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PROFESSOR: Hi.
00:00:34.470 --> 00:00:39.450
Today, we finish up the last
of the basic vector operations.
00:00:39.450 --> 00:00:41.270
Today, what we're
going to discuss
00:00:41.270 --> 00:00:45.330
is something called the vector
product or the cross product.
00:00:45.330 --> 00:00:49.450
And it's a new type of
multiplication, whereby we now
00:00:49.450 --> 00:00:53.890
multiply two vectors
to obtain a vector.
00:00:53.890 --> 00:00:56.150
You see, the dot product
multiplied two vectors,
00:00:56.150 --> 00:00:57.820
but it gave a scalar.
00:00:57.820 --> 00:01:01.070
This is why the dot product
was called a scalar product.
00:01:01.070 --> 00:01:04.510
Now, the idea is, again, we
can go to the physical world
00:01:04.510 --> 00:01:06.270
to seek motivation.
00:01:06.270 --> 00:01:10.520
And my only problem with that is
that some of the examples that
00:01:10.520 --> 00:01:13.430
motivated this are not
quite as elementary
00:01:13.430 --> 00:01:18.860
as the work equals force times
distance formula of last time.
00:01:18.860 --> 00:01:21.370
The cross product
does come up in sort
00:01:21.370 --> 00:01:24.960
of trivial applications,
in terms of moments,
00:01:24.960 --> 00:01:28.960
angular momentum, in
terms of mechanics,
00:01:28.960 --> 00:01:33.350
and in more advanced situations,
in terms of the Maxwell
00:01:33.350 --> 00:01:37.090
equations where, you may
remember, in physics courses,
00:01:37.090 --> 00:01:40.200
going through this
routine of the force
00:01:40.200 --> 00:01:43.280
and the flux and the
intensity and what have you.
00:01:43.280 --> 00:01:46.960
And at any rate, rather than
waste time-- and I mean,
00:01:46.960 --> 00:01:50.070
I use the word waste
here judiciously,
00:01:50.070 --> 00:01:52.050
because, for the
purpose of our course,
00:01:52.050 --> 00:01:56.320
it is not nearly as crucial
to motivate why we invented
00:01:56.320 --> 00:01:58.620
the cross product
physically as it
00:01:58.620 --> 00:02:03.900
is to define it mathematically
and to use it structurally.
00:02:03.900 --> 00:02:05.980
You see, what I
want to do is simply
00:02:05.980 --> 00:02:08.300
get to the crux
of the situation,
00:02:08.300 --> 00:02:11.710
discuss the concept
of a cross product,
00:02:11.710 --> 00:02:15.830
and simply point out that
if A and B are vectors,
00:02:15.830 --> 00:02:21.060
by A cross B-- written this way,
A cross B-- we mean a vector.
00:02:21.060 --> 00:02:22.870
Now, here's the kicker again.
00:02:22.870 --> 00:02:27.180
As soon as I say vector,
that means to define it,
00:02:27.180 --> 00:02:31.820
I have to tell you three
ingredients, namely magnitude,
00:02:31.820 --> 00:02:34.070
direction, and sense.
00:02:34.070 --> 00:02:34.740
Right?
00:02:34.740 --> 00:02:38.660
Now, for the magnitude
of A cross B,
00:02:38.660 --> 00:02:43.290
it's simply the magnitude of
A times the magnitude of B
00:02:43.290 --> 00:02:48.687
times the magnitude of the sine
of the angle between A and B.
00:02:48.687 --> 00:02:50.270
And I'm going to
come back and comment
00:02:50.270 --> 00:02:52.530
on that in a few moments,
about the difference
00:02:52.530 --> 00:02:56.840
between the sine here versus the
cosine of the previous lecture.
00:02:56.840 --> 00:02:59.560
Again, physically, there's
a motivation for this.
00:02:59.560 --> 00:03:03.010
As far as we're concerned,
all we have to do is say,
00:03:03.010 --> 00:03:05.380
this is what the magnitude
is defined to be.
00:03:05.380 --> 00:03:06.130
This is the game.
00:03:06.130 --> 00:03:08.300
We define it this way.
00:03:08.300 --> 00:03:12.620
Next thing I have to do
is tell you its direction.
00:03:12.620 --> 00:03:15.670
And the direction of A
cross B is, by definition,
00:03:15.670 --> 00:03:21.110
perpendicular to both
A and B. All right?
00:03:21.110 --> 00:03:23.080
Another way of saying
that is in the event
00:03:23.080 --> 00:03:26.500
that A and B are not parallel,
then the vectors A and B
00:03:26.500 --> 00:03:28.150
determine a plane.
00:03:28.150 --> 00:03:30.100
And what we're saying
is that A cross B
00:03:30.100 --> 00:03:33.740
is in a vector perpendicular
to the plane determined
00:03:33.740 --> 00:03:36.580
by A and B. All right?
00:03:36.580 --> 00:03:38.880
By the way, if A
and B are parallel,
00:03:38.880 --> 00:03:42.260
they don't determine a
plane, because they coincide.
00:03:42.260 --> 00:03:44.180
They have only one direction.
00:03:44.180 --> 00:03:47.270
But if A and B are parallel,
the angle between them
00:03:47.270 --> 00:03:51.845
is 0 or 180, depending
on whether their sense is
00:03:51.845 --> 00:03:52.760
the same or not.
00:03:52.760 --> 00:03:57.100
Once the angle is either 0 or
180, the sine of that angle
00:03:57.100 --> 00:03:58.410
is 0.
00:03:58.410 --> 00:04:02.750
Once one of the factors here
is 0, the whole product is 0.
00:04:02.750 --> 00:04:06.740
So in the event that A and B
do not determine the plane,
00:04:06.740 --> 00:04:09.080
the magnitude of A
cross B is 0, which
00:04:09.080 --> 00:04:11.300
means that A cross B
is the zero vector.
00:04:11.300 --> 00:04:13.290
And that's a pretty
trivial situation.
00:04:13.290 --> 00:04:15.610
So we just won't even
worry about that.
00:04:15.610 --> 00:04:20.440
And thirdly, I must now give
you the sense of A cross B.
00:04:20.440 --> 00:04:23.460
And the sense is the
usual right-hand rule,
00:04:23.460 --> 00:04:26.730
in other words, the direction
in which a right-handed thread
00:04:26.730 --> 00:04:31.240
screw would turn as
A is rotated into B--
00:04:31.240 --> 00:04:33.310
and now I have to
be careful because
00:04:33.310 --> 00:04:38.410
of this sine situation--
through the smaller angle.
00:04:38.410 --> 00:04:39.880
You see, look it.
00:04:39.880 --> 00:04:42.360
Notice that this sense
is going to depend
00:04:42.360 --> 00:04:45.060
on what-- if you
rotate A into B,
00:04:45.060 --> 00:04:50.620
you get the opposite sense
than if you rotated B into A.
00:04:50.620 --> 00:04:54.130
And not only that, but
if you rotate A into B
00:04:54.130 --> 00:04:56.350
through the larger
of the two angles,
00:04:56.350 --> 00:04:59.250
notice that rotating A
into B through one angle--
00:04:59.250 --> 00:05:02.460
say the smaller one--
gives you a different sense
00:05:02.460 --> 00:05:05.760
than rotating A into B through
the larger of the two angles.
00:05:05.760 --> 00:05:08.310
So there's a certain
ambiguity that comes up here.
00:05:08.310 --> 00:05:11.080
And the ambiguity occurs
because of the fact
00:05:11.080 --> 00:05:13.980
that we're using the sine
rather than the cosine.
00:05:13.980 --> 00:05:16.740
In other words, if you
reverse the sense of an angle,
00:05:16.740 --> 00:05:20.130
if you change its sine,
the algebraic sine,
00:05:20.130 --> 00:05:23.670
the trigonometric
sine, s-i-n changes.
00:05:23.670 --> 00:05:24.480
OK?
00:05:24.480 --> 00:05:27.680
And similarly, the
sine of an angle
00:05:27.680 --> 00:05:33.070
is the negative of the sine of
360 degrees minus that angle.
00:05:33.070 --> 00:05:36.030
But that, at any rate,
is the definition
00:05:36.030 --> 00:05:37.410
of the cross product.
00:05:37.410 --> 00:05:40.590
Given two vectors, their
cross product is what?
00:05:40.590 --> 00:05:44.720
A vector whose magnitude is the
product of the two magnitudes
00:05:44.720 --> 00:05:48.020
times the magnitude of the sine
of the angle between the two
00:05:48.020 --> 00:05:49.040
vectors.
00:05:49.040 --> 00:05:51.440
The cross product is
perpendicular to each
00:05:51.440 --> 00:05:52.850
of the two vectors.
00:05:52.850 --> 00:05:56.100
And it's sense is
the right-hand rule
00:05:56.100 --> 00:05:58.530
as the first is
rotated into the second
00:05:58.530 --> 00:06:01.370
through the smaller
of the two angles.
00:06:03.910 --> 00:06:06.010
This is important
because, you see,
00:06:06.010 --> 00:06:09.380
what happens is that as
important as the cross product
00:06:09.380 --> 00:06:13.220
is from a physical point
of view, arithmetically,
00:06:13.220 --> 00:06:14.350
it's a nuisance.
00:06:14.350 --> 00:06:18.230
We can take very few liberties
with the cross product.
00:06:18.230 --> 00:06:20.440
It turns out that
almost every rule
00:06:20.440 --> 00:06:24.400
that we would like to be true
about numerical multiplication
00:06:24.400 --> 00:06:27.860
is false for vector
cross products.
00:06:27.860 --> 00:06:31.650
For example, among
other things, A cross B
00:06:31.650 --> 00:06:34.250
is not equal to B cross A.
00:06:34.250 --> 00:06:38.500
In fact, A cross B is the
negative of B cross A.
00:06:38.500 --> 00:06:40.710
And the reason for
this is simply,
00:06:40.710 --> 00:06:43.790
as you check this thing
through, A cross B and B
00:06:43.790 --> 00:06:47.920
cross A certainly have
the same magnitude.
00:06:47.920 --> 00:06:50.430
They have the same
direction, because each is
00:06:50.430 --> 00:06:52.840
perpendicular to both A and B.
00:06:52.840 --> 00:06:55.230
But notice that the
sense is different.
00:06:55.230 --> 00:06:59.440
Namely, A cross B
tells us to rotate A
00:06:59.440 --> 00:07:02.680
into B through the
smaller of the two angles.
00:07:02.680 --> 00:07:04.470
And in our particular
diagram that
00:07:04.470 --> 00:07:07.020
is what, the
counterclockwise direction?
00:07:07.020 --> 00:07:11.260
On the other hand, B cross
A says, rotate B into A
00:07:11.260 --> 00:07:13.350
through the smaller
of the two angles.
00:07:13.350 --> 00:07:17.800
And if we do that, notice that
we get the clockwise direction
00:07:17.800 --> 00:07:22.030
so that B cross A and A cross
B have the same magnitude.
00:07:22.030 --> 00:07:23.480
They have the same direction.
00:07:23.480 --> 00:07:25.470
But they have the
opposite sense.
00:07:25.470 --> 00:07:29.260
That is, A cross B is the
negative of B cross A.
00:07:29.260 --> 00:07:31.710
And what that means,
you see, in computations
00:07:31.710 --> 00:07:34.700
is that every time you
change the order of A and B
00:07:34.700 --> 00:07:38.560
in a cross product, you must
remember to change the sign.
00:07:38.560 --> 00:07:40.600
In fact, what's
particularly embarrassing is
00:07:40.600 --> 00:07:42.340
if you forgot to do this.
00:07:42.340 --> 00:07:46.070
If this particular term is going
to be added on to something,
00:07:46.070 --> 00:07:47.600
and you forget to
change its sign,
00:07:47.600 --> 00:07:50.820
you're going to be off by more
than just the error of a sign.
00:07:50.820 --> 00:07:54.650
In other words, if something
was supposed to be plus 4,
00:07:54.650 --> 00:07:57.450
and you have it as minus
4 or vice versa, the error
00:07:57.450 --> 00:08:00.540
is going to be 8 if you add
it rather than subtract it,
00:08:00.540 --> 00:08:01.450
you see.
00:08:01.450 --> 00:08:04.390
So the error gets hidden
in other computations.
00:08:04.390 --> 00:08:07.190
In other words, you can change
the order of the factors,
00:08:07.190 --> 00:08:09.830
but you must remember
to change the sign.
00:08:09.830 --> 00:08:13.830
Another interesting thing is
that unlike in the dot product,
00:08:13.830 --> 00:08:17.380
where A dot (B dot C)
didn't even make sense,
00:08:17.380 --> 00:08:23.430
it is true that A cross (B
cross C) and (A cross B) cross C
00:08:23.430 --> 00:08:25.067
do makes sense.
00:08:25.067 --> 00:08:26.650
And by the way, what
are these things?
00:08:26.650 --> 00:08:27.700
They're vectors.
00:08:27.700 --> 00:08:32.830
B cross C is a vector, which is
perpendicular to both B and C.
00:08:32.830 --> 00:08:34.990
It's perpend-- you see.
00:08:34.990 --> 00:08:36.500
And A is a vector.
00:08:36.500 --> 00:08:38.210
And the cross product
of two vectors
00:08:38.210 --> 00:08:39.780
is, again, a vector, et cetera.
00:08:39.780 --> 00:08:41.059
These are two vectors.
00:08:41.059 --> 00:08:43.580
But in general,
these two vectors
00:08:43.580 --> 00:08:45.230
do not have to be equal.
00:08:45.230 --> 00:08:47.280
In fact, as I shall
show you in a moment,
00:08:47.280 --> 00:08:48.710
these two vectors
don't even have
00:08:48.710 --> 00:08:51.470
to be in the same plane,
let alone being equal.
00:08:51.470 --> 00:08:53.530
They don't even have to
be in parallel planes.
00:08:53.530 --> 00:08:56.960
And again, without making
too much of a fuss over this,
00:08:56.960 --> 00:08:59.060
let's take a look and
see if we can see why.
00:08:59.060 --> 00:09:02.130
Let's look at A
cross (B cross C).
00:09:02.130 --> 00:09:04.510
By definition-- let's just
even look at the direction.
00:09:04.510 --> 00:09:06.093
Let's not even worry
about magnitudes.
00:09:06.093 --> 00:09:09.360
What is the direction
of A cross (B cross C)?
00:09:09.360 --> 00:09:10.830
It must be what?
00:09:10.830 --> 00:09:14.270
It must be, on the one
hand, perpendicular
00:09:14.270 --> 00:09:20.300
to A and on the other hand
perpendicular to B cross C. OK?
00:09:20.300 --> 00:09:24.410
That's the definition of
being a cross product.
00:09:24.410 --> 00:09:29.290
On the other hand, what about
B cross C as a vector itself?
00:09:29.290 --> 00:09:34.070
B cross C is perpendicular
to both B and C.
00:09:34.070 --> 00:09:35.490
So what do you have here?
00:09:35.490 --> 00:09:38.190
You have that whatever
A cross (B cross C)
00:09:38.190 --> 00:09:43.260
is, it's perpendicular to B
cross C, which, in turn, is
00:09:43.260 --> 00:09:46.090
perpendicular to both B and C.
00:09:46.090 --> 00:09:47.810
That sounds like
a tongue twister.
00:09:47.810 --> 00:09:50.050
But if you decipher
it, all it says
00:09:50.050 --> 00:09:54.480
is that A cross (B cross
C) is perpendicular
00:09:54.480 --> 00:09:59.450
to the perpendicular to B and C.
And I call that deciphering it.
00:09:59.450 --> 00:10:02.859
What is the perpendicular
to the perpendicular?
00:10:02.859 --> 00:10:04.400
You say let's just
write it this way.
00:10:04.400 --> 00:10:06.130
Here are two vectors
in the plane.
00:10:06.130 --> 00:10:07.470
Here's a perpendicular.
00:10:07.470 --> 00:10:09.760
I now take a
perpendicular to that.
00:10:09.760 --> 00:10:13.230
And that's going to be
parallel to the original plane.
00:10:13.230 --> 00:10:14.910
This is written up in the text.
00:10:14.910 --> 00:10:17.830
It's written up as one of
our learning exercises.
00:10:17.830 --> 00:10:22.350
But just for a quick run-through
now, all I want you to see
00:10:22.350 --> 00:10:26.340
is that A cross (B
cross C) is parallel
00:10:26.340 --> 00:10:29.200
to the plane
determined by B and C.
00:10:29.200 --> 00:10:31.720
In fact, the easy
way to memorize this
00:10:31.720 --> 00:10:34.520
is look at the vector
that's in parentheses.
00:10:34.520 --> 00:10:36.570
And the cross product is what?
00:10:36.570 --> 00:10:39.800
It's parallel to the plane
determined by the two
00:10:39.800 --> 00:10:41.570
vectors in parentheses.
00:10:41.570 --> 00:10:43.480
The only time these two
vectors, by the way,
00:10:43.480 --> 00:10:46.070
won't determine a plane
is if they're parallel.
00:10:46.070 --> 00:10:47.630
But in that case,
the cross product
00:10:47.630 --> 00:10:49.120
would have been zero anyway.
00:10:49.120 --> 00:10:50.800
In other words,
to summarize what
00:10:50.800 --> 00:10:55.520
I've written on the blackboards
here, A cross (B cross C)
00:10:55.520 --> 00:11:00.150
is a vector which lies in the
plane determined by B and C.
00:11:00.150 --> 00:11:02.300
After all, if the vector
is parallel to the plane,
00:11:02.300 --> 00:11:03.980
you can assume it
lies in the plane,
00:11:03.980 --> 00:11:07.070
because you can shift
it parallel to itself.
00:11:07.070 --> 00:11:10.910
And correspondingly,
(A cross B) cross
00:11:10.910 --> 00:11:14.850
C is a vector in the
plane determined by A
00:11:14.850 --> 00:11:17.810
and B. In particular,
there's no reason
00:11:17.810 --> 00:11:20.520
why the AB plane
and the BC plane
00:11:20.520 --> 00:11:22.050
have to be the same plane.
00:11:22.050 --> 00:11:24.690
No law says that our three
vectors, A, B, and C, all
00:11:24.690 --> 00:11:26.300
lie on the same plane.
00:11:26.300 --> 00:11:30.520
So in particular, this is
why the associative property
00:11:30.520 --> 00:11:32.060
doesn't hold for
a cross product.
00:11:32.060 --> 00:11:34.610
Because as soon as you
shift the parentheses,
00:11:34.610 --> 00:11:40.150
you actually shift the plane
in which the vector exists.
00:11:40.150 --> 00:11:40.650
OK?
00:11:40.650 --> 00:11:42.858
But again, I just want to
go through these highlights
00:11:42.858 --> 00:11:44.760
to make sure that
you don't miss them.
00:11:44.760 --> 00:11:48.410
But these are done in
more detail in the notes.
00:11:48.410 --> 00:11:51.610
Before I start blasting
the cross product too much,
00:11:51.610 --> 00:11:54.490
let me point out that
certain properties that we
00:11:54.490 --> 00:11:58.100
like in numerical arithmetic are
present in the cross product.
00:12:01.250 --> 00:12:03.680
For example, there
is a-- I say this,
00:12:03.680 --> 00:12:06.750
there's a little structure.
00:12:06.750 --> 00:12:09.810
The distributive property
holds for the cross product.
00:12:09.810 --> 00:12:12.440
In other words,
if the vector A is
00:12:12.440 --> 00:12:15.430
crossed with the sum of
the two vectors B and C--
00:12:15.430 --> 00:12:19.310
and by the way, again notice,
if B and C are vectors,
00:12:19.310 --> 00:12:20.345
B plus C is a vector.
00:12:20.345 --> 00:12:23.090
I'll put a v down here
to indicate vector.
00:12:23.090 --> 00:12:24.820
A is a vector.
00:12:24.820 --> 00:12:27.680
And the cross product of
two vectors is a vector.
00:12:27.680 --> 00:12:30.089
Notice that, again,
with the cross product,
00:12:30.089 --> 00:12:31.630
all of the things
that you're working
00:12:31.630 --> 00:12:33.240
with happen to be vectors.
00:12:33.240 --> 00:12:37.570
But my claim is that A
crossed with (B plus C) is
00:12:37.570 --> 00:12:40.930
equal to A crossed
with B plus the product
00:12:40.930 --> 00:12:46.540
of A crossed with C. That's a
very nice structural property.
00:12:46.540 --> 00:12:50.140
Again, as usual,
Cartesian coordinates
00:12:50.140 --> 00:12:51.670
have a lot to offer us.
00:12:51.670 --> 00:12:55.370
If we switch to Cartesian
coordinates, again, as usual,
00:12:55.370 --> 00:12:59.380
writing A as a_1*i
plus a_2*j plus a_3*k,
00:12:59.380 --> 00:13:03.510
and B as b_1*i plus
b_2*j plus b_3*k,
00:13:03.510 --> 00:13:08.350
and multiplying term by term,
we observe a rather interesting
00:13:08.350 --> 00:13:09.750
situation.
00:13:09.750 --> 00:13:12.030
You see, notice
things like this.
00:13:12.030 --> 00:13:16.650
When you cross i with i,
j with j, and k with k,
00:13:16.650 --> 00:13:18.980
you're always going to
get the zero vector.
00:13:18.980 --> 00:13:20.650
And the reason for
that is, we already
00:13:20.650 --> 00:13:24.320
saw that since the sine
of the angle between two
00:13:24.320 --> 00:13:28.020
parallel vectors is 0, the
magnitude of the cross product
00:13:28.020 --> 00:13:31.372
is 0, that means that the cross
product is the zero vector.
00:13:31.372 --> 00:13:32.830
The question comes
up, what happens
00:13:32.830 --> 00:13:36.400
when you cross i with j
or j with i, i with k,
00:13:36.400 --> 00:13:38.820
k with j, et cetera?
00:13:38.820 --> 00:13:40.320
I think it's easy
to see that when
00:13:40.320 --> 00:13:42.650
you cross i and j and
j and i, you're going
00:13:42.650 --> 00:13:44.880
to get either k or minus k.
00:13:44.880 --> 00:13:48.870
The right-handed rule is set
up very, very conveniently,
00:13:48.870 --> 00:13:51.550
again, to help us
with a memory device.
00:13:51.550 --> 00:13:55.830
The easiest way to remember how
these things go is i, j, and k
00:13:55.830 --> 00:13:57.930
were set up to form
a right-handed rule,
00:13:57.930 --> 00:13:59.250
a right-handed system.
00:13:59.250 --> 00:14:01.550
Write i, j, and k sequentially.
00:14:01.550 --> 00:14:04.320
See, i, j, k, i,
j, k, et cetera.
00:14:04.320 --> 00:14:08.290
Call the normal order
the positive direction
00:14:08.290 --> 00:14:10.830
and the opposite order
the negative direction.
00:14:10.830 --> 00:14:14.020
For example, when you
want i cross j, just read
00:14:14.020 --> 00:14:17.990
this list as you go
along: i followed by j
00:14:17.990 --> 00:14:20.530
gives k, which is
in the right order--
00:14:20.530 --> 00:14:24.370
see, i, j, k-- so
i cross j is k.
00:14:24.370 --> 00:14:28.040
On the other hand, j
cross i means what?
00:14:28.040 --> 00:14:30.290
j followed by i.
00:14:30.290 --> 00:14:32.330
j followed by i is this way.
00:14:32.330 --> 00:14:33.980
That also yields k.
00:14:33.980 --> 00:14:36.840
But you're now going in
the negative direction.
00:14:36.840 --> 00:14:39.770
To summarize these
results, i cross j
00:14:39.770 --> 00:14:44.386
is the same as minus
j cross i, which is k.
00:14:44.386 --> 00:14:49.160
k cross i is the negative
of i cross k, which is j.
00:14:49.160 --> 00:14:53.610
i cross i, j cross j,
and k cross k are all 0.
00:14:53.610 --> 00:14:55.290
And notice the use
here of zero vector,
00:14:55.290 --> 00:14:57.260
because we're
dealing with vectors.
00:14:57.260 --> 00:14:59.600
And again, leaving
the details to you.
00:14:59.600 --> 00:15:01.950
They're very simple
to come up with.
00:15:01.950 --> 00:15:03.930
Just carry out the
multiplication.
00:15:03.930 --> 00:15:06.790
And what you find
is that A cross B
00:15:06.790 --> 00:15:09.280
is given by the following.
00:15:09.280 --> 00:15:15.880
The i component of the vector
is a_2*b_3 minus a_3*b_2.
00:15:15.880 --> 00:15:21.330
The j component is
a_3*b_1 minus a_1*b_3.
00:15:21.330 --> 00:15:27.290
And the k component is
a_1*b_2 minus a_2*b_1.
00:15:27.290 --> 00:15:30.380
And if you want a convenient
way of memorizing this,
00:15:30.380 --> 00:15:35.820
again, write 1, 2, 3 in their
correct sequential order.
00:15:35.820 --> 00:15:40.080
And what you're saying
is, each term in A cross B
00:15:40.080 --> 00:15:41.700
involves what?
00:15:41.700 --> 00:15:44.620
An a factor followed
by a b factor.
00:15:44.620 --> 00:15:48.530
The subscripts involve the
component, which isn't present.
00:15:48.530 --> 00:15:52.480
See, for the first component,
i, the subscripts are 2 and 3.
00:15:52.480 --> 00:15:56.770
For the second component,
j, the second component,
00:15:56.770 --> 00:15:58.930
the subscripts are 3 and 1.
00:15:58.930 --> 00:16:02.950
And for the third component,
k, the subscripts are 1 and 2.
00:16:02.950 --> 00:16:05.930
And notice that the way
the signs are determined
00:16:05.930 --> 00:16:09.000
is that if the subscripts
occur in the correct sequential
00:16:09.000 --> 00:16:11.980
order, it's positive,
otherwise negative.
00:16:11.980 --> 00:16:15.840
For example, 2 followed
by 3 is the correct order.
00:16:15.840 --> 00:16:19.970
But 3 followed by 2
is the negative order.
00:16:19.970 --> 00:16:26.020
Notice, in this sequence,
1 follows 3, whereas 1 to 3
00:16:26.020 --> 00:16:27.420
is in the negative order.
00:16:27.420 --> 00:16:28.250
See?
00:16:28.250 --> 00:16:31.990
Just the same, this works
out very, very nicely.
00:16:31.990 --> 00:16:33.530
And the reason I
went through this--
00:16:33.530 --> 00:16:35.071
it may seem like an
awful lot of work
00:16:35.071 --> 00:16:38.990
to go through that to show you
how logical this thing is--
00:16:38.990 --> 00:16:43.030
is that something comes up in
the textbook, which is not bad,
00:16:43.030 --> 00:16:47.610
but which I find as frightening
if you have not had experience
00:16:47.610 --> 00:16:50.070
with something
called determinants.
00:16:50.070 --> 00:16:52.830
You see, the author,
Professor Thomas,
00:16:52.830 --> 00:16:55.870
elects to introduce a
convenient device for memorizing
00:16:55.870 --> 00:17:00.610
this result. And he writes
it as a 3 by 3 determinant.
00:17:00.610 --> 00:17:02.980
Now, let me just write down
the 3 by 3 determinant,
00:17:02.980 --> 00:17:04.940
and then we'll talk
about it later.
00:17:04.940 --> 00:17:08.500
I put convenient in quotation
marks to emphasize to you
00:17:08.500 --> 00:17:11.950
that this is convenient only if
you already know determinants.
00:17:11.950 --> 00:17:13.760
If you don't know
determinants, this
00:17:13.760 --> 00:17:16.640
is not only not convenient,
but it's frightening.
00:17:16.640 --> 00:17:18.200
We're always
frightened of things
00:17:18.200 --> 00:17:19.770
that we don't
understand too well.
00:17:19.770 --> 00:17:23.190
And I call it a memory
device, because you do not
00:17:23.190 --> 00:17:25.490
have to have
determinants to remember
00:17:25.490 --> 00:17:29.640
this particular
result. In fact, if one
00:17:29.640 --> 00:17:33.790
were going to list, in order,
the reasons for inventing
00:17:33.790 --> 00:17:36.830
determinants, and the
man was reading off
00:17:36.830 --> 00:17:40.770
of very, very long list,
and you were getting bored,
00:17:40.770 --> 00:17:44.160
as soon as he came to the reason
that it was convenient to list
00:17:44.160 --> 00:17:47.040
the cross product, you could
heave a sigh of relief,
00:17:47.040 --> 00:17:50.530
because you would know that
he's near the end of the list.
00:17:50.530 --> 00:17:52.050
In other words, to
make a long story
00:17:52.050 --> 00:17:56.110
short, of all the reasons
for inventing determinants,
00:17:56.110 --> 00:17:59.450
using them for a cross
product representation
00:17:59.450 --> 00:18:01.375
is perhaps the
least significant.
00:18:01.375 --> 00:18:04.510
But you see, in Professor
Thomas's approach,
00:18:04.510 --> 00:18:06.630
determinants are
assumed to be known.
00:18:06.630 --> 00:18:09.800
And if you do know them,
it's a good notation.
00:18:09.800 --> 00:18:11.670
If you don't, forget it.
00:18:11.670 --> 00:18:13.620
The trouble is, I
tell you to forget it,
00:18:13.620 --> 00:18:15.119
and you won't forget it.
00:18:15.119 --> 00:18:16.160
It always works that way.
00:18:16.160 --> 00:18:18.743
When you tell the student not
to forget something, he forgets.
00:18:18.743 --> 00:18:20.880
When you say, please
ignore it and forget it,
00:18:20.880 --> 00:18:22.130
he doesn't forget.
00:18:22.130 --> 00:18:25.090
So since you may be a little
bit rusty on determinants,
00:18:25.090 --> 00:18:28.840
let me give you a very, very
brief review of determinants.
00:18:28.840 --> 00:18:32.660
In fact, it's not only very
brief, it's a pseudo-review,
00:18:32.660 --> 00:18:35.060
because I will give you no
logic behind this at all.
00:18:35.060 --> 00:18:37.310
I'm just going to
tell you the recipe.
00:18:37.310 --> 00:18:40.060
First of all, by a
2 by 2 determinant,
00:18:40.060 --> 00:18:43.430
you mean a square array
of numbers consisting
00:18:43.430 --> 00:18:45.470
of two rows and two columns.
00:18:45.470 --> 00:18:49.800
You use absolute value signs
to indicate the determinant.
00:18:49.800 --> 00:18:53.183
And by definition, the
2 by 2 determinant a, b;
00:18:53.183 --> 00:19:00.720
c, d is the product of
the two terms a and d,
00:19:00.720 --> 00:19:03.830
minus the product of
the two terms b and c.
00:19:03.830 --> 00:19:07.530
In other words, if you multiply
the upper left by the lower
00:19:07.530 --> 00:19:12.060
right and subtract from
that the upper right times
00:19:12.060 --> 00:19:15.560
the lower left, that is, by
definition, the value of the 2
00:19:15.560 --> 00:19:19.010
by 2 determinant,
just by definition.
00:19:19.010 --> 00:19:20.720
By the way, in block
seven of our course,
00:19:20.720 --> 00:19:22.780
we're going to do this
in much more detail.
00:19:22.780 --> 00:19:25.380
For now I just want to
give you enough of a hint,
00:19:25.380 --> 00:19:27.590
so that if you are
frightened by determinants,
00:19:27.590 --> 00:19:30.250
you can see what Professor
Thomas is talking about
00:19:30.250 --> 00:19:31.510
in the text here.
00:19:31.510 --> 00:19:34.080
Now, to expand a 3
by 3 determinant,
00:19:34.080 --> 00:19:37.120
one uses a little sign code.
00:19:37.120 --> 00:19:39.480
Starting at the upper
left-hand corner,
00:19:39.480 --> 00:19:41.060
you put a little plus sign.
00:19:41.060 --> 00:19:45.430
Then you alternate plus,
minus, plus, et cetera.
00:19:45.430 --> 00:19:48.220
The plus simply is a
code to tell you, leave
00:19:48.220 --> 00:19:50.030
the sign of the term alone.
00:19:50.030 --> 00:19:53.210
And the minus tells you,
change the sign of the term.
00:19:53.210 --> 00:19:54.910
What that means is this.
00:19:54.910 --> 00:19:58.180
By definition, this
3 by 3 determinant
00:19:58.180 --> 00:20:02.350
is obtained as follows--
take x_1 as it appears
00:20:02.350 --> 00:20:06.370
and strike out the row and
column in which x_1 appears
00:20:06.370 --> 00:20:11.080
and multiply x_1 by the
remaining 2 by 2 determinant.
00:20:11.080 --> 00:20:15.460
Then what you do is you factor
out x_2 with a minus sign,
00:20:15.460 --> 00:20:18.690
strike out the row and
column in which x_2 appears,
00:20:18.690 --> 00:20:22.000
and multiply x_2
by the remaining 2
00:20:22.000 --> 00:20:24.090
by 2 determinant that's left.
00:20:24.090 --> 00:20:30.000
Similarly, take out x_3 with
the sign that it appears with,
00:20:30.000 --> 00:20:32.560
strike out the row and
column in which it appears,
00:20:32.560 --> 00:20:35.840
and multiply by the
remaining 2 by 2 determinant.
00:20:35.840 --> 00:20:39.790
Summarized, to expand
this 3 by 3 determinant,
00:20:39.790 --> 00:20:46.110
it's x_1 times the determinant
y_2, y3; z_2, z_3 minus x_2
00:20:46.110 --> 00:20:50.460
times the determinant
y_1, y_3; z_1, z_3,
00:20:50.460 --> 00:20:55.920
plus x_3 times the determinant
y_1, y_2; z_1, z_2.
00:20:55.920 --> 00:20:58.320
Now, that happens to have
very practical application.
00:20:58.320 --> 00:21:01.160
As I say, for this
particular situation,
00:21:01.160 --> 00:21:04.240
I just want to do it with you
to have you see how it works.
00:21:04.240 --> 00:21:07.390
Let me come back to what
we were doing before,
00:21:07.390 --> 00:21:10.190
and show you how, if
we use this definition,
00:21:10.190 --> 00:21:13.410
we can mechanically write
down A cross B in a hurry.
00:21:13.410 --> 00:21:14.760
What did the rule say?
00:21:14.760 --> 00:21:17.550
It says, to expand the
3 by 3 determinant,
00:21:17.550 --> 00:21:22.680
take out i, as it appears,
and multiply by the 2
00:21:22.680 --> 00:21:25.760
by 2 determinant that's left,
when you strike out the row
00:21:25.760 --> 00:21:27.440
and column that i appears in.
00:21:30.230 --> 00:21:31.630
OK?
00:21:31.630 --> 00:21:32.890
Then it says what?
00:21:32.890 --> 00:21:38.590
Take out j with a minus sign,
strike out the row and column,
00:21:38.590 --> 00:21:42.420
multiply by the remaining
2 by 2 determinant.
00:21:45.240 --> 00:21:51.010
Then it says, factor out k and
strike out the row and column
00:21:51.010 --> 00:21:52.350
that k appears in.
00:21:52.350 --> 00:21:57.570
Multiply by the remaining
2 by 2 determinant.
00:21:57.570 --> 00:22:00.270
Well lookit, by our
rule of multiplying a 2
00:22:00.270 --> 00:22:03.030
by 2 determinant, what
is this thing here?
00:22:03.030 --> 00:22:08.500
This is nothing more than
a_2*b_3 minus a_3*b_2.
00:22:11.499 --> 00:22:13.790
I'm not going to go through
this, because time is short
00:22:13.790 --> 00:22:15.920
and you can carry this
operation out by yourself.
00:22:15.920 --> 00:22:18.200
What I would like
you to see, however,
00:22:18.200 --> 00:22:22.810
is that this term here
is precisely the same
00:22:22.810 --> 00:22:25.280
as the term we have here.
00:22:25.280 --> 00:22:28.890
In other words, if I carry
out this mechanical recipe,
00:22:28.890 --> 00:22:31.585
I am going to be able to
get the same answer that I
00:22:31.585 --> 00:22:35.070
got logically, only without
having to worry about keeping
00:22:35.070 --> 00:22:36.660
track of things in my head.
00:22:36.660 --> 00:22:39.170
In other words, if I
do know determinants,
00:22:39.170 --> 00:22:41.930
it is very convenient
to write down
00:22:41.930 --> 00:22:46.050
the cross product of two vectors
using Cartesian coordinates.
00:22:46.050 --> 00:22:48.985
And just to illustrate that,
let me give you a few examples.
00:22:51.500 --> 00:22:53.720
I call this example one.
00:22:53.720 --> 00:22:56.520
Find the vector
perpendicular to the plane,
00:22:56.520 --> 00:23:00.970
determined by the three
points, (1, 2, 3); (5, 9, 4);
00:23:00.970 --> 00:23:03.100
and (7, 6, 8).
00:23:03.100 --> 00:23:07.430
Now, try to visualize
this problem
00:23:07.430 --> 00:23:09.180
if you didn't know vectors.
00:23:09.180 --> 00:23:12.480
Try to visualize drawing
this thing to scale,
00:23:12.480 --> 00:23:14.910
trying to imagine what
slope the plane has
00:23:14.910 --> 00:23:17.445
and how you find a
vector, or a line
00:23:17.445 --> 00:23:18.570
perpendicular to the plane.
00:23:18.570 --> 00:23:20.986
And by the way, I don't even
need the word vector in here.
00:23:20.986 --> 00:23:23.600
Later I could say,
find a direction.
00:23:23.600 --> 00:23:26.870
But we'll worry about
that more next time.
00:23:26.870 --> 00:23:29.440
The thing that I want
to emphasize here
00:23:29.440 --> 00:23:32.130
is, I just very
quickly draw myself
00:23:32.130 --> 00:23:33.730
a little convenient device.
00:23:33.730 --> 00:23:35.200
I don't draw it to scale.
00:23:35.200 --> 00:23:37.410
I assume that the points
A, B, and C are not
00:23:37.410 --> 00:23:38.760
in the same straight line.
00:23:38.760 --> 00:23:41.290
And if they are, it'll
become very interesting
00:23:41.290 --> 00:23:44.690
that I simply won't get--
I'll get 0 for a cross product
00:23:44.690 --> 00:23:48.940
to tell me that they don't
form two non-parallel vectors.
00:23:48.940 --> 00:23:52.670
But I mark off A, B, and C.
And the first thing I do,
00:23:52.670 --> 00:23:58.160
as always, is I vectorize--
I draw the lines AB and AC,
00:23:58.160 --> 00:23:59.600
and I vectorize them.
00:23:59.600 --> 00:24:04.050
That forms for me, as usual,
the vector AB and the vector AC.
00:24:04.050 --> 00:24:07.000
Remembering how I
form a vector from two
00:24:07.000 --> 00:24:09.370
points in Cartesian
coordinates, again,
00:24:09.370 --> 00:24:12.540
subtracting the
coordinates of the point
00:24:12.540 --> 00:24:14.630
that I'm going-- from the
point that I'm going to,
00:24:14.630 --> 00:24:17.300
I subtract away the coordinates
of a point that I'm leaving.
00:24:17.300 --> 00:24:22.190
In other words, 5 minus
1, 9 minus 2, 4 minus 3.
00:24:22.190 --> 00:24:27.510
The vector AB, very simply,
is 4i plus 7j plus k.
00:24:27.510 --> 00:24:31.650
And similarly, the vector
AC, very conveniently,
00:24:31.650 --> 00:24:34.700
is 6i plus 4j plus 5k.
00:24:34.700 --> 00:24:39.132
You see, 7 minus 1,
6 minus 2, 8 minus 3.
00:24:39.132 --> 00:24:40.840
Now, I better stop
checking these things.
00:24:40.840 --> 00:24:43.089
Because, with my luck, I've
probably subtracted wrong,
00:24:43.089 --> 00:24:45.680
and you might not notice it if
we just go through it rapidly.
00:24:45.680 --> 00:24:49.220
But look at how quickly
I've got AB and AC now.
00:24:49.220 --> 00:24:50.770
Now, here's the beauty.
00:24:50.770 --> 00:24:52.150
What do I want?
00:24:52.150 --> 00:24:54.680
I want a vector
perpendicular to the plane
00:24:54.680 --> 00:24:56.810
determined by A, B, and C.
00:24:56.810 --> 00:25:01.040
In particular, if I can find a
vector perpendicular to both AB
00:25:01.040 --> 00:25:04.860
and AC, namely if a vector
is perpendicular to two
00:25:04.860 --> 00:25:07.540
lines in a plane, it's
perpendicular to the plane
00:25:07.540 --> 00:25:08.250
itself.
00:25:08.250 --> 00:25:11.280
If I can find a vector
perpendicular to both AB
00:25:11.280 --> 00:25:13.240
and AC, I'm home free.
00:25:13.240 --> 00:25:14.960
That's the vector
I'm looking for,
00:25:14.960 --> 00:25:17.150
or at least a vector
that I'm looking for.
00:25:17.150 --> 00:25:20.090
Do I know, conveniently,
a vector perpendicular
00:25:20.090 --> 00:25:21.560
to two given vectors?
00:25:21.560 --> 00:25:24.300
And the answer had darn
well better hinge somehow
00:25:24.300 --> 00:25:27.770
on the title of today's lecture,
being the cross product.
00:25:27.770 --> 00:25:31.760
Yes, by definition, if I
cross AB and AC, whatever
00:25:31.760 --> 00:25:35.720
vector I get is going to be
perpendicular to the plane
00:25:35.720 --> 00:25:37.470
determined by A, B, and C.
00:25:37.470 --> 00:25:39.800
And now what I'm saying
is that the beauty
00:25:39.800 --> 00:25:43.710
of Cartesian coordinates is that
I can now very quickly write
00:25:43.710 --> 00:25:46.940
down what AB cross AC
is as a determinant.
00:25:46.940 --> 00:25:49.410
Namely, I write down i, j, k.
00:25:49.410 --> 00:25:51.870
Then I write down
the components of AB.
00:25:51.870 --> 00:25:54.277
Then I write down
the components of AC.
00:25:54.277 --> 00:25:57.370
See, 4, 7, 1; 6, 4, 5.
00:25:57.370 --> 00:25:58.860
And now I do what?
00:25:58.860 --> 00:26:03.090
I strike out the row and
column in which i appears.
00:26:03.090 --> 00:26:05.770
And the remaining 2 by
2 determinant is what?
00:26:05.770 --> 00:26:11.640
7 times 5 minus 1 times 4--
in other words, 35 minus 4.
00:26:11.640 --> 00:26:14.950
I take out minus j, strike
out the row and column
00:26:14.950 --> 00:26:16.440
in which j appears.
00:26:16.440 --> 00:26:19.570
The remaining 2 by 2
determinant is 4 times
00:26:19.570 --> 00:26:24.330
5, minus 1 times 6-- in
other words, 20 minus 6.
00:26:24.330 --> 00:26:27.470
I strike out the row and
column in which k appears.
00:26:27.470 --> 00:26:34.680
The remaining 2 by 2 determinant
is 4 times 4 minus 7 times 6.
00:26:34.680 --> 00:26:36.800
That's 16 minus 42.
00:26:36.800 --> 00:26:39.370
And now simplifying,
what do I have?
00:26:39.370 --> 00:26:47.230
I have 31i minus 14j minus 26k.
00:26:47.230 --> 00:26:52.920
In other words, the vector
31i minus 14j minus 26k
00:26:52.920 --> 00:26:55.070
is the vector that
I'm looking for.
00:26:55.070 --> 00:26:58.290
It's a vector
perpendicular to the plane.
00:26:58.290 --> 00:27:02.490
And what I would like you
to see is how convenient
00:27:02.490 --> 00:27:06.990
this vector arithmetic is
to studying space geometry.
00:27:06.990 --> 00:27:08.390
And in the next
lesson, I'm going
00:27:08.390 --> 00:27:10.760
to say some more
about space geometry.
00:27:10.760 --> 00:27:13.850
But for now I thought
maybe another nice example
00:27:13.850 --> 00:27:16.850
before we finish up,
and let it go at that.
00:27:16.850 --> 00:27:20.520
Let me just do one
more example for you.
00:27:20.520 --> 00:27:22.520
And to simplify
the computation, I
00:27:22.520 --> 00:27:25.040
might as well use the
vectors I've already used.
00:27:25.040 --> 00:27:26.700
Let's take the same
three points we had
00:27:26.700 --> 00:27:31.540
before, (1, 2, 3);
(5, 9, 4); (7, 6, 8).
00:27:31.540 --> 00:27:35.110
And we're going to ask, what
is the area of the triangle
00:27:35.110 --> 00:27:37.520
determined by
these three points?
00:27:37.520 --> 00:27:39.910
And the subtlety that I
wanted to bring out here
00:27:39.910 --> 00:27:43.790
was an excuse to show
you how the magnitude
00:27:43.790 --> 00:27:46.860
of the cross product of two
vectors can be interpreted.
00:27:46.860 --> 00:27:50.230
Namely, look at two vectors.
00:27:50.230 --> 00:27:50.730
All right?
00:27:50.730 --> 00:27:52.500
Call them x and y.
00:27:52.500 --> 00:27:54.810
Draw them in to scale
from a common point.
00:27:54.810 --> 00:27:58.090
And let theta be the angle
formed between these two
00:27:58.090 --> 00:28:00.120
vectors.
00:28:00.120 --> 00:28:03.390
Consider the parallelogram
formed by these two vectors.
00:28:03.390 --> 00:28:06.040
The area of a
parallelogram is what?
00:28:06.040 --> 00:28:08.640
It's the base times the height.
00:28:08.640 --> 00:28:10.210
Now, lookit.
00:28:10.210 --> 00:28:15.840
The base of this parallelogram
is the magnitude of y.
00:28:15.840 --> 00:28:17.640
And the height of
this parallelogram--
00:28:17.640 --> 00:28:19.720
remember, this magnitude is x.
00:28:19.720 --> 00:28:22.250
So this height is x sine theta.
00:28:27.720 --> 00:28:32.880
This is the area of
the parallelogram.
00:28:32.880 --> 00:28:35.280
But notice that this
is also, by definition,
00:28:35.280 --> 00:28:37.750
the magnitude of x cross y.
00:28:37.750 --> 00:28:43.390
In other words, notice that
this is, by definition,
00:28:43.390 --> 00:28:45.360
the magnitude of
x cross y. x cross
00:28:45.360 --> 00:28:47.870
y is a vector whose
magnitude is this.
00:28:47.870 --> 00:28:50.640
In other words, the
magnitude of x cross y
00:28:50.640 --> 00:28:55.370
is the area of the parallelogram
determined by x and y,
00:28:55.370 --> 00:28:58.680
which gives you a rather nice,
interesting interpretation.
00:28:58.680 --> 00:29:02.360
Imagine, for example, that x
and y don't change in length.
00:29:02.360 --> 00:29:06.710
Is it clear to you that as
I change the angle between x
00:29:06.710 --> 00:29:09.400
and y, the area of
the parallelogram
00:29:09.400 --> 00:29:12.030
formed by x and y changes?
00:29:12.030 --> 00:29:13.860
In other words,
one extreme case is
00:29:13.860 --> 00:29:16.990
when the angle between x
and y is 0, in which case
00:29:16.990 --> 00:29:18.970
the parallelogram has no area.
00:29:18.970 --> 00:29:22.860
Another extreme case is when
the angle between x and y
00:29:22.860 --> 00:29:23.920
is 90 degrees.
00:29:23.920 --> 00:29:26.410
In other words, x and y are
perpendicular to each other.
00:29:26.410 --> 00:29:29.220
But if the lengths
of x and y remain
00:29:29.220 --> 00:29:32.900
fixed, as I alter the
angle between them,
00:29:32.900 --> 00:29:36.880
I alter the area of the
parallelogram determined
00:29:36.880 --> 00:29:37.550
by them.
00:29:37.550 --> 00:29:38.400
All right?
00:29:38.400 --> 00:29:41.220
I just thought that's a
cute little device whereby
00:29:41.220 --> 00:29:44.630
you can visualize how the
magnitude of the cross product
00:29:44.630 --> 00:29:46.340
of two vectors varies.
00:29:46.340 --> 00:29:50.090
Think of the two vectors
at a common origin,
00:29:50.090 --> 00:29:55.990
and the magnitude is affected by
the area of that parallelogram
00:29:55.990 --> 00:29:57.561
as the angle changes.
00:29:57.561 --> 00:29:58.060
OK?
00:29:58.060 --> 00:29:59.460
In other words, if
you change the angle,
00:29:59.460 --> 00:30:01.130
the area of the
parallelogram changes.
00:30:01.130 --> 00:30:03.750
But at any rate, getting
back to our original problem,
00:30:03.750 --> 00:30:06.040
the corollary to that
is the following.
00:30:06.040 --> 00:30:08.510
We already know what
the magnitude of AB
00:30:08.510 --> 00:30:11.620
cross AC is from example one.
00:30:11.620 --> 00:30:13.900
But what does that mean
now, geometrically?
00:30:13.900 --> 00:30:17.440
From what we've just
seen, the magnitude of AB
00:30:17.440 --> 00:30:22.060
cross AC is the area of
the parallelogram, which
00:30:22.060 --> 00:30:25.980
has AB and AC as edges.
00:30:25.980 --> 00:30:31.975
Notice that that parallelogram
is precisely one-half-- I
00:30:31.975 --> 00:30:34.320
shouldn't say that-- the
triangle that we're looking for
00:30:34.320 --> 00:30:37.060
is precisely one-half
the parallelogram.
00:30:37.060 --> 00:30:40.360
In other words, notice that
the area of the triangle
00:30:40.360 --> 00:30:44.420
is simply 1/2 the
magnitude of AB cross AC.
00:30:44.420 --> 00:30:47.020
How do we find the
magnitude of AB cross C?
00:30:47.020 --> 00:30:50.020
We square each of
its components,
00:30:50.020 --> 00:30:54.480
add them, and extract the
positive square root, as usual.
00:30:54.480 --> 00:30:55.950
Without going
through the details,
00:30:55.950 --> 00:30:57.740
it's simply going to be what?
00:30:57.740 --> 00:31:00.960
This particular amount.
00:31:00.960 --> 00:31:04.820
And the rest is arithmetic's
baby, so to speak.
00:31:04.820 --> 00:31:05.590
All right?
00:31:05.590 --> 00:31:09.930
Now, what I'm hoping is that
by this stage of the game,
00:31:09.930 --> 00:31:12.970
you now have a better
feeling as to what
00:31:12.970 --> 00:31:15.400
the cross product means.
00:31:15.400 --> 00:31:17.150
Going back to the
beginning of my lecture,
00:31:17.150 --> 00:31:20.040
notice that I played down
the physical applications.
00:31:20.040 --> 00:31:22.540
Because one of the things
that I wanted you to see
00:31:22.540 --> 00:31:26.030
is that the cross product
has tremendous application,
00:31:26.030 --> 00:31:30.180
just to space geometry alone,
quite apart from anything else.
00:31:30.180 --> 00:31:32.240
Well, at any rate, what
I'm going to do next time
00:31:32.240 --> 00:31:34.970
is-- now that we've talked
about lines and planes
00:31:34.970 --> 00:31:39.290
and what have you-- to get down
to the Cartesian representation
00:31:39.290 --> 00:31:41.440
of what lines and
planes would look like.
00:31:41.440 --> 00:31:44.440
They play a very vital role
in the study of functions
00:31:44.440 --> 00:31:45.530
of several variables.
00:31:45.530 --> 00:31:49.130
We'll see more about that as
the course continues to unfold.
00:31:49.130 --> 00:31:52.740
So until next time, goodbye.
00:31:52.740 --> 00:31:55.100
Funding for the
publication of this video
00:31:55.100 --> 00:31:59.990
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:31:59.990 --> 00:32:04.160
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00:32:04.160 --> 00:32:08.578
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