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PROFESSOR: Hi.
00:00:49.650 --> 00:00:52.380
Our unit today
concerns something
00:00:52.380 --> 00:00:54.490
called the dot product.
00:00:54.490 --> 00:00:56.980
Now it's very easy
to just mechanically
00:00:56.980 --> 00:00:59.080
give a definition
of the dot product,
00:00:59.080 --> 00:01:02.380
but in keeping with the
spirit both of our game
00:01:02.380 --> 00:01:05.030
and of our correlation
between the real
00:01:05.030 --> 00:01:08.820
and the abstract world, let's
keep in mind that when it came
00:01:08.820 --> 00:01:11.980
time to define vector
addition, we chose,
00:01:11.980 --> 00:01:15.220
as our definition, a
concept already used
00:01:15.220 --> 00:01:19.780
in the physical world-- namely,
that of a resultant vector.
00:01:19.780 --> 00:01:21.840
And now what we
would like to do is
00:01:21.840 --> 00:01:25.680
to introduce a further structure
in our game of vectors.
00:01:25.680 --> 00:01:28.100
And by the way, keep
in mind that I do not
00:01:28.100 --> 00:01:30.760
need any more physical
interpretations
00:01:30.760 --> 00:01:33.550
to add more
complexities to my game,
00:01:33.550 --> 00:01:35.970
but that perhaps
if we do this, it
00:01:35.970 --> 00:01:38.490
makes the subject
more meaningful,
00:01:38.490 --> 00:01:41.950
both to the applied person
and to the theoretical person.
00:01:41.950 --> 00:01:45.380
The idea that I wanted to
keep in mind for motivating
00:01:45.380 --> 00:01:50.500
today's lesson was the old
high school idea of work
00:01:50.500 --> 00:01:55.720
equals force times distance in
the elementary physics course.
00:01:55.720 --> 00:01:58.980
Well rather than talk on like
this, let's just take a look
00:01:58.980 --> 00:02:01.380
and see what's going on.
00:02:01.380 --> 00:02:04.260
Let's say the lecture is
called "The Dot Product,"
00:02:04.260 --> 00:02:08.410
and our physical
motivation is the recipe
00:02:08.410 --> 00:02:12.870
from elementary physics, work
equals force times distance.
00:02:12.870 --> 00:02:16.340
Now this is a deceptive
little formula.
00:02:16.340 --> 00:02:18.120
First of all, what
it really meant was,
00:02:18.120 --> 00:02:21.460
when we learned this
recipe, was that the force
00:02:21.460 --> 00:02:24.650
was taking place in
the same direction
00:02:24.650 --> 00:02:26.740
as the object was moving.
00:02:26.740 --> 00:02:31.150
Now what happened next was
the following situation.
00:02:31.150 --> 00:02:35.590
We have an object, say,
being moved by a force.
00:02:35.590 --> 00:02:39.370
The object is being moved
along a tabletop, say.
00:02:39.370 --> 00:02:43.440
The force is represented
now by a vector, an arrow.
00:02:43.440 --> 00:02:45.440
And we're assuming
enough friction
00:02:45.440 --> 00:02:49.310
here so that the block
moves along the table
00:02:49.310 --> 00:02:50.540
and doesn't rise.
00:02:50.540 --> 00:02:54.100
And the question is, how much
work is done on the object
00:02:54.100 --> 00:02:57.660
as the object moves, say,
from this point to this point?
00:02:57.660 --> 00:02:59.830
Now without worrying
about what motivates
00:02:59.830 --> 00:03:02.620
this thing physically,
the important thing was,
00:03:02.620 --> 00:03:05.020
is that people observe
that physically,
00:03:05.020 --> 00:03:07.320
the only thing that
went into the work
00:03:07.320 --> 00:03:10.820
was the component of the
force in the direction
00:03:10.820 --> 00:03:12.000
of the displacement.
00:03:12.000 --> 00:03:18.580
In other words,
this was the force
00:03:18.580 --> 00:03:21.890
that one found had to be
multiplied by the displacement.
00:03:21.890 --> 00:03:25.070
In other words, to find
the work being done,
00:03:25.070 --> 00:03:30.420
one took not the magnitude of
F, but rather the component of F
00:03:30.420 --> 00:03:31.830
in the direction of s.
00:03:31.830 --> 00:03:34.550
And in this diagram,
that's just what?
00:03:34.550 --> 00:03:39.430
It's the magnitude of
F times cosine theta.
00:03:39.430 --> 00:03:42.420
And that quantity, which was
called the effective force,
00:03:42.420 --> 00:03:46.510
was then multiplied
by the displacement.
00:03:46.510 --> 00:03:49.340
And to write that in
more suggestive form,
00:03:49.340 --> 00:03:52.100
notice that the work
was not the magnitude
00:03:52.100 --> 00:03:54.980
of the force times the
magnitude of the displacement.
00:03:54.980 --> 00:03:57.780
That was only true
in the special case
00:03:57.780 --> 00:03:59.640
where the force and
the displacement
00:03:59.640 --> 00:04:01.020
were in the same direction.
00:04:01.020 --> 00:04:03.470
But that rather
the work is what?
00:04:03.470 --> 00:04:06.460
It's the magnitude of the
force times the magnitude
00:04:06.460 --> 00:04:08.790
of the displacement
times the cosine
00:04:08.790 --> 00:04:13.520
of the angle between the
force and the displacement.
00:04:13.520 --> 00:04:16.440
And notice by the way,
the very special cases,
00:04:16.440 --> 00:04:20.440
that if F and s happen
to be parallel and have
00:04:20.440 --> 00:04:27.270
the same sense, then the
angle between F and s is 0.
00:04:27.270 --> 00:04:31.670
The cosine of 0 is 1, in which
case that you have the work
00:04:31.670 --> 00:04:35.170
is the magnitude of the
force times the displacement.
00:04:35.170 --> 00:04:38.050
The other extreme
case magnitude-wise
00:04:38.050 --> 00:04:42.750
is if s and F happen to be
perpendicular, in which case
00:04:42.750 --> 00:04:44.760
the angle, of course,
is 90 degrees.
00:04:44.760 --> 00:04:48.150
The cosine of a
90-degree angle is 0.
00:04:48.150 --> 00:04:49.230
In which case, what?
00:04:49.230 --> 00:04:52.360
If the force was at right
angles to the displacement,
00:04:52.360 --> 00:04:54.340
the work was 0.
00:04:54.340 --> 00:04:56.550
At any rate then,
whether we understand
00:04:56.550 --> 00:04:59.120
this physical motivation
or not is irrelevant.
00:04:59.120 --> 00:05:00.930
The important point
is, if we want
00:05:00.930 --> 00:05:05.300
to keep this structure or
this particular motivation
00:05:05.300 --> 00:05:09.230
in a structural form, we now
generalize this as follows.
00:05:09.230 --> 00:05:14.000
We simply say, let A and B
be any two vectors, arrows.
00:05:14.000 --> 00:05:18.370
And we will define A dot B to
be the magnitude of A times
00:05:18.370 --> 00:05:23.320
the magnitude of B times the
cosine of the angle between A
00:05:23.320 --> 00:05:27.310
and B. And I write this this
way to indicate an ordering.
00:05:27.310 --> 00:05:31.360
In other words, don't think
of A and B as being A and B.
00:05:31.360 --> 00:05:33.860
Think of A as denoting
the first and B
00:05:33.860 --> 00:05:35.620
as denoting the second vector.
00:05:35.620 --> 00:05:37.650
And what we're saying
is the first dotted
00:05:37.650 --> 00:05:40.550
with the second is the
magnitude of the first,
00:05:40.550 --> 00:05:42.550
times the magnitude
of the second,
00:05:42.550 --> 00:05:46.640
times the cosine of the angle
as you rotate the first vector
00:05:46.640 --> 00:05:48.090
into the second.
00:05:48.090 --> 00:05:50.710
And the beauty of having
a cosine over here
00:05:50.710 --> 00:05:52.100
is the fact that what?
00:05:52.100 --> 00:05:55.910
If you reverse the angle of
rotation-- in other words,
00:05:55.910 --> 00:06:00.950
from B into A-- notice that you
change the sign of the angle,
00:06:00.950 --> 00:06:04.820
but the cosine of theta is the
same as cosine minus theta,
00:06:04.820 --> 00:06:07.570
so no harm is done
this particular way.
00:06:07.570 --> 00:06:10.990
On the other hand, had we been
dealing with sine of the angle,
00:06:10.990 --> 00:06:14.550
as we will in our next lecture,
this will make a difference.
00:06:14.550 --> 00:06:17.380
But be this as it
may, we define A dot
00:06:17.380 --> 00:06:20.630
B to be the magnitude of
A times the magnitude of B
00:06:20.630 --> 00:06:24.560
times the cosine of the angle
between A and B. All right?
00:06:24.560 --> 00:06:26.300
The two extreme
cases being what?
00:06:26.300 --> 00:06:30.750
If A and B are perpendicular,
the dot product is 0.
00:06:30.750 --> 00:06:34.360
If A and B are parallel,
the dot product
00:06:34.360 --> 00:06:40.190
is equal to the magnitude of the
product of the two magnitudes.
00:06:40.190 --> 00:06:43.460
Now the only difficult thing
here, or what we sometimes
00:06:43.460 --> 00:06:45.570
call undesirable--
and maybe this
00:06:45.570 --> 00:06:50.280
is what separates the new
three-dimensional geometry
00:06:50.280 --> 00:06:52.480
from the old
three-dimensional geometry--
00:06:52.480 --> 00:06:55.342
is that the cosine of an angle
is particularly difficult
00:06:55.342 --> 00:06:57.300
to keep track of, especially
when the lines are
00:06:57.300 --> 00:06:58.750
in three-dimensional space.
00:06:58.750 --> 00:07:00.890
How do you measure
an angle this way?
00:07:00.890 --> 00:07:02.620
You see, in the
plane, it's simple.
00:07:02.620 --> 00:07:04.050
You draw the thing to scale.
00:07:04.050 --> 00:07:06.860
But in three-space, this
can be rather difficult.
00:07:06.860 --> 00:07:09.650
So what we would like
to do is to eliminate
00:07:09.650 --> 00:07:10.720
this particular term.
00:07:10.720 --> 00:07:13.890
We would like to find an
expression for A dot B
00:07:13.890 --> 00:07:18.620
that doesn't involve the cosine
of an angle, at least directly.
00:07:18.620 --> 00:07:21.670
And to do this, we simply
draw a little diagram.
00:07:21.670 --> 00:07:25.420
And notice that even if A and B
are three-dimensional vectors,
00:07:25.420 --> 00:07:29.010
since A and B are lines,
if they are not parallel,
00:07:29.010 --> 00:07:31.190
if they emanate
at a common point,
00:07:31.190 --> 00:07:32.560
they form a plane of their own.
00:07:32.560 --> 00:07:34.640
Let's call that the
plane of the blackboard.
00:07:34.640 --> 00:07:37.190
The third side of the
triangle is either
00:07:37.190 --> 00:07:39.730
A minus B or B minus
A, depending on where
00:07:39.730 --> 00:07:40.900
you put the arrowhead here.
00:07:40.900 --> 00:07:43.204
But we've already
discussed that idea.
00:07:43.204 --> 00:07:44.870
And the interesting
point here is notice
00:07:44.870 --> 00:07:50.050
that A minus B very subtly
includes the angle theta.
00:07:50.050 --> 00:07:52.930
In other words, imagine
the magnitudes of A and B
00:07:52.930 --> 00:07:54.070
to be fixed.
00:07:54.070 --> 00:07:59.040
And now fan out A and B.
As you fan out A and B,
00:07:59.040 --> 00:08:00.270
what you're doing is what?
00:08:00.270 --> 00:08:01.790
Just changing this angle.
00:08:01.790 --> 00:08:05.540
As you change this angle,
notice that A minus B changes.
00:08:05.540 --> 00:08:08.170
Namely, as these
fan out, the vector
00:08:08.170 --> 00:08:10.940
that joins the two
arrowheads here
00:08:10.940 --> 00:08:14.550
becomes a different vector,
both in magnitude and direction.
00:08:14.550 --> 00:08:16.460
In other words, whether
it looks it or not,
00:08:16.460 --> 00:08:19.070
one of the beauties
of our vector notation
00:08:19.070 --> 00:08:21.630
is that the cosine
of the angle theta
00:08:21.630 --> 00:08:25.400
is indirectly
included in A minus B.
00:08:25.400 --> 00:08:28.590
But now you see, once we
have this triangle here,
00:08:28.590 --> 00:08:31.550
notice that the Law
of Cosines tells us
00:08:31.550 --> 00:08:35.370
how to relate the third side
of a triangle in terms of two
00:08:35.370 --> 00:08:37.580
sides and the included angle.
00:08:37.580 --> 00:08:38.179
OK?
00:08:38.179 --> 00:08:41.309
And in fact, when we
use the Law of Cosines,
00:08:41.309 --> 00:08:44.250
notice that one of the
terms is going to be what?
00:08:44.250 --> 00:08:47.480
The product of the
magnitudes of two sides
00:08:47.480 --> 00:08:49.950
times the cosine of
the included angle.
00:08:49.950 --> 00:08:55.340
And that, roughly speaking, is
just what we mean by A dot B.
00:08:55.340 --> 00:08:57.590
So without any
further ado, what we
00:08:57.590 --> 00:09:00.040
do now is we just write
the Law of Cosines
00:09:00.040 --> 00:09:01.840
down here, which says what?
00:09:01.840 --> 00:09:04.730
The magnitude of
A minus B squared
00:09:04.730 --> 00:09:08.610
is equal to the magnitude of A
squared plus the magnitude of B
00:09:08.610 --> 00:09:11.870
squared minus twice the
magnitude of A times
00:09:11.870 --> 00:09:14.990
the magnitude of B times the
cosine of the angle between A
00:09:14.990 --> 00:09:18.410
and B. And then,
you see, we simply
00:09:18.410 --> 00:09:21.900
recognize that this term
here is, by definition,
00:09:21.900 --> 00:09:25.880
A dot B. We can now
take this equation
00:09:25.880 --> 00:09:28.260
and solve for A dot
B-- which, by the way,
00:09:28.260 --> 00:09:30.230
this is very, very
important to notice.
00:09:30.230 --> 00:09:32.030
I should have pointed
this out sooner.
00:09:32.030 --> 00:09:34.440
But A dot B is a number.
00:09:34.440 --> 00:09:37.810
It's a product of two
magnitudes times the cosine
00:09:37.810 --> 00:09:39.520
of an angle, which is a number.
00:09:39.520 --> 00:09:41.790
This is a numerical equation.
00:09:41.790 --> 00:09:46.460
We can therefore solve for A
dot B. And we wind up with what?
00:09:46.460 --> 00:09:50.140
That A dot B is the magnitude
of A minus B squared
00:09:50.140 --> 00:09:53.940
minus the magnitude of A
squared minus the magnitude of B
00:09:53.940 --> 00:09:56.210
squared, all divided by 2.
00:09:56.210 --> 00:10:00.570
And the beauty now is that we
have expressed A dot B solely
00:10:00.570 --> 00:10:03.200
in terms of magnitudes.
00:10:03.200 --> 00:10:05.110
And notice especially
in Cartesian
00:10:05.110 --> 00:10:07.120
coordinates-- and
I'll do that next--
00:10:07.120 --> 00:10:09.270
but in terms of
Cartesian coordinates,
00:10:09.270 --> 00:10:12.540
notice that magnitudes are
particularly simple to find.
00:10:12.540 --> 00:10:15.090
We just subtract
corresponding components
00:10:15.090 --> 00:10:16.850
and square, et cetera.
00:10:16.850 --> 00:10:18.520
But the important
point is that even
00:10:18.520 --> 00:10:21.780
without Cartesian coordinates,
this particular result,
00:10:21.780 --> 00:10:26.970
expressed as A dot B in terms
of the magnitudes of A, B, and A
00:10:26.970 --> 00:10:29.690
minus B, and is
a result which is
00:10:29.690 --> 00:10:32.320
independent of any
coordinate system.
00:10:32.320 --> 00:10:35.570
However-- and this is done
very simply in the text,
00:10:35.570 --> 00:10:38.190
reinforced in our
exercises-- if you
00:10:38.190 --> 00:10:44.550
elect to write A, B and A minus
B in Cartesian coordinates
00:10:44.550 --> 00:10:48.200
and use this particularly
straightforward recipe, what
00:10:48.200 --> 00:10:51.770
we wind up with is a
rather elegant result--
00:10:51.770 --> 00:10:54.160
elegant in terms of
simplicity, at least.
00:10:54.160 --> 00:10:56.740
And that is-- remember
in Cartesian coordinates,
00:10:56.740 --> 00:11:01.062
we would write A as a_1*i
plus a_2*j plus a_3*k.
00:11:01.062 --> 00:11:05.120
B would be b_1*i plus
b_2*j plus b_3*k.
00:11:05.120 --> 00:11:06.450
And then the beauty is what?
00:11:06.450 --> 00:11:11.060
That A dot B turns out to be
very simply and conveniently
00:11:11.060 --> 00:11:15.900
a_1*b_1 plus a_2*b_2
plus a_3*b_3.
00:11:15.900 --> 00:11:19.280
In other words,
that to dot A and B,
00:11:19.280 --> 00:11:21.380
if the vectors are
written in Cartesian
00:11:21.380 --> 00:11:23.170
coordinates-- and
this is crucial.
00:11:23.170 --> 00:11:25.680
If this is not done in
Cartesian coordinates,
00:11:25.680 --> 00:11:27.930
you can get into
a heck of a mess.
00:11:27.930 --> 00:11:31.620
And I have deliberately made
an exercise on this unit,
00:11:31.620 --> 00:11:35.040
get you into that mess, if you
fall into that particular trap.
00:11:35.040 --> 00:11:37.970
But if we have
Cartesian coordinates,
00:11:37.970 --> 00:11:41.880
it turns out that to dot two
vectors, you simply do what?
00:11:41.880 --> 00:11:44.500
Multiply the two i
components together.
00:11:44.500 --> 00:11:47.110
Multiply the two j
components together.
00:11:47.110 --> 00:11:50.695
Multiply the two k
components together, and add.
00:11:50.695 --> 00:11:51.740
All right?
00:11:51.740 --> 00:11:54.460
By the way, to show
you why this works
00:11:54.460 --> 00:11:56.610
from a structural point of
view, without belaboring
00:11:56.610 --> 00:11:58.830
this point right now,
notice that if you
00:11:58.830 --> 00:12:01.730
were to multiply in the
usual sense of the word
00:12:01.730 --> 00:12:04.520
"multiplication," form
the dot product here,
00:12:04.520 --> 00:12:06.210
you would expect
to get nine terms.
00:12:06.210 --> 00:12:08.820
In other words, each
of the terms in A
00:12:08.820 --> 00:12:12.420
multiplies each of
the three terms in B.
00:12:12.420 --> 00:12:15.002
So that altogether you
would expect nine terms.
00:12:15.002 --> 00:12:16.460
The thing that's
rather interesting
00:12:16.460 --> 00:12:21.400
here is that notice that i
dot i, j dot j, and k dot
00:12:21.400 --> 00:12:23.580
k all happen to be 1.
00:12:23.580 --> 00:12:26.050
Because after all, the
magnitudes of these vectors
00:12:26.050 --> 00:12:27.060
are each 1.
00:12:27.060 --> 00:12:30.580
The angle between our i
and i is 0, j and j is 0.
00:12:30.580 --> 00:12:33.040
The angle between k and k is 0.
00:12:33.040 --> 00:12:37.710
So that i dot i, j dot
j, and k dot k are all 1.
00:12:37.710 --> 00:12:40.940
Whereas on the other hand, when
you take mixed terms, notice
00:12:40.940 --> 00:12:45.320
that because i and j,
i and k, and j and k
00:12:45.320 --> 00:12:49.360
are all at right angles,
i dot j, j dot k,
00:12:49.360 --> 00:12:52.270
i dot k are all going to be 0.
00:12:52.270 --> 00:12:55.520
And that therefore, those
other six terms will drop out.
00:12:55.520 --> 00:12:58.140
In other words, structurally
what's happening here
00:12:58.140 --> 00:13:01.210
is the fact that the three
vectors that we're using here
00:13:01.210 --> 00:13:03.650
all happen to have unit length.
00:13:03.650 --> 00:13:07.170
And they happen to be
mutually perpendicular.
00:13:07.170 --> 00:13:11.000
If they were not perpendicular,
these mixed terms
00:13:11.000 --> 00:13:12.020
would appear in here.
00:13:12.020 --> 00:13:13.780
In other words, in
general, when you
00:13:13.780 --> 00:13:17.100
dot two vectors in
three-space, depending
00:13:17.100 --> 00:13:19.760
on the coordinate
system, you can expect up
00:13:19.760 --> 00:13:22.370
to nine terms in your answer.
00:13:22.370 --> 00:13:24.430
But the beauty is
that as long as we
00:13:24.430 --> 00:13:26.150
have Cartesian
coordinates, there
00:13:26.150 --> 00:13:29.910
happens to be a particularly
simple, beautiful recipe
00:13:29.910 --> 00:13:31.460
to compute A dot B.
00:13:31.460 --> 00:13:34.800
Now keep in mind, the A dot B
that we're talking about here
00:13:34.800 --> 00:13:37.030
is the same one that
we defined before.
00:13:37.030 --> 00:13:39.210
It's the magnitude of
A times the magnitude
00:13:39.210 --> 00:13:41.830
of B times the cosine
of the angle between A
00:13:41.830 --> 00:13:43.940
and B. All we're
saying is that if we
00:13:43.940 --> 00:13:46.950
use Cartesian coordinates,
we can compute it
00:13:46.950 --> 00:13:49.340
almost as fast as we can read.
00:13:49.340 --> 00:13:53.260
And let me show you that
in terms of some examples.
00:13:53.260 --> 00:13:55.670
My first example
is the following.
00:13:55.670 --> 00:13:57.790
Let's imagine that
we have three points
00:13:57.790 --> 00:13:59.500
in Cartesian three-space.
00:13:59.500 --> 00:14:02.110
A is the point 1
comma 2 comma 3,
00:14:02.110 --> 00:14:04.680
B is the point 2
comma 4 comma 1,
00:14:04.680 --> 00:14:07.730
and C is the point
3 comma 0 comma 4.
00:14:07.730 --> 00:14:10.920
We draw the straight
lines AB and AC,
00:14:10.920 --> 00:14:14.880
and we would like to find the
angle BAC-- in other words,
00:14:14.880 --> 00:14:16.980
the angle theta.
00:14:16.980 --> 00:14:19.200
The first thing that
we do-- and this
00:14:19.200 --> 00:14:22.490
is one of the beauties of how
vectors are used in geometry--
00:14:22.490 --> 00:14:25.470
is that we vectorize
the lines A and B.
00:14:25.470 --> 00:14:27.070
We put arrowheads on them.
00:14:27.070 --> 00:14:29.450
That immediately
makes them vectors.
00:14:29.450 --> 00:14:31.860
We already know
from last time how
00:14:31.860 --> 00:14:34.950
to read the vectors AB and AC.
00:14:34.950 --> 00:14:39.910
Namely, AB is the vector i
plus 2j-- see, just subtract
00:14:39.910 --> 00:14:41.700
component by component.
00:14:41.700 --> 00:14:43.180
2 minus 1 is 1.
00:14:43.180 --> 00:14:45.300
4 minus 2 is 2.
00:14:45.300 --> 00:14:48.220
1 minus 3 is minus 3, et cetera.
00:14:48.220 --> 00:14:53.890
So that the vector AB is the
vector i plus 2j minus 2k.
00:14:53.890 --> 00:14:57.940
And the vector AC, working
in a similar way, is 2i
00:14:57.940 --> 00:15:00.320
minus 2j plus k.
00:15:00.320 --> 00:15:03.230
Now the beauty is that we can
compute these magnitudes very,
00:15:03.230 --> 00:15:04.870
very quickly by recipe.
00:15:04.870 --> 00:15:06.270
And we've just
learned the recipe
00:15:06.270 --> 00:15:08.216
for finding A dot B in a hurry.
00:15:08.216 --> 00:15:09.632
I mean, well in
this case, I don't
00:15:09.632 --> 00:15:14.010
mean A dot B. I mean the vector
AB dotted with the vector AC.
00:15:14.010 --> 00:15:16.190
And going through the
computational details
00:15:16.190 --> 00:15:19.310
here, we square the
components of AB,
00:15:19.310 --> 00:15:21.460
extract the positive
square root,
00:15:21.460 --> 00:15:26.150
and we find very easily that
the magnitude of AB is 3.
00:15:26.150 --> 00:15:29.320
And the hardest part of these
problems for me, as a teacher,
00:15:29.320 --> 00:15:33.010
is to find ones where I find
the sum of 3 squares coming out
00:15:33.010 --> 00:15:34.130
to be a whole number.
00:15:34.130 --> 00:15:36.500
So I always use the
vector [1, 2, 2],
00:15:36.500 --> 00:15:38.150
because that's a
nice vector that way.
00:15:38.150 --> 00:15:43.740
Similarly, the vector AC also
happens to have magnitude 3.
00:15:43.740 --> 00:15:44.720
OK?
00:15:44.720 --> 00:15:48.326
And to find AB dot
AC, what do we do?
00:15:48.326 --> 00:15:49.950
Let's just come back
here and make sure
00:15:49.950 --> 00:15:51.350
we know what we're doing now.
00:15:51.350 --> 00:15:54.260
We simply dot
component by component.
00:15:54.260 --> 00:16:00.240
It's 1 times 2, plus 2 times
minus 2, plus minus 2 times 1.
00:16:00.240 --> 00:16:04.550
In other words, AB dot
AC is 2 minus 4 minus 2,
00:16:04.550 --> 00:16:06.490
which is minus 4.
00:16:06.490 --> 00:16:09.620
Now using our
recipe, we see what?
00:16:09.620 --> 00:16:15.090
That cosine theta
is AB dot AC divided
00:16:15.090 --> 00:16:18.150
by the product of the
magnitudes of AB and AC,
00:16:18.150 --> 00:16:20.440
from which we very
quickly conclude
00:16:20.440 --> 00:16:24.390
that the cosine of
theta is minus 4/9.
00:16:24.390 --> 00:16:27.280
And if you're still mixed up as
to what that minus sign means,
00:16:27.280 --> 00:16:30.490
just by way of a quick review
of the inverse trigonometric
00:16:30.490 --> 00:16:34.110
functions, you locate
the point minus 4 comma
00:16:34.110 --> 00:16:37.630
9 in the xy-plane,
and your angle theta
00:16:37.630 --> 00:16:40.230
is this particular
angle here, which
00:16:40.230 --> 00:16:42.740
means that in terms
of principal values,
00:16:42.740 --> 00:16:45.730
if you look up the angle in
the tables whose cosine is
00:16:45.730 --> 00:16:48.600
4/9, that will give
you this angle here.
00:16:48.600 --> 00:16:50.530
Subtract that from 180.
00:16:50.530 --> 00:16:52.650
And that's the angle
that you're looking for.
00:16:52.650 --> 00:16:54.120
But the beauty is what?
00:16:54.120 --> 00:16:59.720
That you can now find an angle
between two lines in space
00:16:59.720 --> 00:17:02.770
without having to
geometrically worry about what
00:17:02.770 --> 00:17:03.690
the angle looks like.
00:17:03.690 --> 00:17:06.839
The algebra in Cartesian
coordinates takes care of this
00:17:06.839 --> 00:17:08.040
by itself.
00:17:08.040 --> 00:17:10.839
The same thing happens when
you're looking for projections
00:17:10.839 --> 00:17:12.069
in three-dimensional space.
00:17:12.069 --> 00:17:14.720
Suppose you have a
force and a displacement
00:17:14.720 --> 00:17:16.240
in three-dimensional
space, and you
00:17:16.240 --> 00:17:21.390
want to project a force
onto a line, a direction.
00:17:21.390 --> 00:17:25.359
And our next example shows
how the dot product can
00:17:25.359 --> 00:17:27.170
be used to find projections.
00:17:27.170 --> 00:17:30.120
Namely here's a vector
A, here's a vector B.
00:17:30.120 --> 00:17:33.262
And I would like to project
the vector A onto the vector B.
00:17:33.262 --> 00:17:35.720
And I would like to find what
the length of that projection
00:17:35.720 --> 00:17:36.450
is.
00:17:36.450 --> 00:17:39.390
Well, from elementary
trigonometry,
00:17:39.390 --> 00:17:41.500
I know that the length
of this projection
00:17:41.500 --> 00:17:45.360
is just the magnitude of A
times the cosine of theta.
00:17:45.360 --> 00:17:48.580
And by the way, notice if theta
were greater than 90 degrees,
00:17:48.580 --> 00:17:50.510
cosine theta would be negative.
00:17:50.510 --> 00:17:53.040
And the minus sign would
not affect the length.
00:17:53.040 --> 00:17:56.550
It would simply tell us
that the projection was
00:17:56.550 --> 00:18:00.090
in the opposite sense of B.
That's all that would mean.
00:18:00.090 --> 00:18:02.960
But here's the
interesting point.
00:18:02.960 --> 00:18:06.400
If you look at the magnitude
of A times the cosine of theta,
00:18:06.400 --> 00:18:09.230
it almost looks
like a dot product.
00:18:09.230 --> 00:18:12.960
After all, theta is the
angle between A and B.
00:18:12.960 --> 00:18:15.680
And if the magnitude
of B were in here,
00:18:15.680 --> 00:18:19.220
this would just be A dot
B. But the magnitude of B
00:18:19.220 --> 00:18:20.520
isn't in here.
00:18:20.520 --> 00:18:23.590
Of course if the magnitude
of B happened to be 1,
00:18:23.590 --> 00:18:25.280
that would be fine.
00:18:25.280 --> 00:18:27.830
But the magnitude
of B might not be 1.
00:18:27.830 --> 00:18:29.820
And the most honest
way to make it 1
00:18:29.820 --> 00:18:32.840
is to divide B by its magnitude.
00:18:32.840 --> 00:18:34.120
And what will that give you?
00:18:34.120 --> 00:18:37.080
If you divide any
vector by its magnitude,
00:18:37.080 --> 00:18:39.790
that automatically
gives you a unit vector
00:18:39.790 --> 00:18:43.729
having the same direction and
sense as the vector that you
00:18:43.729 --> 00:18:44.270
started with.
00:18:44.270 --> 00:18:46.880
In other words,
let u sub B, which
00:18:46.880 --> 00:18:51.174
is B divided by its
magnitude, be the unit vector
00:18:51.174 --> 00:18:52.590
in the direction--
and by the way,
00:18:52.590 --> 00:18:56.280
here direction includes
sense-- in the direction of B.
00:18:56.280 --> 00:19:00.160
And notice that the unit
vector in the direction of B
00:19:00.160 --> 00:19:02.620
has the same
direction as B itself.
00:19:02.620 --> 00:19:05.740
Therefore, to find the
angle between u sub B
00:19:05.740 --> 00:19:09.830
and A is the same as finding
the angle between B and A.
00:19:09.830 --> 00:19:11.580
In other words, the
kicker now seems
00:19:11.580 --> 00:19:14.860
to be that I take
this length, which
00:19:14.860 --> 00:19:17.590
is the magnitude of
A times cosine theta,
00:19:17.590 --> 00:19:19.440
and rewrite that as follows.
00:19:19.440 --> 00:19:22.610
It's the magnitude of
A-- and now remembering
00:19:22.610 --> 00:19:27.420
that u sub B has
unit length, I just
00:19:27.420 --> 00:19:30.310
throw that in as a
factor-- and theta,
00:19:30.310 --> 00:19:32.620
being the angle between
A and B, is also
00:19:32.620 --> 00:19:37.160
the angle between A and u sub
B. But this, by definition, is
00:19:37.160 --> 00:19:40.030
A dot u sub B.
00:19:40.030 --> 00:19:44.030
You see, in other words, to
find the projection of A onto B,
00:19:44.030 --> 00:19:47.390
all you have to do is
dot A with the unit
00:19:47.390 --> 00:19:51.010
vector in the direction of
B. In fact, to summarize
00:19:51.010 --> 00:19:53.980
that without the u sub B
in there, all I'm saying
00:19:53.980 --> 00:19:58.650
is given two vectors A
and B, if you dot A and B
00:19:58.650 --> 00:20:01.480
and then divide by the
magnitude of B, that
00:20:01.480 --> 00:20:06.350
will be the projection of A
in the direction of B. OK?
00:20:06.350 --> 00:20:09.760
The projection of A in the
direction of B. And of course,
00:20:09.760 --> 00:20:11.260
if you want it the
other way around,
00:20:11.260 --> 00:20:13.330
you have to reverse
the roles of A and B.
00:20:13.330 --> 00:20:16.560
The beauty of this unit,
in Cartesian coordinates,
00:20:16.560 --> 00:20:20.370
is how easy it is to compute A
dot B in Cartesian coordinates.
00:20:20.370 --> 00:20:23.620
Oh, another example that you
might be interested in, that I
00:20:23.620 --> 00:20:26.190
think is very
interesting, and that's
00:20:26.190 --> 00:20:28.130
the special case
where A and B already
00:20:28.130 --> 00:20:30.030
happen to be unit vectors.
00:20:30.030 --> 00:20:32.870
If A and B already happen
to be unit vectors,
00:20:32.870 --> 00:20:36.950
then if we use our recipe
for the formula for A dot B,
00:20:36.950 --> 00:20:40.920
we observe that, in this case,
by definition of unit vectors,
00:20:40.920 --> 00:20:43.730
both the magnitudes
of A and B are 1.
00:20:43.730 --> 00:20:47.020
And we find that A dot
B is then the cosine
00:20:47.020 --> 00:20:50.270
of the angle between
A and B. Which
00:20:50.270 --> 00:20:53.890
means that if A and B happen to
be unit vectors, as soon as you
00:20:53.890 --> 00:20:56.490
dot them, you have
automatically found
00:20:56.490 --> 00:21:00.620
the cosine of the angle
between the two vectors, which
00:21:00.620 --> 00:21:03.930
suggests a rather
general type of approach.
00:21:03.930 --> 00:21:08.250
Given any two vectors, divide
each by the magnitude, right?
00:21:08.250 --> 00:21:10.200
That gives you unit vectors.
00:21:10.200 --> 00:21:12.220
Dot them, and that
gives you the cosine
00:21:12.220 --> 00:21:14.620
of the angle between them.
00:21:14.620 --> 00:21:15.350
You see?
00:21:15.350 --> 00:21:17.720
In particular, and here's
an interesting thing.
00:21:17.720 --> 00:21:19.680
You know, I don't know
if it's that funny,
00:21:19.680 --> 00:21:21.440
it just struck me as funny.
00:21:21.440 --> 00:21:24.540
Last night at supper as we
were sitting down to eat,
00:21:24.540 --> 00:21:26.600
my four-year-old
looked at me and said,
00:21:26.600 --> 00:21:28.750
"Dad, did they
have baked potatoes
00:21:28.750 --> 00:21:30.500
when you was a little boy?"
00:21:30.500 --> 00:21:32.150
And you get the
feeling sometimes
00:21:32.150 --> 00:21:34.560
that people think that
the modern world really
00:21:34.560 --> 00:21:38.690
changed the old in certain basic
ways that didn't happen at all.
00:21:38.690 --> 00:21:40.170
And one of the
interesting points
00:21:40.170 --> 00:21:44.100
is that long before vector
geometry was invented,
00:21:44.100 --> 00:21:47.390
people were doing
three-dimensional geometry
00:21:47.390 --> 00:21:49.660
using non-vector methods.
00:21:49.660 --> 00:21:51.780
And one technique that
happened to be used
00:21:51.780 --> 00:21:54.160
were things called
directional cosines.
00:21:54.160 --> 00:21:57.650
Namely, suppose you were
given a line in space.
00:21:57.650 --> 00:21:59.210
OK?
00:21:59.210 --> 00:22:01.610
As a vector, if you wish
to look at it that way--
00:22:01.610 --> 00:22:03.610
or if you didn't want to
look at it as a vector,
00:22:03.610 --> 00:22:06.290
imagine the line parallel
to the given line
00:22:06.290 --> 00:22:08.570
that goes through the origin.
00:22:08.570 --> 00:22:11.410
As soon as I know what
that line looks like,
00:22:11.410 --> 00:22:14.770
I can compute the angle that it
makes with the positive x-axis.
00:22:14.770 --> 00:22:18.720
I can compute the angle that it
makes with the positive y-axis.
00:22:18.720 --> 00:22:22.130
I can compute the angle that it
makes with the positive z-axis.
00:22:22.130 --> 00:22:26.070
And those three angles
uniquely determine the position
00:22:26.070 --> 00:22:29.470
of the line in space, the
direction of the line.
00:22:29.470 --> 00:22:30.960
OK?
00:22:30.960 --> 00:22:34.567
And those were called
the directional angles.
00:22:34.567 --> 00:22:35.400
You understand that?
00:22:35.400 --> 00:22:38.530
That was the three-dimensional
analog of slope.
00:22:38.530 --> 00:22:40.590
In other words, to find
the slope of a line
00:22:40.590 --> 00:22:44.060
in three-dimensional space, draw
the line parallel to that line
00:22:44.060 --> 00:22:46.950
that goes through the origin,
and measure each of the three
00:22:46.950 --> 00:22:51.160
angles that that line makes
with the positive x-, y-,
00:22:51.160 --> 00:22:52.780
and z-axes.
00:22:52.780 --> 00:22:55.100
And what the beauty was
of the dot product was
00:22:55.100 --> 00:22:57.880
it just gave us a simpler
way of doing that.
00:22:57.880 --> 00:23:03.790
Namely, if A is any vector,
I divide A by its magnitude.
00:23:03.790 --> 00:23:06.870
That gives me the unit
vector in the direction of A.
00:23:06.870 --> 00:23:09.730
If I now dot the unit
vector in the direction of A
00:23:09.730 --> 00:23:12.400
with i-- and after
all, what is i?
00:23:12.400 --> 00:23:14.360
i is the unit vector
in the direction
00:23:14.360 --> 00:23:16.424
of the positive x-axis.
00:23:16.424 --> 00:23:17.840
Since these are
both unit vectors,
00:23:17.840 --> 00:23:22.090
this would be the cosine of
the angle between i and A.
00:23:22.090 --> 00:23:23.900
And that's just what?
00:23:23.900 --> 00:23:26.630
The cosine of the
angle that A makes
00:23:26.630 --> 00:23:29.800
with the positive
x-axis-- traditionally,
00:23:29.800 --> 00:23:31.810
that angle was called alpha.
00:23:31.810 --> 00:23:35.490
So u_A dot i is
simply cosine alpha.
00:23:35.490 --> 00:23:38.880
Correspondingly, u_A
dot j is the cosine
00:23:38.880 --> 00:23:41.080
of the angle between A and j.
00:23:41.080 --> 00:23:44.140
That's the cosine
of the angle that A
00:23:44.140 --> 00:23:46.230
makes with the positive y-axis.
00:23:46.230 --> 00:23:48.230
That angle was called beta.
00:23:48.230 --> 00:23:49.960
This is cosine beta.
00:23:49.960 --> 00:23:53.660
And u sub A dot k is
cosine gamma where
00:23:53.660 --> 00:23:57.040
gamma is the angle that A makes
with the positive k direction.
00:23:57.040 --> 00:23:59.500
And these, being the
cosines of the angles,
00:23:59.500 --> 00:24:01.840
these were called the
directional cosines,
00:24:01.840 --> 00:24:03.910
and they yielded
the slope of lines.
00:24:03.910 --> 00:24:08.110
But one must not believe
that one needed vectors
00:24:08.110 --> 00:24:10.370
before he could do
three-dimensional geometry.
00:24:10.370 --> 00:24:13.860
What did happen was that
vector techniques greatly
00:24:13.860 --> 00:24:17.840
simplified many of the aspects
of three-dimensional geometry.
00:24:17.840 --> 00:24:21.530
Well, let's leave
this part for a moment
00:24:21.530 --> 00:24:26.190
and close for today by
coming back to our game idea.
00:24:26.190 --> 00:24:30.680
Remember that ultimately, all
we will ever use, once we get
00:24:30.680 --> 00:24:34.050
started with our game,
all we will ever use
00:24:34.050 --> 00:24:36.920
are the structural properties.
00:24:36.920 --> 00:24:39.960
Now I've gone through
these in the notes.
00:24:39.960 --> 00:24:41.990
I've gone through
them-- well you
00:24:41.990 --> 00:24:43.495
go through them
with me in the text
00:24:43.495 --> 00:24:45.180
or with yourselves in the text.
00:24:45.180 --> 00:24:49.760
Let me just point out certain
properties of the dot product
00:24:49.760 --> 00:24:53.080
that are shared by regular
arithmetic as well.
00:24:53.080 --> 00:24:56.440
For example, A dot
B equals B dot A.
00:24:56.440 --> 00:24:58.380
The dot product is commutative.
00:24:58.380 --> 00:24:59.490
Why is that?
00:24:59.490 --> 00:25:01.580
Well think of what A dot B is.
00:25:01.580 --> 00:25:04.670
It's the magnitude of A
times the magnitude of B
00:25:04.670 --> 00:25:07.600
times the cosine of the
angle between A and B.
00:25:07.600 --> 00:25:09.400
But that's the same as what?
00:25:09.400 --> 00:25:11.840
The magnitude of B times
the magnitude of A--
00:25:11.840 --> 00:25:15.300
after all, numbers, we know are
commutative when you multiply
00:25:15.300 --> 00:25:18.650
them-- and the cosine of
the angle between A and B
00:25:18.650 --> 00:25:21.760
is the same as the cosine of
the angle between B and A.
00:25:21.760 --> 00:25:27.320
So these are equal
numerical quantities.
00:25:27.320 --> 00:25:29.730
Again, without going
through the proof here,
00:25:29.730 --> 00:25:31.680
it turns out that
if you want to dot
00:25:31.680 --> 00:25:34.330
a vector with the sum
of two given vectors,
00:25:34.330 --> 00:25:35.840
the distributive property holds.
00:25:35.840 --> 00:25:42.650
Namely, A dot B plus C is A
dot B plus A dot C. By the way,
00:25:42.650 --> 00:25:46.420
if you did want to prove this,
all you would have to do,
00:25:46.420 --> 00:25:48.480
if you couldn't see
it geometrically,
00:25:48.480 --> 00:25:50.330
is to argue as follows.
00:25:50.330 --> 00:25:51.450
You say, you know?
00:25:51.450 --> 00:25:54.420
The easiest way to
add and dot vectors
00:25:54.420 --> 00:25:56.430
is in Cartesian coordinates.
00:25:56.430 --> 00:25:58.360
So let me prove
that this result is
00:25:58.360 --> 00:26:00.650
true in Cartesian coordinates.
00:26:00.650 --> 00:26:04.660
Carry out the details, and if it
works in Cartesian coordinates,
00:26:04.660 --> 00:26:08.130
since the result doesn't depend
on the coordinate system,
00:26:08.130 --> 00:26:09.740
the result must be
true, regardless
00:26:09.740 --> 00:26:11.130
of the coordinate system.
00:26:11.130 --> 00:26:15.130
But it's a very simple exercise
to actually write A, B, and C
00:26:15.130 --> 00:26:18.220
in terms of i, j, k
components, compute
00:26:18.220 --> 00:26:21.680
both sides of this
expression, and show
00:26:21.680 --> 00:26:23.080
that they're numerically equal.
00:26:23.080 --> 00:26:26.140
I say "numerically" because
it's crucial to notice
00:26:26.140 --> 00:26:29.980
that both expressions on
either side of the equal sign
00:26:29.980 --> 00:26:31.080
are numbers.
00:26:31.080 --> 00:26:32.770
B plus C is a vector.
00:26:32.770 --> 00:26:33.750
A is a vector.
00:26:33.750 --> 00:26:36.940
When you dot two vectors,
you get a number.
00:26:36.940 --> 00:26:40.030
Finally, a scalar
multiple of a vector
00:26:40.030 --> 00:26:43.870
dotted with another vector has
the property that you can leave
00:26:43.870 --> 00:26:45.600
the scalar multiple outside.
00:26:45.600 --> 00:26:48.870
In other words, you can
first dot the two vectors
00:26:48.870 --> 00:26:50.930
and then multiply by the scalar.
00:26:50.930 --> 00:26:53.120
In other words, in
a way, you don't
00:26:53.120 --> 00:26:55.160
have to worry about
voice inflection
00:26:55.160 --> 00:26:56.910
when you have a scalar multiple.
00:26:56.910 --> 00:26:59.910
And we will talk more
about these as we go along.
00:26:59.910 --> 00:27:02.290
However, what's very
crucial is to notice
00:27:02.290 --> 00:27:05.510
that the dot product does
have some difficulties not
00:27:05.510 --> 00:27:08.450
associated with
ordinary multiplication.
00:27:08.450 --> 00:27:11.010
So I say, beware.
00:27:11.010 --> 00:27:12.940
For example,
somebody might say, I
00:27:12.940 --> 00:27:15.970
wonder if the dot
product is associative.
00:27:15.970 --> 00:27:19.270
I wonder if A dot
B dotted with C
00:27:19.270 --> 00:27:23.070
is the same as A dotted
with B dotted with C.
00:27:23.070 --> 00:27:25.350
And this is nonsensical.
00:27:25.350 --> 00:27:27.407
I wanted to do this with
great dramatic gesture,
00:27:27.407 --> 00:27:29.990
but I probably would have broken
two fingers against the board
00:27:29.990 --> 00:27:30.860
here.
00:27:30.860 --> 00:27:33.070
Let's just cross this
out, so that you won't
00:27:33.070 --> 00:27:34.620
be inclined to remember that.
00:27:34.620 --> 00:27:37.660
This not only is
false, it's stupid.
00:27:37.660 --> 00:27:39.480
And the reason
that it's stupid is
00:27:39.480 --> 00:27:42.800
that it's nonsensical, that
these things don't make sense.
00:27:42.800 --> 00:27:45.460
Namely, the dot product
has been defined
00:27:45.460 --> 00:27:50.130
to be an operation between two
vectors that yields a number.
00:27:50.130 --> 00:27:53.090
Notice that as soon
as you dot A and B,
00:27:53.090 --> 00:27:54.440
you no longer have a vector.
00:27:54.440 --> 00:27:55.540
You have a number.
00:27:55.540 --> 00:27:58.400
And you cannot dot a
number with a vector.
00:27:58.400 --> 00:28:01.036
In other words,
neither (A dot B)
00:28:01.036 --> 00:28:06.110
dot C nor A dot (B
dot C) is defined.
00:28:06.110 --> 00:28:10.270
Because you see, a number is
never dotted with a vector.
00:28:10.270 --> 00:28:12.100
All right?
00:28:12.100 --> 00:28:15.860
And finally, a closing
note, as we've already seen,
00:28:15.860 --> 00:28:18.880
if A is perpendicular
to B, then the cosine
00:28:18.880 --> 00:28:21.760
of the angle between
A and B is 0.
00:28:21.760 --> 00:28:24.470
And that says that if A
is perpendicular to B,
00:28:24.470 --> 00:28:25.770
then A dot B is 0.
00:28:25.770 --> 00:28:28.730
In fact, this is one of
the most common usages
00:28:28.730 --> 00:28:31.000
on the elementary level,
of the dot product,
00:28:31.000 --> 00:28:33.440
is to prove that two
vectors are perpendicular.
00:28:33.440 --> 00:28:36.740
But, whereas that's
a nice property,
00:28:36.740 --> 00:28:39.900
what causes great hardship here
is to notice the following,
00:28:39.900 --> 00:28:42.990
in particular, that
if A dot B is 0,
00:28:42.990 --> 00:28:47.000
we cannot conclude that either
A is the zero vector or B is
00:28:47.000 --> 00:28:48.010
the zero vector.
00:28:48.010 --> 00:28:50.490
For example, i dot j is 0.
00:28:50.490 --> 00:28:53.890
But neither i nor j
is the zero vector.
00:28:53.890 --> 00:28:55.800
You see, with
ordinary arithmetic,
00:28:55.800 --> 00:28:57.980
we had the cancellation
rule, things
00:28:57.980 --> 00:29:01.100
that said if the product of
two numbers is 0, at least one
00:29:01.100 --> 00:29:02.570
of the factors must be 0.
00:29:02.570 --> 00:29:06.010
With the dot product,
this need not be true.
00:29:06.010 --> 00:29:08.302
The thing I want you to
get from this lesson more
00:29:08.302 --> 00:29:10.260
than anything else, other
than the applications
00:29:10.260 --> 00:29:13.190
that you can get from the
book and from the exercises,
00:29:13.190 --> 00:29:15.810
is to learn to get a
feeling for the structure.
00:29:15.810 --> 00:29:18.530
Don't be upset that certain
vector properties are
00:29:18.530 --> 00:29:21.250
different than
arithmetic properties,
00:29:21.250 --> 00:29:22.890
and certain ones are the same.
00:29:22.890 --> 00:29:26.920
Notice in terms of the
game, we take our rules
00:29:26.920 --> 00:29:29.840
as they may apply and
just carry them out
00:29:29.840 --> 00:29:32.230
towards inescapable conclusions.
00:29:32.230 --> 00:29:35.120
But I think that will become
clearer as you read the text
00:29:35.120 --> 00:29:36.800
and do the exercises.
00:29:36.800 --> 00:29:40.710
And until next time, when
we'll talk about a new vector
00:29:40.710 --> 00:29:42.870
product, let's
just say, so long.
00:29:45.410 --> 00:29:47.780
Funding for the
publication of this video
00:29:47.780 --> 00:29:52.660
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:29:52.660 --> 00:29:56.830
Help OCW continue to provide
free and open access to MIT
00:29:56.830 --> 00:30:01.248
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