that is perpendicular to both.

00:06:41.400 --> 00:06:43.070
And this is really where it's
going to help, because all of

00:06:43.070 --> 00:06:45.660
these right here, these are all
scalar quantities, right?

00:06:45.660 --> 00:06:47.380
So these don't specify
the direction.

00:06:47.380 --> 00:06:50.990
The direction is completely
specified by this unit vector,

00:06:50.990 --> 00:06:53.420
and a unit vector is just a
vector of magnitude 1 that's

00:06:53.420 --> 00:06:55.750
pointing in some direction.

00:06:55.750 --> 00:06:58.250
Well, look, this cross product,
this part of it, the

00:06:58.250 --> 00:07:02.000
part that just gives us
magnitudes, we just calculated

00:07:02.000 --> 00:07:05.200
that using what we knew
before of torques.

00:07:05.200 --> 00:07:08.730
The magnitude of our force
vector times sine of theta,

00:07:08.730 --> 00:07:11.110
that gave us the component of
the force vector that is

00:07:11.110 --> 00:07:13.110
perpendicular to the arm.

00:07:13.110 --> 00:07:15.430
And we just multiply that times
the magnitude of r, and

00:07:15.430 --> 00:07:21.110
we got the magnitude of the
torque vector, which was 15.

00:07:21.110 --> 00:07:23.960
We can leave out the newton
meters for now.

00:07:23.960 --> 00:07:27.910
15, and then its direction
is this vector that we

00:07:27.910 --> 00:07:28.880
specified by n.

00:07:28.880 --> 00:07:30.500
We can call it the
normal vector.

00:07:30.500 --> 00:07:31.560
And what do we know
about this vector?

00:07:31.560 --> 00:07:36.590
It's perpendicular to both r--
this is r, right-- and it's

00:07:36.590 --> 00:07:37.960
perpendicular to F.

00:07:37.960 --> 00:07:40.780
And the only way that I
can visualize in our

00:07:40.780 --> 00:07:43.990
three-dimensional universe, a
vector that's perpendicular to

00:07:43.990 --> 00:07:47.120
both this and this is
if it pops in or out

00:07:47.120 --> 00:07:48.960
of this page, right?

00:07:48.960 --> 00:07:51.690
Because both of these vectors
are in the plane that are

00:07:51.690 --> 00:07:53.650
defined by our video.

00:07:53.650 --> 00:07:58.320
So if I'm a vector that is
perpendicular to your screen,

00:07:58.320 --> 00:08:00.350
whatever you're watching this
on, then it's going to be

00:08:00.350 --> 00:08:03.050
perpendicular to both
of these vectors.

00:08:03.050 --> 00:08:06.150
And how do we figure out if that
vector pops out or pops

00:08:06.150 --> 00:08:07.240
into the page?

00:08:07.240 --> 00:08:10.035
We use the right hand
rule, right?

00:08:10.035 --> 00:08:14.340
In the right hand rule, we
take-- r is our index finger,

00:08:14.340 --> 00:08:17.410
F is our middle finger, and
whichever direction our thumb

00:08:17.410 --> 00:08:23.340
points in tells us whether or
not we are-- the direction of

00:08:23.340 --> 00:08:24.980
the cross product.

00:08:24.980 --> 00:08:26.930
So let's draw it.

00:08:26.930 --> 00:08:31.350
Let me see if I can do a
good job right here.

00:08:31.350 --> 00:08:43.409
So if that is my index finger,
and you could imagine your

00:08:43.409 --> 00:08:47.130
hand sitting on top
of this screen.

00:08:47.130 --> 00:08:51.160
So that's my index finger
representing r, and this is my

00:08:51.160 --> 00:08:51.610
right hand.

00:08:51.610 --> 00:08:52.850
Remember, it only works
with your right hand.

00:08:52.850 --> 00:08:56.440
If you do your left hand, it's
going to be the opposite.

00:08:56.440 --> 00:09:01.220
And then my middle finger is
going to go in the direction

00:09:01.220 --> 00:09:06.460
of F, and then the rest of my
fingers are-- and I encourage

00:09:06.460 --> 00:09:07.570
you to draw this.

00:09:07.570 --> 00:09:10.300
So if I were to draw it-- let
me draw my nails just so you

00:09:10.300 --> 00:09:11.200
know what this is.

00:09:11.200 --> 00:09:13.470
So this is the nail on
my index finger.

00:09:13.470 --> 00:09:15.470
This is the nail on
my middle finger.

00:09:15.470 --> 00:09:17.960
And so in this situation, where
is my thumb going to be?

00:09:17.960 --> 00:09:19.550
My thumb is going to
be popping out.

00:09:19.550 --> 00:09:23.380
I wish I could-- that's
the nail of my thumb.

00:09:23.380 --> 00:09:25.400
Hopefully, that makes
some sense, right?

00:09:25.400 --> 00:09:26.870
That's the palm of my hand.

00:09:26.870 --> 00:09:29.940
That's the other side of my--
and I could keep drawing, but

00:09:29.940 --> 00:09:32.250
hopefully, that makes
some sense.

00:09:32.250 --> 00:09:33.220
This is my index finger.

00:09:33.220 --> 00:09:34.080
This is the middle finger.

00:09:34.080 --> 00:09:37.440
My thumb is pointing out of
the page, so that tells us

00:09:37.440 --> 00:09:40.280
that the torque is actually
pointing out of the page.

00:09:40.280 --> 00:09:43.780
So the direction of this unit
vector n is going to be out of

00:09:43.780 --> 00:09:49.080
the page, and we could signify
that by a circle with a dot.

00:09:49.080 --> 00:09:51.950
And I'm almost at my time limit,
and so there you have

00:09:51.950 --> 00:09:57.980
it: the cross product as it
is applied to torque.

00:09:57.980 --> 00:09:59.820
See you in the next video.


Cross product 2

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:03.380
Let's see if we can get a little
bit more practice and

00:00:03.380 --> 00:00:06.890
intuition of what cross products
are all about.

00:00:06.890 --> 00:00:09.450
So in the last example,
we took a cross b.

00:00:09.450 --> 00:00:13.580
Let's see what happens when
we take b cross a.

00:00:13.580 --> 00:00:18.250
So let me erase some of this.

00:00:18.250 --> 00:00:20.230
I don't want to erase all of it
because it might be useful

00:00:20.230 --> 00:00:21.590
to give us some intuition
to compare.

00:00:25.672 --> 00:00:27.170
I'm going to keep that.

00:00:31.430 --> 00:00:33.285
Actually, I can erase
this, I think.

00:00:39.600 --> 00:00:44.100
So the things I have drawn
here, this was a cross b.

00:00:44.100 --> 00:00:51.530
Let me cordon it off so you
don't get confused.

00:00:51.530 --> 00:00:55.970
So that was me using the right
hand rule when I tried to do a

00:00:55.970 --> 00:01:02.140
cross b, and then we saw that
the magnitude of this was 25,

00:01:02.140 --> 00:01:06.320
and n, the direction,
pointed downwards.

00:01:06.320 --> 00:01:09.330
Or when I drew it here, it would
point into the page.

00:01:09.330 --> 00:01:13.650
So let's see what happens with
b cross a, so I'm just

00:01:13.650 --> 00:01:15.000
switching the order.

00:01:15.000 --> 00:01:18.760
b cross a.

00:01:18.760 --> 00:01:21.090
Well, the magnitude is going to
be the same thing, right?

00:01:21.090 --> 00:01:24.610
Because I'm still going to take
the magnitude of b times

00:01:24.610 --> 00:01:28.720
the magnitude of a times the
sine of the angle between

00:01:28.720 --> 00:01:35.100
them, which was pi over 6
radians and then times some

00:01:35.100 --> 00:01:36.585
unit vector n.

00:01:36.585 --> 00:01:37.930
But this is going
to be the same.

00:01:37.930 --> 00:01:39.830
When I multiply scalar
quantities, it doesn't matter

00:01:39.830 --> 00:01:41.510
what order I multiply
them in, right?

00:01:41.510 --> 00:01:45.030
So this is still going to be
25, whatever my units might

00:01:45.030 --> 00:01:47.900
have been, times
some vector n.

00:01:47.900 --> 00:01:50.920
And we still know that that
vector n has to be

00:01:50.920 --> 00:01:54.020
perpendicular to both a and b,
and now we have to figure out,

00:01:54.020 --> 00:01:56.080
well, is it, in being
perpendicular, it can either

00:01:56.080 --> 00:01:59.050
kind of point into the page here
or it could pop out of

00:01:59.050 --> 00:02:00.450
the page, or point
out of the page.

00:02:00.450 --> 00:02:01.570
So which one is it?

00:02:01.570 --> 00:02:05.290
And then we take our right hand
out, and we try it again.

00:02:05.290 --> 00:02:09.560
So what we do is we take
our right hand.

00:02:09.560 --> 00:02:12.990
I'm actually using my right hand
right now, although you

00:02:12.990 --> 00:02:16.920
can't see it, just to make sure
I draw the right thing.

00:02:16.920 --> 00:02:21.800
So in this example, if I take
my right hand, I take the

00:02:21.800 --> 00:02:25.400
index finger in the
direction of b.

00:02:25.400 --> 00:02:28.140
I take my middle finger in the
direction of a, so my middle

00:02:28.140 --> 00:02:32.230
figure is going to look
something like that, right?

00:02:32.230 --> 00:02:36.310
And then I have two leftover
fingers there.

00:02:36.310 --> 00:02:39.610
Then the thumb goes in the
direction of the cross

00:02:39.610 --> 00:02:43.060
product, right?

00:02:43.060 --> 00:02:45.025
Because your thumb has a right
angle right there.

00:02:45.025 --> 00:02:46.920
That's the right angle
of your thumb.

00:02:46.920 --> 00:02:54.770
So in this example, that's the
direction of a, this is the

00:02:54.770 --> 00:02:59.480
direction of b, and we're
doing b cross a.

00:02:59.480 --> 00:03:02.020
That's why b gets your
index finger.

00:03:02.020 --> 00:03:05.410
The index finger gets the first
term, your middle finger

00:03:05.410 --> 00:03:09.010
gets the second term, and the
thumb gets the direction of

00:03:09.010 --> 00:03:10.330
the cross product.

00:03:10.330 --> 00:03:13.280
So in this example, the
direction of the cross product

00:03:13.280 --> 00:03:14.820
is upwards.

00:03:14.820 --> 00:03:18.840
Or when we're drawing it in two
dimensions right here, the

00:03:18.840 --> 00:03:20.440
cross product would actually
pop out of the

00:03:20.440 --> 00:03:22.340
page for b cross a.

00:03:22.340 --> 00:03:24.250
So I'll draw it over.

00:03:24.250 --> 00:03:26.080
It would be the circle
with the dot.

00:03:26.080 --> 00:03:30.790
Or if I were to draw it
analogous to this, so this

00:03:30.790 --> 00:03:34.650
right here, that
was a cross b.

00:03:34.650 --> 00:03:37.150
And then b cross a is the exact
same magnitude, but it

00:03:37.150 --> 00:03:39.420
goes in the other direction.

00:03:39.420 --> 00:03:41.486
That's b cross a.

00:03:41.486 --> 00:03:44.020
It just flips in the
opposite direction.

00:03:44.020 --> 00:03:46.340
And that's why you have to use
your right hand, because you

00:03:46.340 --> 00:03:48.240
might know that, oh, something's
going to pop in or

00:03:48.240 --> 00:03:49.960
out of the page, et cetera, et
cetera, but you need to know

00:03:49.960 --> 00:03:52.110
your right hand to know
whether it goes in

00:03:52.110 --> 00:03:54.350
or out of the page.

00:03:54.350 --> 00:03:56.380
Anyway, let's see if we can
get a little bit more

00:03:56.380 --> 00:04:00.900
intuition of what this is all
about because this is all

00:04:00.900 --> 00:04:01.540
about intuition.

00:04:01.540 --> 00:04:05.320
And frankly, I'll tell you, the
cross product comes into

00:04:05.320 --> 00:04:08.430
use in a lot of concepts that
frankly we don't have a lot of

00:04:08.430 --> 00:04:11.590
real-life intuition, with
electrons flying through a

00:04:11.590 --> 00:04:15.780
magnetic field or magnetic
fields through a coil.

00:04:15.780 --> 00:04:18.880
A lot of things in our everyday
life experience,

00:04:18.880 --> 00:04:23.430
maybe if we were metal filings
living in a magnetic field--

00:04:23.430 --> 00:04:24.570
well, we do live in
a magnetic field.

00:04:24.570 --> 00:04:26.380
In a strong magnetic field,
maybe we would get an

00:04:26.380 --> 00:04:29.600
intuition, but it's hard to have
as deep of an intuition

00:04:29.600 --> 00:04:33.780
as we do for, say, falling
objects, or friction, or

00:04:33.780 --> 00:04:37.310
forces, or fluid dynamics even,
because we've all played

00:04:37.310 --> 00:04:39.440
with water.

00:04:39.440 --> 00:04:40.810
But anyway, let's get a little
bit more intuition.

00:04:40.810 --> 00:04:43.800
And let's think about why is
there that sine of theta?

00:04:43.800 --> 00:04:46.180
Why not just multiply the
magnitudes times each other

00:04:46.180 --> 00:04:48.520
and use the right hand rule and
figure out a direction?

00:04:48.520 --> 00:04:50.170
What is that sine of
theta all about?

00:04:50.170 --> 00:04:57.840
I think I need to clear this up
a little bit just so this

00:04:57.840 --> 00:05:00.400
could be useful.

00:05:00.400 --> 00:05:02.390
So why is that sine
of theta there?

00:05:07.480 --> 00:05:09.780
Let me redraw some vectors.

00:05:09.780 --> 00:05:12.200
I'll draw them a
little fatter.

00:05:12.200 --> 00:05:22.870
So let's say that's a,
that's a, this is b.

00:05:22.870 --> 00:05:25.500
b doesn't always have
to be longer than a.

00:05:25.500 --> 00:05:29.630
So this is a and this is b.

00:05:29.630 --> 00:05:30.850
Now, we can think of
it a little bit.

00:05:30.850 --> 00:05:33.550
We could say, well, this is the
same thing as a sine theta

00:05:33.550 --> 00:05:37.650
times b, or we could say this
is b sine theta times a.

00:05:37.650 --> 00:05:40.200
I hope I'm not confusing-- all
I'm saying is you could

00:05:40.200 --> 00:05:43.020
interpret this as--
because these are

00:05:43.020 --> 00:05:44.090
just magnitudes, right?

00:05:44.090 --> 00:05:46.950
So it doesn't matter what order
you multiply them in.

00:05:46.950 --> 00:05:52.790
You could say this is a sine
theta times the magnitude of

00:05:52.790 --> 00:05:55.600
b, all of that in the direction
of the normal

00:05:55.600 --> 00:05:57.630
vector, or you could put the
sine theta the other way.

00:05:57.630 --> 00:05:58.890
But let's think about what
this would mean.

00:05:58.890 --> 00:06:02.050
a sine theta, if
this is theta.

00:06:02.050 --> 00:06:04.410
What is a sine theta?

00:06:04.410 --> 00:06:08.210
Sine is opposite over
hypotenuse, right?

00:06:08.210 --> 00:06:11.450
So opposite over hypotenuse.

00:06:11.450 --> 00:06:17.470
So this would be the
magnitude of a.

00:06:17.470 --> 00:06:18.935
Let me draw something.

00:06:18.935 --> 00:06:24.160
Let me draw a line here and
make it a real line.

00:06:24.160 --> 00:06:26.400
Let me draw a line there,
so I have a right angle.

00:06:32.030 --> 00:06:34.370
So what's a sine theta?

00:06:34.370 --> 00:06:36.760
This is the opposite side.

00:06:36.760 --> 00:06:40.110
So a sine theta is a, and sine
of theta is opposite over

00:06:40.110 --> 00:06:41.150
hypotenuse.

00:06:41.150 --> 00:06:44.230
The hypotenuse is the magnitude
of a, right?

00:06:44.230 --> 00:06:47.880
So sine of theta is equal to
this side, which I call o for

00:06:47.880 --> 00:06:51.720
opposite, over the
magnitude of a.

00:06:51.720 --> 00:06:56.850
So it's opposite over
the magnitude of a.

00:06:56.850 --> 00:07:01.440
So this term a sine theta is
actually just the magnitude of

00:07:01.440 --> 00:07:04.170
this line right here.

00:07:04.170 --> 00:07:07.220
Another way you could--
let me redraw it.

00:07:07.220 --> 00:07:08.940
It doesn't matter where the
vectors start from.

00:07:08.940 --> 00:07:10.540
All you care about is this
magnitude and direction, so

00:07:10.540 --> 00:07:12.970
you could shift vectors
around.

00:07:12.970 --> 00:07:15.830
So this vector right here, and
you could call it this

00:07:15.830 --> 00:07:19.880
opposite vector, that's the
same thing as this vector.

00:07:19.880 --> 00:07:21.610
That's the same thing as this.

00:07:21.610 --> 00:07:23.035
I just shifted it away.

00:07:26.160 --> 00:07:28.470
And so another way to think
about it is, it is the

00:07:28.470 --> 00:07:30.820
component of vector a, right?

00:07:30.820 --> 00:07:35.260
We're used to taking a vector
and splitting it up into x-

00:07:35.260 --> 00:07:38.360
and y-components, but now we're
taking a vector a, and

00:07:38.360 --> 00:07:41.140
we're splitting it up into--
you can think of it as a

00:07:41.140 --> 00:07:45.350
component that's parallel to
vector b and a component that

00:07:45.350 --> 00:07:48.430
is perpendicular to vector b.

00:07:48.430 --> 00:07:54.460
So a sine theta is the magnitude
of the component of

00:07:54.460 --> 00:07:57.250
vector a that is perpendicular
to b.

00:07:57.250 --> 00:08:00.060
So when you're taking the cross
product of two numbers,

00:08:00.060 --> 00:08:02.500
you're saying, well, I don't
care about the entire

00:08:02.500 --> 00:08:06.100
magnitude of vector a in this
example, I care about the

00:08:06.100 --> 00:08:10.330
magnitude of vector a that is
perpendicular to vector b, and

00:08:10.330 --> 00:08:12.860
those are the two numbers that
I want to multiply and then

00:08:12.860 --> 00:08:15.120
give it that direction
as specified by

00:08:15.120 --> 00:08:16.090
the right hand rule.

00:08:16.090 --> 00:08:17.340
And I'll show you some
applications.

00:08:17.340 --> 00:08:21.390
This is especially important--
well, we'll use it in torque

00:08:21.390 --> 00:08:25.290
and we'll also use it in
magnetic fields, but it's

00:08:25.290 --> 00:08:27.500
important in both of those
applications to figure out the

00:08:27.500 --> 00:08:31.610
components of the vector that
are perpendicular to either a

00:08:31.610 --> 00:08:33.600
force or a radius in question.

00:08:33.600 --> 00:08:36.659
So that's why this cross product
has the sine theta

00:08:36.659 --> 00:08:39.070
because we're taking-- so in
this, if you view it as

00:08:39.070 --> 00:08:42.210
magnitude of a sine theta
times b, this is kind of

00:08:42.210 --> 00:08:47.030
saying this is the magnitude
of the component of a

00:08:47.030 --> 00:08:49.435
perpendicular to b, or
you could interpret

00:08:49.435 --> 00:08:51.060
it the other way.

00:08:51.060 --> 00:08:59.120
You could interpret it as a
times b sine theta, right?

00:08:59.120 --> 00:09:00.400
Put a parentheses here.

00:09:00.400 --> 00:09:01.500
And then you could view
it the other way.

00:09:01.500 --> 00:09:05.610
You could say, well, b sine
theta is the component of b

00:09:05.610 --> 00:09:06.710
that is perpendicular to a.

00:09:06.710 --> 00:09:11.110
Let me draw that, just to
hit the point home.

00:09:11.110 --> 00:09:16.800
So that's my a, that's my b.

00:09:23.260 --> 00:09:25.950
This is a, this is b.

00:09:25.950 --> 00:09:29.510
So b has some component of it
that is perpendicular to a,

00:09:29.510 --> 00:09:32.190
and that is going to look
something like-- well, I've

00:09:32.190 --> 00:09:33.480
run out of space.

00:09:33.480 --> 00:09:34.640
Let me draw it here.

00:09:34.640 --> 00:09:41.210
If that's a, that's b, the
component of b that is

00:09:41.210 --> 00:09:42.760
perpendicular to a is going
to look like this.

00:09:46.300 --> 00:09:48.370
It's going to be perpendicular
to a, and it's going to go

00:09:48.370 --> 00:09:51.580
that far, right?

00:09:51.580 --> 00:09:54.230
And then you could go back to
SOH CAH TOA and you could

00:09:54.230 --> 00:09:58.850
prove to yourself that the
magnitude of this vector is b

00:09:58.850 --> 00:09:59.650
sine theta.

00:09:59.650 --> 00:10:01.410
So that is where the sine
theta comes from.

00:10:01.410 --> 00:10:03.990
It makes sure that we're not
just multiplying the vectors.

00:10:03.990 --> 00:10:06.320
It makes sure we're multiplying
the components of

00:10:06.320 --> 00:10:09.540
the vectors that are
perpendicular to each other to

00:10:09.540 --> 00:10:13.650
get a third vector that is
perpendicular to both of them.

00:10:13.650 --> 00:10:17.310
And then the people who invented
the cross product

00:10:17.310 --> 00:10:19.010
said, well, it's still ambiguous
because it doesn't

00:10:19.010 --> 00:10:21.475
tell us-- there's always two
vectors that are perpendicular

00:10:21.475 --> 00:10:22.300
to these two.

00:10:22.300 --> 00:10:24.050
One goes in, one goes out.

00:10:24.050 --> 00:10:25.340
They're in opposite
directions.

00:10:25.340 --> 00:10:26.950
And that's where the right
hand rule comes in.

00:10:26.950 --> 00:10:29.080
They'll say, OK, well, we're
just going to say a convention

00:10:29.080 --> 00:10:32.195
that you use your right hand,
point it like a gun, make all

00:10:32.195 --> 00:10:34.410
your fingers perpendicular,
and then you know what

00:10:34.410 --> 00:10:36.760
direction that vector
points in.

00:10:36.760 --> 00:10:38.830
Anyway, hopefully, you're
not confused.

00:10:38.830 --> 00:10:40.670
Now I want you to watch
the next video.

00:10:40.670 --> 00:10:43.890
This is actually going to be
some physics on electricity,

00:10:43.890 --> 00:10:46.400
magnetism and torque, and
that's essentially the

00:10:46.400 --> 00:10:48.250
applications of the cross
product, and it'll give you a

00:10:48.250 --> 00:10:49.770
little bit more intuition
of how to use it.

00:10:49.770 --> 00:10:51.470
See you soon.


Cross product 1

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:03.890
I've been requested to do a
video on the cross product,

00:00:03.890 --> 00:00:08.390
and its special circumstances,
because I was at the point on

00:00:08.390 --> 00:00:11.580
the physics playlist where I had
to teach magnetism anyway,

00:00:11.580 --> 00:00:15.050
so this is as good a time as any
to introduce the notion of

00:00:15.050 --> 00:00:15.870
the cross product.

00:00:15.870 --> 00:00:18.170
So what's the cross product?

00:00:18.170 --> 00:00:21.890
Well, we know about vector
addition, vector subtraction,

00:00:21.890 --> 00:00:23.840
but what happens when you
multiply vectors?

00:00:23.840 --> 00:00:25.960
And there's actually two ways to
do it: with the dot product

00:00:25.960 --> 00:00:27.200
or the cross product.

00:00:27.200 --> 00:00:30.300
And just keep in mind these
are-- well, really, every

00:00:30.300 --> 00:00:33.140
operation we've learned is
defined by human beings for

00:00:33.140 --> 00:00:35.500
some other purpose, and there's
nothing different

00:00:35.500 --> 00:00:36.700
about the cross product.

00:00:36.700 --> 00:00:38.830
I take the time to say that
here because the cross

00:00:38.830 --> 00:00:41.130
product, at least when I first
learned it, seemed a little

00:00:41.130 --> 00:00:41.810
bit unnatural.

00:00:41.810 --> 00:00:42.740
Anyway, enough talk.

00:00:42.740 --> 00:00:44.180
Let me show you what it is.

00:00:44.180 --> 00:00:47.140
So the cross product of two
vectors: Let's say I have

00:00:47.140 --> 00:00:53.600
vector a cross vector b, and the
notation is literally like

00:00:53.600 --> 00:00:56.150
the times sign that you knew
before you started taking

00:00:56.150 --> 00:00:58.130
algebra and using dots and
parentheses, so it's

00:00:58.130 --> 00:00:59.770
literally just an x.

00:00:59.770 --> 00:01:05.400
So the cross product of vectors
a and b is equal to--

00:01:05.400 --> 00:01:08.130
and this is going to seem very
bizarre at first, but

00:01:08.130 --> 00:01:10.670
hopefully, we can get a little
bit of a visual feel of what

00:01:10.670 --> 00:01:12.180
this means.

00:01:12.180 --> 00:01:19.830
It equals the magnitude of
vector a times the magnitude

00:01:19.830 --> 00:01:28.350
of vector b times the sine of
the angle between them, the

00:01:28.350 --> 00:01:30.430
smallest angle between them.

00:01:30.430 --> 00:01:34.580
And now, this is the kicker, and
this quantity is not going

00:01:34.580 --> 00:01:36.040
to be just a scalar quantity.

00:01:36.040 --> 00:01:37.340
It's not just going
to have magnitude.

00:01:37.340 --> 00:01:41.190
It actually has direction, and
that direction we specify by

00:01:41.190 --> 00:01:43.760
the vector n, the
unit vector n.

00:01:43.760 --> 00:01:45.630
We could put a little
cap on it to show

00:01:45.630 --> 00:01:48.540
that it's a unit vector.

00:01:48.540 --> 00:01:50.790
There are a couple of things
that are special about this

00:01:50.790 --> 00:01:54.370
direction that's
specified by n.

00:01:54.370 --> 00:01:59.730
One, n is perpendicular to
both of these vectors.

00:01:59.730 --> 00:02:02.890
It is orthogonal to both of
these vectors, so we'll think

00:02:02.890 --> 00:02:04.140
about it in a second
what that implies

00:02:04.140 --> 00:02:05.580
about it just visually.

00:02:05.580 --> 00:02:09.789
And then the other thing is the
direction of this vector

00:02:09.789 --> 00:02:11.840
is defined by the right
hand rule, and we'll

00:02:11.840 --> 00:02:12.620
see that in a second.

00:02:12.620 --> 00:02:15.290
So let's try to think
about this visually.

00:02:15.290 --> 00:02:18.780
And I have to give you an
important caveat: You can only

00:02:18.780 --> 00:02:22.050
take a cross product when
we are dealing in three

00:02:22.050 --> 00:02:22.890
dimensions.

00:02:22.890 --> 00:02:25.970
A cross product really has--
maybe you could define a use

00:02:25.970 --> 00:02:28.760
for it in other dimensions or a
way to take a cross product

00:02:28.760 --> 00:02:31.230
in other dimensions, but it
really only has a use in three

00:02:31.230 --> 00:02:33.820
dimensions, and that's useful,
because we live in a

00:02:33.820 --> 00:02:35.420
three-dimensional world.

00:02:35.420 --> 00:02:36.120
So let's see.

00:02:36.120 --> 00:02:37.280
Let's take some cross
products.

00:02:37.280 --> 00:02:39.300
I think when you see it
visually, it will make a

00:02:39.300 --> 00:02:41.490
little bit more sense,
especially once you get used

00:02:41.490 --> 00:02:43.332
to the right hand rule.

00:02:43.332 --> 00:02:49.250
So let's say that
that's vector b.

00:02:49.250 --> 00:02:50.550
I don't have to draw a
straight line, but it

00:02:50.550 --> 00:02:53.130
doesn't hurt to.

00:02:53.130 --> 00:02:54.610
I don't have to draw
it neatly.

00:02:57.370 --> 00:02:59.320
OK, here we go.

00:02:59.320 --> 00:03:04.650
Let's say that that is vector
a, and we want to take the

00:03:04.650 --> 00:03:06.690
cross product of them.

00:03:06.690 --> 00:03:09.040
This is vector a.

00:03:09.040 --> 00:03:10.230
This is b.

00:03:10.230 --> 00:03:12.250
I'll probably just switch to one
color because it's hard to

00:03:12.250 --> 00:03:13.550
keep switching between them.

00:03:13.550 --> 00:03:18.690
And then the angle between
them is theta.

00:03:18.690 --> 00:03:23.350
Now, let's say the length of a
is-- I don't know, let's say

00:03:23.350 --> 00:03:28.060
magnitude of a is equal to
5, and let's say that the

00:03:28.060 --> 00:03:32.815
magnitude of b is equal to 10.

00:03:32.815 --> 00:03:34.310
It looks about double that.

00:03:34.310 --> 00:03:36.060
I'm just making up the
numbers on the fly.

00:03:36.060 --> 00:03:37.730
So what's the cross product?

00:03:37.730 --> 00:03:39.910
Well, the magnitude
part is easy.

00:03:39.910 --> 00:03:45.660
Let's say this angle is
equal to 30 degrees.

00:03:45.660 --> 00:03:48.880
30 degrees, or if we wanted
to write it in radians, I

00:03:48.880 --> 00:03:51.200
always-- just because we grow
up in a world of degrees, I

00:03:51.200 --> 00:03:53.400
always find it easier to
visualize degrees, but we

00:03:53.400 --> 00:03:55.780
could think about it in terms
of radians as well.

00:03:55.780 --> 00:04:01.030
30 degrees is-- let's see,
there's 3, 6-- it's pi over 6,

00:04:01.030 --> 00:04:04.670
so we could also write
pi over 6 radians.

00:04:04.670 --> 00:04:07.280
But anyway, this is a 30-degree
angle, so what will

00:04:07.280 --> 00:04:09.810
be a cross b?

00:04:09.810 --> 00:04:16.120
a cross b is going to equal
the magnitude of a for the

00:04:16.120 --> 00:04:21.000
length of this vector, so it's
going to be equal to 5 times

00:04:21.000 --> 00:04:26.950
the length of this b vector, so
times 10, times the sine of

00:04:26.950 --> 00:04:28.150
the angle between them.

00:04:28.150 --> 00:04:29.690
And, of course, you
could've taken the

00:04:29.690 --> 00:04:31.100
larger, the obtuse angle.

00:04:31.100 --> 00:04:33.970
You could have said this was the
angle between them, but I

00:04:33.970 --> 00:04:36.780
said earlier that it was the
smaller, the acute, angle

00:04:36.780 --> 00:04:38.505
between them up to 90 degrees.

00:04:38.505 --> 00:04:46.900
This is going to be sine of 30
degrees times this vector n.

00:04:46.900 --> 00:04:50.690
And it's a unit vector, so I'll
go over what direction

00:04:50.690 --> 00:04:51.770
it's actually pointing
in a second.

00:04:51.770 --> 00:04:54.100
Let's just figure out
its magnitude.

00:04:54.100 --> 00:04:57.910
So this is equal to 50, and
what's sine of 30 degrees?

00:04:57.910 --> 00:04:59.930
Sine of 30 degrees is 1/2.

00:04:59.930 --> 00:05:02.520
You could type it in your
calculator if you're not sure.

00:05:02.520 --> 00:05:11.780
So it's 5 times 10 times 1/2
times the unit vector, so that

00:05:11.780 --> 00:05:14.870
equals 25 times the
unit vector.

00:05:14.870 --> 00:05:17.310
Now, this is where it gets,
depending on your point of

00:05:17.310 --> 00:05:20.250
view, either interesting
or confusing.

00:05:20.250 --> 00:05:24.090
So what direction is this
unit vector pointing in?

00:05:24.090 --> 00:05:25.310
So what I said earlier is, it's

00:05:25.310 --> 00:05:27.140
perpendicular to both of these.

00:05:27.140 --> 00:05:28.300
So how can something be

00:05:28.300 --> 00:05:30.710
perpendicular to both of these?

00:05:30.710 --> 00:05:32.480
It seems like I can't
draw one.

00:05:32.480 --> 00:05:35.240
Well, that's because right here,
where I drew a and b,

00:05:35.240 --> 00:05:37.340
I'm operating in
two dimensions.

00:05:37.340 --> 00:05:40.270
But if I have a third dimension,
if I could go in or

00:05:40.270 --> 00:05:45.540
out of my writing pad or, from
your point of view, your

00:05:45.540 --> 00:05:50.350
screen, then I have a vector
that is perpendicular to both.

00:05:50.350 --> 00:05:53.190
So imagine of vector that's-- I
wish I could draw it-- that

00:05:53.190 --> 00:05:56.620
is literally going straight in
at this point or straight out

00:05:56.620 --> 00:05:57.290
at this point.

00:05:57.290 --> 00:05:58.350
Hopefully, you're seeing it.

00:05:58.350 --> 00:05:59.590
Let me show you the
notation for that.

00:05:59.590 --> 00:06:04.010
So if I draw a vector like this,
if I draw a circle with

00:06:04.010 --> 00:06:09.670
an x in it like that, that is a
vector that's going into the

00:06:09.670 --> 00:06:12.290
page or into the screen.

00:06:12.290 --> 00:06:17.690
And if I draw this, that is a
vector that's popping out of

00:06:17.690 --> 00:06:18.190
the screen.

00:06:18.190 --> 00:06:20.106
And where does that convention
come from?

00:06:20.106 --> 00:06:21.920
It's from an arrowhead,
because what does

00:06:21.920 --> 00:06:23.810
an arrow look like?

00:06:23.810 --> 00:06:27.900
An arrow, which is our
convention for drawing

00:06:27.900 --> 00:06:34.410
vectors, looks something like
this: The tip of an arrow is

00:06:34.410 --> 00:06:37.680
circular and it comes to a
point, so that's the tip, if

00:06:37.680 --> 00:06:40.520
you look at it head-on, if it
was popping out of the video.

00:06:40.520 --> 00:06:41.980
And what does the tail of
an arrow look like?

00:06:41.980 --> 00:06:43.742
It has fins, right?

00:06:43.742 --> 00:06:46.020
There would be one fin here
and there'd be another fin

00:06:46.020 --> 00:06:47.680
right there.

00:06:47.680 --> 00:06:51.840
And so if you took this arrow
and you were to go into the

00:06:51.840 --> 00:06:55.070
page and just see the back of
the arrow or the behind of the

00:06:55.070 --> 00:06:57.050
arrow, it would look
like that.

00:06:57.050 --> 00:06:59.140
So this is a vector that's going
into the page and this

00:06:59.140 --> 00:07:01.650
is a vector that's going
out of the page.

00:07:01.650 --> 00:07:06.390
So we know that n is
perpendicular to both a and b,

00:07:06.390 --> 00:07:07.890
and so the only way you can
get a vector that's

00:07:07.890 --> 00:07:11.110
perpendicular to both of these,
it essentially has to

00:07:11.110 --> 00:07:16.750
be perpendicular, or normal,
or orthogonal to the plane

00:07:16.750 --> 00:07:19.260
that's your computer screen.

00:07:19.260 --> 00:07:23.020
But how do we know if it's going
into the screen or how

00:07:23.020 --> 00:07:26.130
do we know if it's coming out of
the screen, this vector n?

00:07:26.130 --> 00:07:28.950
And this is where the right hand
rule-- I know this is a

00:07:28.950 --> 00:07:29.680
little bit overwhelming.

00:07:29.680 --> 00:07:32.550
We'll do a bunch of example
problems. But the right hand

00:07:32.550 --> 00:07:37.620
rule, what you do is you take
your right hand-- that's why

00:07:37.620 --> 00:07:40.560
it's called the right hand
rule-- and you take your index

00:07:40.560 --> 00:07:44.010
finger and you point it in the
direction of the first vector

00:07:44.010 --> 00:07:45.960
in your cross product,
and order matters.

00:07:45.960 --> 00:07:47.410
So let's do that.

00:07:47.410 --> 00:07:49.940
So you have to take your finger
and put it in the

00:07:49.940 --> 00:07:53.330
direction of the first arrow,
which is a, and then you have

00:07:53.330 --> 00:07:56.800
to take your middle finger and
point it in that direction of

00:07:56.800 --> 00:07:58.290
the second arrow, b.

00:07:58.290 --> 00:08:00.770
So in this case, your
hand would look

00:08:00.770 --> 00:08:01.850
something like this.

00:08:01.850 --> 00:08:03.110
I'm going to try to draw it.

00:08:11.760 --> 00:08:15.490
This is pushing the abilities
of my art skills.

00:08:19.950 --> 00:08:21.100
So that's my right hand.

00:08:21.100 --> 00:08:24.510
My thumb is going to be
coming down, right?

00:08:24.510 --> 00:08:26.530
That is my right hand
that I drew.

00:08:26.530 --> 00:08:29.510
This is my index finger, and
I'm pointing it in the

00:08:29.510 --> 00:08:30.200
direction of a.

00:08:30.200 --> 00:08:34.760
Maybe it goes a little bit more
in this direction, right?

00:08:34.760 --> 00:08:37.850
Then I put my middle finger, and
I kind of make an L with

00:08:37.850 --> 00:08:40.169
it, or you could kind of say
it almost looks like you're

00:08:40.169 --> 00:08:42.150
shooting a gun.

00:08:42.150 --> 00:08:44.520
And I point that in the
direction of b, and then

00:08:44.520 --> 00:08:47.540
whichever direction that your
thumb faces in, so in this

00:08:47.540 --> 00:08:50.220
case, your thumb is going
into the page, right?

00:08:50.220 --> 00:08:53.040
Your thumb would be going down
if you took your right hand

00:08:53.040 --> 00:08:55.150
into this configuration.

00:08:55.150 --> 00:08:58.700
So that tells us that the vector
n points into the page.

00:08:58.700 --> 00:09:03.110
So the vector n has magnitude
25, and it points into the

00:09:03.110 --> 00:09:05.810
page, so we could draw it
like that with an x.

00:09:05.810 --> 00:09:07.720
If I were to attempt to draw
it in three dimensions, it

00:09:07.720 --> 00:09:09.890
would look something
like this.

00:09:09.890 --> 00:09:11.250
Vector a.

00:09:11.250 --> 00:09:13.150
Let me see if I can give
some perspective.

00:09:16.030 --> 00:09:22.060
If this was straight down, if
that's vector n, then a could

00:09:22.060 --> 00:09:23.085
look something like that.

00:09:23.085 --> 00:09:25.610
Let me draw it in the
same color as a.

00:09:25.610 --> 00:09:29.670
a could look something like
that, and then b would look

00:09:29.670 --> 00:09:30.230
something like that.

00:09:30.230 --> 00:09:32.170
I'm trying to draw a
three-dimensional figure on

00:09:32.170 --> 00:09:34.320
two dimensions, so it might look
a little different, but I

00:09:34.320 --> 00:09:34.900
think you get the point.

00:09:34.900 --> 00:09:36.630
Here I drew a and
b on the plane.

00:09:36.630 --> 00:09:38.300
Here I have perspective
where I was able to

00:09:38.300 --> 00:09:40.770
draw n going down.

00:09:40.770 --> 00:09:44.940
But this is the definition
of a cross product.

00:09:44.940 --> 00:09:46.910
Now, I'm going to leave it
there, just because for some

00:09:46.910 --> 00:09:49.020
reason, YouTube hasn't been
letting me go over the limit

00:09:49.020 --> 00:09:51.490
as much, and I will do another
video where I do several

00:09:51.490 --> 00:09:53.880
problems, and actually, in the
process, I'm going to explain

00:09:53.880 --> 00:09:55.600
a little bit about magnetism.

00:09:55.600 --> 00:09:57.960
And we'll take the cross product
of several things, and

00:09:57.960 --> 00:09:59.810
hopefully, you'll get a little
bit better intuition.

00:09:59.810 --> 00:10:01.760
See you soon.


More chain rule and implicit differentiation intuition

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.230
Let's try to get a better
intuition of the chain rule,

00:00:04.230 --> 00:00:06.920
and in the process, we'll get
a better intuition of how

00:00:06.920 --> 00:00:10.700
it applies to implicit
differentiation or vice versa.

00:00:10.700 --> 00:00:13.710
So let's say-- and I'm going to
get out of the world of x's and

00:00:13.710 --> 00:00:16.890
y's and f of x's for a second,
just so that you see that those

00:00:16.890 --> 00:00:18.810
are just letters and there's
nothing special about them.

00:00:18.810 --> 00:00:21.720
And I think sometimes it
helps develop the intuition.

00:00:21.720 --> 00:00:32.000
So let's say that a is equal
to b squared and that b is

00:00:32.000 --> 00:00:36.530
equal to the sine of c.

00:00:36.530 --> 00:00:40.240
And my question to you is
what is the derivative

00:00:40.240 --> 00:00:44.310
of a with respect to c?

00:00:49.180 --> 00:00:52.475
We have a defined in terms of
b, so it's not like we can

00:00:52.475 --> 00:00:53.530
just take the derivative.

00:00:53.530 --> 00:00:54.950
I mean, how do we do that?

00:00:54.950 --> 00:00:56.530
Well, that's where the
chain rule comes in.

00:00:56.530 --> 00:01:00.890
And the chain rule just tells
us that the derivative of a

00:01:00.890 --> 00:01:05.550
with respect to c is equal to
the derivative of a with

00:01:05.550 --> 00:01:09.930
respect to b times
the derivative of b

00:01:09.930 --> 00:01:10.980
with respect to c.

00:01:10.980 --> 00:01:13.250
And you shouldn't really--
you don't even have

00:01:13.250 --> 00:01:14.370
to memorize this.

00:01:14.370 --> 00:01:17.320
We're just multiplying two
fractions where the denominator

00:01:17.320 --> 00:01:19.430
in the first one is the same as
the numerator in

00:01:19.430 --> 00:01:19.990
the second one.

00:01:19.990 --> 00:01:21.600
So if you were to multiply
fractions, they would

00:01:21.600 --> 00:01:23.240
cancel out, right?

00:01:23.240 --> 00:01:26.100
Or if you were to actually
multiply these fractions

00:01:26.100 --> 00:01:30.430
out, you'd get da times
db on the numerator.

00:01:30.430 --> 00:01:35.240
In the denominator, you would
get-- da-- sorry, db times dc

00:01:35.240 --> 00:01:37.670
in the numerator, and then
these would cancel out, and

00:01:37.670 --> 00:01:40.860
then you're back
with da over dc.

00:01:40.860 --> 00:01:43.630
So there's nothing fancy,
really, about the chain rule

00:01:43.630 --> 00:01:45.710
when you actually view
them for what they are.

00:01:45.710 --> 00:01:50.380
They're fractions, but the
values in the numerator and the

00:01:50.380 --> 00:01:53.090
denominator, they are these
differentials, these infinitely

00:01:53.090 --> 00:01:55.820
small changes in
these variables.

00:01:55.820 --> 00:01:59.400
But anyway, the chain rule
tells us: change of a with

00:01:59.400 --> 00:02:01.740
respect to c is equal to the
change of a with respect to

00:02:01.740 --> 00:02:05.920
b times the change of
b with respect to c.

00:02:05.920 --> 00:02:09.080
And these are pretty easy
to calculate based on

00:02:09.080 --> 00:02:11.110
the information that
I've just given you.

00:02:11.110 --> 00:02:13.140
What is the change of
a with respect to b?

00:02:13.140 --> 00:02:15.320
The derivative of a
with respect to b?

00:02:15.320 --> 00:02:16.960
Well, that's pretty
straightforward, right?

00:02:16.960 --> 00:02:23.350
The derivative a with respect
to b is equal to 2b.

00:02:23.350 --> 00:02:25.030
So this is 2b.

00:02:25.030 --> 00:02:27.970
And what's the derivative
of b with respect to c?

00:02:27.970 --> 00:02:31.770
Derivative of b with
respect to c, well, that's

00:02:31.770 --> 00:02:33.070
equal to cosine of c.

00:02:36.950 --> 00:02:37.910
And so we're done.

00:02:37.910 --> 00:02:45.810
The derivative of a with
respect to c is equal to this,

00:02:45.810 --> 00:02:49.340
which is this, 2b, the
derivative of a with respect to

00:02:49.340 --> 00:02:51.360
b times the derivative
of b with respect to c.

00:02:51.360 --> 00:02:56.510
So times cosine of c.

00:02:56.510 --> 00:02:58.660
And then if you don't like it
in this format, if you don't

00:02:58.660 --> 00:03:02.470
like the derivative of a with
respect to c being dependent or

00:03:02.470 --> 00:03:06.420
being a function of both b and
c, you can just substitute for

00:03:06.420 --> 00:03:08.640
b because b is also
a function of c.

00:03:08.640 --> 00:03:09.430
So let's do that.

00:03:09.430 --> 00:03:13.880
So the derivative of a with
respect to c is equal to 2b.

00:03:13.880 --> 00:03:14.950
But what's b?

00:03:14.950 --> 00:03:16.880
b is just sine of c.

00:03:16.880 --> 00:03:24.930
So 2-- just substitute sine of
c for b-- 2 sine of c, and then

00:03:24.930 --> 00:03:26.570
we have the remainder, right?

00:03:26.570 --> 00:03:34.020
This is this, and then
times cosine of c,

00:03:34.020 --> 00:03:35.640
and that's just that.

00:03:35.640 --> 00:03:36.860
And we're done.

00:03:36.860 --> 00:03:40.880
And I want to now try to make
the connection between this and

00:03:40.880 --> 00:03:44.790
the kind of-- I don't want to
say-- well, you know, kind of

00:03:44.790 --> 00:03:47.060
the plain English way I
talked about doing that

00:03:47.060 --> 00:03:48.280
chain rule before.

00:03:48.280 --> 00:03:51.200
So all of the chain rule
problems that I think I've done

00:03:51.200 --> 00:03:54.410
so far would have had a
explicitly defined in terms of

00:03:54.410 --> 00:03:57.460
c, but you're saying, oh, but
then, you know, I said outside

00:03:57.460 --> 00:03:58.850
and inside and et cetera.

00:03:58.850 --> 00:04:00.080
And we can do that.

00:04:00.080 --> 00:04:03.570
We can make a explicitly
defined in terms of c,

00:04:03.570 --> 00:04:04.160
and how do we do that?

00:04:04.160 --> 00:04:06.520
We could substitute before we
even take the derivative.

00:04:06.520 --> 00:04:10.600
So we can substitute this
for here, and we get what?

00:04:10.600 --> 00:04:16.370
a is equal to b squared,
b is sine of c, so it's

00:04:16.370 --> 00:04:21.770
sine of c squared.

00:04:21.770 --> 00:04:23.930
And now we could take the
derivative of a with respect

00:04:23.930 --> 00:04:25.210
to c using the chain rule.

00:04:25.210 --> 00:04:27.050
And this is the more
traditional way that I showed

00:04:27.050 --> 00:04:29.750
it, but maybe it gave
you more intuition.

00:04:29.750 --> 00:04:31.650
It probably gave you more
intuition of how to actually do

00:04:31.650 --> 00:04:34.210
the problem, but it probably
gave you less intuition of how

00:04:34.210 --> 00:04:35.230
it applies to the chain rule.

00:04:35.230 --> 00:04:37.510
So hopefully, this will
connect everything together.

00:04:37.510 --> 00:04:44.570
So the derivative of a with
respect to c is equal to-- I

00:04:44.570 --> 00:04:46.670
think sometimes I said that I
like to take the derivative of

00:04:46.670 --> 00:04:48.400
the inside and then multiply
that times the derivative of

00:04:48.400 --> 00:04:50.520
the outside, but we could
do it in either order.

00:04:50.520 --> 00:04:52.370
So let's take the
derivative of the outside.

00:04:52.370 --> 00:04:53.930
So what's the derivative
of the outside?

00:04:53.930 --> 00:05:01.230
sine of c squared, well,
it's 2 sine of c times the

00:05:01.230 --> 00:05:02.260
derivative of the inside.

00:05:02.260 --> 00:05:04.370
The derivative of the
inside is cosine of c.

00:05:07.060 --> 00:05:08.600
The phone is ringing.

00:05:08.600 --> 00:05:10.910
I'm not going to answer it.

00:05:10.910 --> 00:05:14.540
And so we ended up with the
exact same result, and it makes

00:05:14.540 --> 00:05:17.960
sense, because when I say the
derivative of the outside, when

00:05:17.960 --> 00:05:20.770
I take the derivative of sine c
squared and I get 2 sine of c,

00:05:20.770 --> 00:05:23.430
I'm really just taking the
derivative of a with

00:05:23.430 --> 00:05:25.000
respect to b, right?

00:05:25.000 --> 00:05:26.400
Because the derivative
of a with respect to b

00:05:26.400 --> 00:05:28.590
is 2b or 2 sine of c.

00:05:28.590 --> 00:05:30.800
And then when I take the
derivative of the inside, I'm

00:05:30.800 --> 00:05:34.480
really taking the derivative
of b with respect to c.

00:05:34.480 --> 00:05:36.750
So hopefully, that gives
you a little intuition.

00:05:36.750 --> 00:05:40.530
So now let's take that
intuition and tackle some

00:05:40.530 --> 00:05:44.110
implicit differentiation
problems.

00:05:44.110 --> 00:05:46.190
And if you think about it,
this kind of was an implicit

00:05:46.190 --> 00:05:49.440
differentiation problem.

00:05:49.440 --> 00:05:54.780
So let's say I want to take the
derivative with respect to x

00:05:54.780 --> 00:06:02.900
of y squared x to the third.

00:06:02.900 --> 00:06:06.610
And some of you all might know
how to mechanically do this.

00:06:06.610 --> 00:06:08.080
Well, I won't go into the
mechanics because the

00:06:08.080 --> 00:06:09.550
whole point of this
video is the intuition.

00:06:09.550 --> 00:06:10.860
And maybe you have the
intuition in which case you

00:06:10.860 --> 00:06:12.630
might not have to watch this
video, but let me give

00:06:12.630 --> 00:06:13.270
you the intuition.

00:06:13.270 --> 00:06:15.060
And I think the best way
to give the intuition

00:06:15.060 --> 00:06:18.760
is to actually do some
variable substitution.

00:06:18.760 --> 00:06:26.050
Let's define a is equal to y
squared and b is equal to

00:06:26.050 --> 00:06:28.810
x to the third, right?

00:06:28.810 --> 00:06:31.580
Then this is the same thing,
and I just made this stuff--

00:06:31.580 --> 00:06:33.190
you know, this could
have been any letters.

00:06:33.190 --> 00:06:36.430
This is the same thing as
the derivative with respect

00:06:36.430 --> 00:06:42.620
to x of a times b, right?

00:06:42.620 --> 00:06:45.080
And what does the
chain rule tell us?

00:06:45.080 --> 00:06:47.070
Well, actually, we could just
do the product rule here, so

00:06:47.070 --> 00:06:48.230
let's do the product rule.

00:06:48.230 --> 00:06:51.340
So we're taking the derivative
with respect to x of a times b.

00:06:51.340 --> 00:06:52.340
So let's do that.

00:06:52.340 --> 00:06:54.900
This is just the product rule
so it's the derivative of a

00:06:54.900 --> 00:07:03.480
with respect to x times b plus
the derivative of b with

00:07:03.480 --> 00:07:09.220
respect to x times a.

00:07:09.220 --> 00:07:10.760
That's just the product rule.

00:07:10.760 --> 00:07:12.750
But what's the derivative of a
with respect to x and what's

00:07:12.750 --> 00:07:15.560
the derivative of b
with respect to x?

00:07:15.560 --> 00:07:18.280
So the derivative of a with
respect to x-- and this is

00:07:18.280 --> 00:07:21.890
where we get into
the chain rule.

00:07:21.890 --> 00:07:27.070
The derivative of a with
respect to x is equal to the

00:07:27.070 --> 00:07:33.080
derivative of a with respect to
y times the derivative of y

00:07:33.080 --> 00:07:34.950
with respect to x, right?

00:07:34.950 --> 00:07:36.100
This is just the chain rule.

00:07:36.100 --> 00:07:39.160
These dy in the denominator
and the numerator,

00:07:39.160 --> 00:07:40.760
they'll cancel out.

00:07:40.760 --> 00:07:42.270
So this is just the chain rule.

00:07:42.270 --> 00:07:42.940
And what is this?

00:07:42.940 --> 00:07:44.800
What's the derivative of
a with respect to y?

00:07:44.800 --> 00:07:51.070
Well, a is equal to y squared,
so that is equal to 2y times

00:07:51.070 --> 00:07:53.130
the derivative of y
with respect to x.

00:07:53.130 --> 00:07:55.825
Well, we haven't explicitly
defined y in terms of x, so

00:07:55.825 --> 00:07:57.250
we're just going to have
to leave it like that.

00:07:57.250 --> 00:08:01.230
So times dy/dx.

00:08:01.230 --> 00:08:04.190
So this, right here, the
derivative of a with respect

00:08:04.190 --> 00:08:07.010
to x, is just that.

00:08:07.010 --> 00:08:09.450
And what's the derivative
of b with respect to x?

00:08:09.450 --> 00:08:12.660
Well, here b is explicitly
defined in terms of x, so this

00:08:12.660 --> 00:08:14.460
is pretty straightforward.

00:08:14.460 --> 00:08:17.600
There's no chain rule
application here, so that is

00:08:17.600 --> 00:08:20.480
just going to be 3x squared.

00:08:20.480 --> 00:08:23.210
And so what's our final answer?

00:08:23.210 --> 00:08:26.910
The final answer is the
derivative with respect to x of

00:08:26.910 --> 00:08:32.870
y squared x to the third is
equal to derivative of a with

00:08:32.870 --> 00:08:35.670
respect to x, or you could view
that as the derivative of y

00:08:35.670 --> 00:08:44.920
squared with respect to x,
which is 2y dy/dx, right?

00:08:44.920 --> 00:08:47.060
You could say this is the
derivative of a with respect to

00:08:47.060 --> 00:08:50.100
x or the derivative of y
squared with respect to x, and

00:08:50.100 --> 00:08:51.650
this is just from the
chain rule, right?

00:08:51.650 --> 00:08:53.010
This was the chain rule.

00:08:53.010 --> 00:08:57.070
2y times dy/dx and
then times b.

00:08:57.070 --> 00:09:04.200
Well, what's b? b is x to the
third plus the derivative of b

00:09:04.200 --> 00:09:08.900
with respect to x, well, that's
3x squared, times a.

00:09:08.900 --> 00:09:12.090
a is just y squared.

00:09:12.090 --> 00:09:14.750
So hopefully, that gave you the
intuition and made it a little

00:09:14.750 --> 00:09:17.630
bit clearer of why implicit
differentiation-- you're really

00:09:17.630 --> 00:09:18.710
just applying the chain rule.

00:09:18.710 --> 00:09:20.830
And then, you know, just the
way I've been doing in all the

00:09:20.830 --> 00:09:23.040
problems without going through
all of this mess just to show

00:09:23.040 --> 00:09:25.700
you that it isn't that hard of
a problem, that you could

00:09:25.700 --> 00:09:27.710
just do it straight up.

00:09:27.710 --> 00:09:29.520
You could just say, OK,
this is the product rule.

00:09:29.520 --> 00:09:32.750
It's the derivative of the
first expression times the

00:09:32.750 --> 00:09:34.390
second expression plus the
derivative of the second

00:09:34.390 --> 00:09:35.740
expression times the first.

00:09:35.740 --> 00:09:38.450
What's the derivative of this
first expression y squared?

00:09:38.450 --> 00:09:40.480
Well, it's the derivative of
that expression with respect

00:09:40.480 --> 00:09:44.630
to y times the derivative
of y with respect to x.

00:09:44.630 --> 00:09:46.160
And it's only because we're
taking the derivative with

00:09:46.160 --> 00:09:51.040
respect to x times the second
expression, x to the third,

00:09:51.040 --> 00:09:53.130
plus the derivative of the
second expression with respect

00:09:53.130 --> 00:10:00.070
to x, 3x squared, times the
first expression, y squared.

00:10:00.070 --> 00:10:01.950
Well, anyway, I'm
at my time limit.

00:10:01.950 --> 00:10:04.440
Hopefully, that helped a little
bit and didn't confuse things,

00:10:04.440 --> 00:10:08.610
and let me know if you want me
to do even more videos on this.

00:10:08.610 --> 00:10:10.112
See you soon.