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PROFESSOR: Hi.
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In the last unit, we
introduced the notion
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of the structure of vectors in
terms of arrows in the plane.
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And today, we want to talk about
the added geometric property
00:00:49.740 --> 00:00:52.800
of what I could call
three-dimensional vectors
00:00:52.800 --> 00:00:55.040
or three-dimensional arrows.
00:00:55.040 --> 00:00:57.750
Now before I say that-- I
guess I've already said it--
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but before I say
any more about it,
00:00:59.470 --> 00:01:01.220
let me point out,
of course, that this
00:01:01.220 --> 00:01:03.250
is a matter of semantics.
00:01:03.250 --> 00:01:08.010
Obviously an arrow, being a
line, has only one dimension,
00:01:08.010 --> 00:01:12.050
be it in the direction of the
x-axis, be it in the x-y plane,
00:01:12.050 --> 00:01:14.530
or be it in
three-dimensional space.
00:01:14.530 --> 00:01:17.010
When we say
"three-dimensional vectors,"
00:01:17.010 --> 00:01:21.270
we do not negate the fact that
we're talking about arrows,
00:01:21.270 --> 00:01:24.870
but rather that we are dealing
with a coordinate system which
00:01:24.870 --> 00:01:28.480
takes into consideration the
fact that the line is drawn
00:01:28.480 --> 00:01:30.640
through three-dimensional space.
00:01:30.640 --> 00:01:32.620
In other words, where
as I call the lecture
00:01:32.620 --> 00:01:36.230
three-dimensional vectors
or arrows-- and by the way,
00:01:36.230 --> 00:01:39.080
both in the notes
and as I'm lecturing,
00:01:39.080 --> 00:01:42.920
I will very often, whenever
I write the word "vector,"
00:01:42.920 --> 00:01:45.120
put "arrow" in
parentheses, whenever
00:01:45.120 --> 00:01:47.660
I write the word "arrow,"
put "vector" in parentheses
00:01:47.660 --> 00:01:50.190
so that you can see
the juxtaposition
00:01:50.190 --> 00:01:54.330
between these two, the
geometry versus the physical or
00:01:54.330 --> 00:01:55.850
mathematical concept.
00:01:55.850 --> 00:01:57.770
But the idea is
something like this.
00:01:57.770 --> 00:02:01.320
In three-dimensional space,
we have a natural extension
00:02:01.320 --> 00:02:02.980
of Cartesian coordinates.
00:02:02.980 --> 00:02:05.450
Rather than talk
about the x-y plane,
00:02:05.450 --> 00:02:09.699
we pick a third axis, a
third number line, which
00:02:09.699 --> 00:02:13.530
goes through the origin,
perpendicular to the x-y plane,
00:02:13.530 --> 00:02:17.010
and which has the sense
that if the x-axis is
00:02:17.010 --> 00:02:22.350
rotated into the y-axis through
the positive 90-degree angle
00:02:22.350 --> 00:02:25.420
here, that the z-axis
is in the direction
00:02:25.420 --> 00:02:28.950
in which a right-handed
screw would turn.
00:02:28.950 --> 00:02:30.410
I won't belabor this.
00:02:30.410 --> 00:02:32.370
It's done very, very
nicely in the textbook.
00:02:32.370 --> 00:02:35.710
And I'm sure that you have seen
this type of coordinate system
00:02:35.710 --> 00:02:36.360
before.
00:02:36.360 --> 00:02:37.818
If you haven't,
it's something that
00:02:37.818 --> 00:02:40.910
takes a matter of some 20
or 30 seconds to pick up.
00:02:40.910 --> 00:02:42.850
But at any rate,
what we're saying
00:02:42.850 --> 00:02:45.820
is, let's imagine this
three-dimensional coordinate
00:02:45.820 --> 00:02:48.930
system, three-dimensional
Cartesian coordinates.
00:02:48.930 --> 00:02:52.800
The convention is that
just as, in the plane,
00:02:52.800 --> 00:02:57.480
we label the point by its x and
y components, in three space,
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a point is labeled by its
x, y, and z components.
00:03:02.070 --> 00:03:06.070
So for example, if I take a
vector in three-space-- meaning
00:03:06.070 --> 00:03:06.570
what?
00:03:06.570 --> 00:03:09.580
It's a line that goes through
three-dimensional space-- I
00:03:09.580 --> 00:03:11.310
again shift it
parallel to itself,
00:03:11.310 --> 00:03:13.350
so it begins at my origin.
00:03:13.350 --> 00:03:16.340
And at the risk of causing
some confusion here,
00:03:16.340 --> 00:03:18.150
I still think it's worth doing.
00:03:18.150 --> 00:03:20.660
Let me call the
vector A, and let
00:03:20.660 --> 00:03:24.680
me call the point at which the
vector terminates the point A.
00:03:24.680 --> 00:03:29.290
And let's suppose that the point
A has coordinates A_1, A_2,
00:03:29.290 --> 00:03:30.130
A_3.
00:03:30.130 --> 00:03:32.790
The reason that this
doesn't make much difference
00:03:32.790 --> 00:03:36.390
if you get confused is
that if the vector, which
00:03:36.390 --> 00:03:39.150
originates at the
origin, terminates
00:03:39.150 --> 00:03:41.850
at the point whose
coordinates a_1, a_2,
00:03:41.850 --> 00:03:46.810
and a_3, then the components
of that vector will have what?
00:03:46.810 --> 00:03:49.070
The i component will be a_1.
00:03:49.070 --> 00:03:51.480
The j component will be a_2.
00:03:51.480 --> 00:03:54.940
And by the way, we let
k be the unit vector
00:03:54.940 --> 00:03:56.910
in the positive z direction.
00:03:56.910 --> 00:04:00.100
And the k component will be a_3.
00:04:00.100 --> 00:04:03.260
To see this thing
pictorially, notice
00:04:03.260 --> 00:04:05.440
the three-dimensional
effect of the vector A.
00:04:05.440 --> 00:04:07.830
It means that if we drop a
perpendicular from the head
00:04:07.830 --> 00:04:15.300
of A to the x-y plane, then,
you see, this distance is a_1.
00:04:15.300 --> 00:04:17.300
This distance is a_2.
00:04:17.300 --> 00:04:19.540
And this height is a_3.
00:04:19.540 --> 00:04:23.810
And notice, again, in terms of
adding vectors head to tail et
00:04:23.810 --> 00:04:29.200
cetera, notice that as a vector,
this would be the vector a_1*i.
00:04:29.200 --> 00:04:33.460
This would be the vector a_2*j.
00:04:33.460 --> 00:04:37.420
And this would be
the vector a_3*k.
00:04:37.420 --> 00:04:41.350
In other words, the vector A
is the sum of the three vectors
00:04:41.350 --> 00:04:45.560
a_1*i, a_2*j, and a_3*k.
00:04:45.560 --> 00:04:47.430
In other words,
to summarize this,
00:04:47.430 --> 00:04:49.260
the vector A is simply what?
00:04:49.260 --> 00:04:52.660
a_1*i, plus a_2*j, plus a_3*k.
00:04:52.660 --> 00:04:55.630
And again, notice the
juxtapositioning, if you wish,
00:04:55.630 --> 00:04:58.910
between the components
of the vector
00:04:58.910 --> 00:05:02.810
and the coordinates
of the point.
00:05:02.810 --> 00:05:06.640
Also, we can get ahold of
the magnitude of this vector.
00:05:06.640 --> 00:05:09.300
Remember, the magnitude
is the length.
00:05:09.300 --> 00:05:12.270
And how can we figure out
the length of this vector?
00:05:12.270 --> 00:05:13.930
Notice that, by the
way, the geometry
00:05:13.930 --> 00:05:16.275
gets much tougher in
three-dimensional space.
00:05:16.275 --> 00:05:18.650
But I want to show you something
interesting in a minute.
00:05:18.650 --> 00:05:21.190
Let's suffer through the
geometry for a minute.
00:05:21.190 --> 00:05:23.690
Let's see how we can find
the length of the vector A.
00:05:23.690 --> 00:05:27.670
Well, notice, by the way,
that A happens to be--
00:05:27.670 --> 00:05:30.660
or the magnitude of A
happens to be the hypotenuse
00:05:30.660 --> 00:05:32.160
of a right triangle.
00:05:32.160 --> 00:05:34.190
What right triangle is it?
00:05:34.190 --> 00:05:37.110
It's the right
triangle that joins
00:05:37.110 --> 00:05:42.010
the origin-- let me call
this point here B-- and A.
00:05:42.010 --> 00:05:45.270
In other words, triangle
OBA is a right triangle
00:05:45.270 --> 00:05:48.840
because OB is in the
x-y plane and AB is
00:05:48.840 --> 00:05:51.300
perpendicular to the x-y plane.
00:05:51.300 --> 00:05:55.320
Therefore, by the Pythagorean
theorem, the magnitude of A
00:05:55.320 --> 00:05:59.950
will be the square root of the
square of the magnitude of OB
00:05:59.950 --> 00:06:02.730
plus the square of
the magnitude of AB.
00:06:02.730 --> 00:06:10.750
On the other hand, notice that
OCB is also a right triangle.
00:06:10.750 --> 00:06:14.480
So by the Pythagorean
theorem, the length of OB
00:06:14.480 --> 00:06:20.510
is the square root of a_1
squared plus a_2 squared.
00:06:20.510 --> 00:06:23.800
And therefore, you see, by
the Pythagorean theorem,
00:06:23.800 --> 00:06:26.250
the square of the
magnitude of A is
00:06:26.250 --> 00:06:28.310
this squared plus this squared.
00:06:28.310 --> 00:06:30.820
In other words,
the magnitude of A
00:06:30.820 --> 00:06:34.550
is the square root of a_1
squared plus a_2 squared
00:06:34.550 --> 00:06:35.990
plus a_3 squared.
00:06:35.990 --> 00:06:38.150
In other words, in
Cartesian coordinates,
00:06:38.150 --> 00:06:40.250
to find the magnitude
of a vector,
00:06:40.250 --> 00:06:43.780
you need only take the
positive square root
00:06:43.780 --> 00:06:46.070
of the sum of the squares
of the components.
00:06:46.070 --> 00:06:48.750
It's a regular distance formula,
the Pythagorean theorem.
00:06:48.750 --> 00:06:50.930
By the way, just a
very, very quick review
00:06:50.930 --> 00:06:52.970
from part one of our
course-- remember
00:06:52.970 --> 00:06:55.160
that technically
speaking, the square root
00:06:55.160 --> 00:06:57.130
is a double-valued function.
00:06:57.130 --> 00:07:00.820
Namely, any positive number
has two square roots.
00:07:00.820 --> 00:07:01.320
All right?
00:07:01.320 --> 00:07:04.200
In other words, the
square root of 16 is 4.
00:07:04.200 --> 00:07:06.150
It's also negative 4.
00:07:06.150 --> 00:07:09.470
But by convention, when
we don't indicate a sign,
00:07:09.470 --> 00:07:12.180
we're always referring to
the positive square root.
00:07:12.180 --> 00:07:13.280
And that's fine.
00:07:13.280 --> 00:07:15.220
Because after all,
the magnitude--
00:07:15.220 --> 00:07:17.010
the length of an
arrow-- is always
00:07:17.010 --> 00:07:18.540
looked upon as being what?
00:07:18.540 --> 00:07:20.550
Some non-negative number.
00:07:20.550 --> 00:07:24.390
Notice that the negativeness
is taken care of by the sense.
00:07:24.390 --> 00:07:25.130
OK?
00:07:25.130 --> 00:07:26.900
And this is what's
rather interesting.
00:07:26.900 --> 00:07:28.380
I think it's worth noting.
00:07:28.380 --> 00:07:31.260
If we use Cartesian
coordinates in one dimension,
00:07:31.260 --> 00:07:34.480
all you have is the
x-axis, in which case
00:07:34.480 --> 00:07:38.720
any vector is some scale or
multiple of the unit vector i.
00:07:38.720 --> 00:07:42.190
In other words, A
is equal to a_1*i.
00:07:42.190 --> 00:07:45.750
In this case, the magnitude
of A is just what?
00:07:45.750 --> 00:07:48.470
It's the square
root of a_1 squared.
00:07:48.470 --> 00:07:50.890
And notice that means the
positive square root--
00:07:50.890 --> 00:07:53.330
notice, by the way, that
that's the usual meaning
00:07:53.330 --> 00:07:54.570
of absolute value.
00:07:54.570 --> 00:07:57.380
After all, if a_1
were already positive,
00:07:57.380 --> 00:08:00.640
taking the positive square
root of its square root--
00:08:00.640 --> 00:08:01.564
of its square.
00:08:01.564 --> 00:08:02.480
These tongue twisters.
00:08:02.480 --> 00:08:04.250
The positive square
root of the square
00:08:04.250 --> 00:08:05.920
would give you the
number back again.
00:08:05.920 --> 00:08:08.080
On the other hand,
if a_1 were negative,
00:08:08.080 --> 00:08:11.620
and then you square it, and then
take the positive square root,
00:08:11.620 --> 00:08:14.780
all you do is change
the sign of a_1.
00:08:14.780 --> 00:08:16.940
And since it was negative,
changing its sign
00:08:16.940 --> 00:08:18.420
makes it positive.
00:08:18.420 --> 00:08:21.020
So this is the usual
absolute value, all right?
00:08:21.020 --> 00:08:22.940
Now in terms of
the previous unit,
00:08:22.940 --> 00:08:25.240
we saw that in
two-dimensional space,
00:08:25.240 --> 00:08:27.620
using i and j as
our basic vectors,
00:08:27.620 --> 00:08:31.520
that if A was the
vector a_1*i plus a_2*j,
00:08:31.520 --> 00:08:34.970
then the magnitude of A, again,
by the Pythagorean theorem,
00:08:34.970 --> 00:08:40.130
was just a square root of
a_1 squared plus a_2 squared.
00:08:40.130 --> 00:08:42.330
And now we've seen
that structurally--
00:08:42.330 --> 00:08:46.010
even though geometrically
it was harder to show--
00:08:46.010 --> 00:08:50.230
that if A in three-dimensional
space is a_1*i plus a_2*j plus
00:08:50.230 --> 00:08:53.250
a_3*k, then A is what?
00:08:53.250 --> 00:08:55.190
The positive square
root of a_1 squared
00:08:55.190 --> 00:08:57.410
plus a_2 squared
plus a_3 squared.
00:08:57.410 --> 00:09:01.030
And notice you see this
structural resemblance.
00:09:01.030 --> 00:09:03.230
This is a beautiful
structural resemblance.
00:09:03.230 --> 00:09:05.610
Why is it a beautiful
structural resemblance?
00:09:05.610 --> 00:09:07.950
Because now, maybe we're
getting the feeling
00:09:07.950 --> 00:09:10.510
that once our rules and
recipes are made up,
00:09:10.510 --> 00:09:13.610
we don't have to worry about
how difficult the geometry is.
00:09:13.610 --> 00:09:15.510
In other words,
except for the fact
00:09:15.510 --> 00:09:17.910
that you have an extra
component to worry about here,
00:09:17.910 --> 00:09:20.320
there is no basic
difference structurally
00:09:20.320 --> 00:09:23.500
between finding the length of a
vector in one-dimensional space
00:09:23.500 --> 00:09:25.500
or in three-dimensional space.
00:09:25.500 --> 00:09:28.610
And in fact, another reason that
I'm harping on this is the fact
00:09:28.610 --> 00:09:31.740
that in a little while-- meaning
within the next few lessons--
00:09:31.740 --> 00:09:34.730
we are going to be talking
about some horrible thing called
00:09:34.730 --> 00:09:38.620
n-dimensional space, which
geometrically cannot be drawn,
00:09:38.620 --> 00:09:41.940
but which analytically preserves
the same structure that
00:09:41.940 --> 00:09:43.310
we're talking about here.
00:09:43.310 --> 00:09:46.520
You see, the whole bag is
still this idea of structure.
00:09:46.520 --> 00:09:49.580
We're trying to emphasize
that the recipes remain
00:09:49.580 --> 00:09:52.650
the same independently
of the dimension.
00:09:52.650 --> 00:09:56.970
And just to carry this idea a
few steps further, what we're
00:09:56.970 --> 00:10:00.620
saying is, remember in the last
unit, we talked about the fact
00:10:00.620 --> 00:10:02.780
that the nice thing
about i and j components
00:10:02.780 --> 00:10:05.070
was that the definition
for adding two vectors
00:10:05.070 --> 00:10:08.680
was that if they were in
terms of i and j components,
00:10:08.680 --> 00:10:10.860
you just add it
component by component?
00:10:10.860 --> 00:10:13.500
Well again, without
belaboring the point
00:10:13.500 --> 00:10:18.900
and leaving the details to
the text and to the exercises,
00:10:18.900 --> 00:10:21.650
let's keep in mind that
it's relatively easy to show
00:10:21.650 --> 00:10:24.780
that the same formal definition
of addition-- namely,
00:10:24.780 --> 00:10:27.680
placing the arrows head
to tail, et cetera--
00:10:27.680 --> 00:10:29.764
forgetting about Cartesian
coordinates, but notice
00:10:29.764 --> 00:10:31.180
the definition of
addition doesn't
00:10:31.180 --> 00:10:32.530
mention the coordinate system.
00:10:32.530 --> 00:10:35.300
It just says put the vectors
head to tail, et cetera.
00:10:35.300 --> 00:10:37.730
And what I'm saying
is it's easy to show,
00:10:37.730 --> 00:10:41.530
under those conditions, that
if A is the vector a_1*i plus
00:10:41.530 --> 00:10:46.640
a_2*j plus a_3*k, and B is the
vector b_1*i plus b_2*j plus
00:10:46.640 --> 00:10:49.710
b_3*k, then you still
add vectors the same way
00:10:49.710 --> 00:10:52.300
in three-space as
you did in two-space.
00:10:52.300 --> 00:10:55.220
Again, the geometric arguments
are a little bit more
00:10:55.220 --> 00:10:57.570
sophisticated, if only
because it's harder
00:10:57.570 --> 00:10:59.620
to draw three
dimensions to scale
00:10:59.620 --> 00:11:01.210
than it is two dimensions.
00:11:01.210 --> 00:11:02.350
But the idea is what?
00:11:02.350 --> 00:11:05.620
A plus B is just
obtained by adding what?
00:11:05.620 --> 00:11:08.740
The two i components,
the two j components,
00:11:08.740 --> 00:11:11.010
the two k components.
00:11:11.010 --> 00:11:13.740
And similarly, for
scalar multiplication,
00:11:13.740 --> 00:11:16.490
in the same way in terms
of i and j components,
00:11:16.490 --> 00:11:19.740
you simply multiplied each
complement by the scalar c.
00:11:19.740 --> 00:11:23.940
If c is any number and A is
still the vector a_1*i plus
00:11:23.940 --> 00:11:28.330
a_2*j plus a_3k, the scalar
multiple c times A turns out
00:11:28.330 --> 00:11:32.270
to be c*a_1*i plus
c*a_3*j plus c*a_3*k.
00:11:32.270 --> 00:11:35.470
That you just multiply
component by component.
00:11:35.470 --> 00:11:38.990
And by the way, don't lose
track of this important point,
00:11:38.990 --> 00:11:43.820
that the scalar multiple c*A
means the same thing in two
00:11:43.820 --> 00:11:45.770
dimensions as it did three.
00:11:45.770 --> 00:11:49.480
Because after all, when you draw
that vector, once that arrow
00:11:49.480 --> 00:11:54.100
is drawn, it's a one-dimensional
thing if you pick as your axis
00:11:54.100 --> 00:11:56.919
the line of action of
the particular arrow.
00:11:56.919 --> 00:11:58.460
In other words,
scalar multiplication
00:11:58.460 --> 00:11:59.450
means the same thing.
00:11:59.450 --> 00:12:02.280
What's happening is that
you're looking at it in terms
00:12:02.280 --> 00:12:03.820
of different components.
00:12:03.820 --> 00:12:06.780
You're looking at it in terms of
i, j, and k, rather than just i
00:12:06.780 --> 00:12:09.740
and j, or in terms of any
other coordinate system.
00:12:09.740 --> 00:12:11.930
But carrying this
out in particular,
00:12:11.930 --> 00:12:15.180
notice then, the connection
between the vector minus B--
00:12:15.180 --> 00:12:19.360
negative B-- and the
vector negative 1 times
00:12:19.360 --> 00:12:23.790
B. Namely, the vector negative
B, which has the same magnitude
00:12:23.790 --> 00:12:27.690
and direction as B,
but the opposite sense,
00:12:27.690 --> 00:12:29.560
is the same way as saying what?
00:12:29.560 --> 00:12:31.337
Multiply B by minus 1.
00:12:31.337 --> 00:12:33.920
Because that gives you a vector
which has the same magnitude--
00:12:33.920 --> 00:12:37.780
namely 1 times as much as
B-- the same direction,
00:12:37.780 --> 00:12:38.810
and the opposite sense.
00:12:38.810 --> 00:12:40.720
In other words, the
vector negative B,
00:12:40.720 --> 00:12:45.610
in three-dimensional space,
is minus b_1 i, minus b_2 j,
00:12:45.610 --> 00:12:47.440
minus b_3 k.
00:12:47.440 --> 00:12:52.490
And therefore, since A minus B
still means A plus negative B,
00:12:52.490 --> 00:12:56.370
the vector A minus B
is simply obtained how?
00:12:56.370 --> 00:12:58.680
You subtract the
same way as we did
00:12:58.680 --> 00:13:02.390
in ordinary one-dimensional
analytic geometry,
00:13:02.390 --> 00:13:03.610
in terms of a number line.
00:13:03.610 --> 00:13:06.390
In other words, you
subtract component
00:13:06.390 --> 00:13:08.800
by component-- subtracting what?
00:13:14.570 --> 00:13:18.430
The first vector minus the
corresponding component
00:13:18.430 --> 00:13:19.660
of the second vector.
00:13:19.660 --> 00:13:22.470
In other words, in Cartesian
coordinates, the vector A
00:13:22.470 --> 00:13:27.910
minus B-- once we know a_1, a_2,
and a_3 as the components of A,
00:13:27.910 --> 00:13:30.740
b_1, b_2, and b_3 as
the components of B,
00:13:30.740 --> 00:13:33.530
we just subtract
component by component
00:13:33.530 --> 00:13:35.770
to get this particular
result. And this is, I
00:13:35.770 --> 00:13:38.010
think, very important
to understand.
00:13:38.010 --> 00:13:40.900
By the same token,
as simple as this is,
00:13:40.900 --> 00:13:42.320
it's going to
cause-- if we're not
00:13:42.320 --> 00:13:45.230
careful-- great
misinterpretation.
00:13:45.230 --> 00:13:50.170
Namely, the vector arithmetic
that we're talking about
00:13:50.170 --> 00:13:53.420
does not depend on
Cartesian coordinates.
00:13:53.420 --> 00:13:56.580
Vectors or arrows, as
we're looking at them,
00:13:56.580 --> 00:13:58.970
and the arrows exist no
matter what coordinate system
00:13:58.970 --> 00:13:59.710
we're using.
00:13:59.710 --> 00:14:02.470
What is particularly
important, however,
00:14:02.470 --> 00:14:06.410
is that many of the luxuries
of using vector arithmetic
00:14:06.410 --> 00:14:08.890
hinge on using
Cartesian coordinates.
00:14:08.890 --> 00:14:10.590
Now what do I mean by that?
00:14:10.590 --> 00:14:14.340
Well, let me just
emphasize: Note the need
00:14:14.340 --> 00:14:16.950
for Cartesian coordinates.
00:14:16.950 --> 00:14:19.420
Let me talk instead,
you see, about something
00:14:19.420 --> 00:14:21.110
called polar coordinates.
00:14:21.110 --> 00:14:23.060
Now later on,
we're going to talk
00:14:23.060 --> 00:14:26.560
about polar coordinates
in more detail-- in fact,
00:14:26.560 --> 00:14:28.540
in very, very much more detail.
00:14:28.540 --> 00:14:30.930
But for the time
being, let's view polar
00:14:30.930 --> 00:14:34.270
coordinates simply as
a radar-type thing,
00:14:34.270 --> 00:14:37.590
as a range-and-bearing
type navigation.
00:14:37.590 --> 00:14:39.870
Namely, in polar
coordinates, the way
00:14:39.870 --> 00:14:42.600
you specify a vector
in the plane, say,
00:14:42.600 --> 00:14:49.020
is you specify its length, which
we'll call r, and the angle
00:14:49.020 --> 00:14:52.980
that it makes with the positive
x-axis, that we'll call theta.
00:14:52.980 --> 00:14:57.800
In other words, in polar
coordinates, the vector A,
00:14:57.800 --> 00:15:00.420
which I've labeled
r_2 comma theta_2 ,
00:15:00.420 --> 00:15:06.030
that would indicate that
the magnitude of A is r_2.
00:15:06.030 --> 00:15:09.340
And that the angle that A
makes with the positive x-axis
00:15:09.340 --> 00:15:13.050
is theta_2.
00:15:13.050 --> 00:15:16.510
And when I label the vector
B as r_1 comma theta_1,
00:15:16.510 --> 00:15:19.060
it's simply my way of
saying that the magnitude
00:15:19.060 --> 00:15:22.500
of the vector B is some
length, which we'll call r_1.
00:15:22.500 --> 00:15:26.790
And the angle that B makes
with the positive x-axis
00:15:26.790 --> 00:15:29.110
is some angle which
we'll call theta_1.
00:15:29.110 --> 00:15:30.160
OK?
00:15:30.160 --> 00:15:32.940
Perfectly well-defined
system, isn't it?
00:15:32.940 --> 00:15:36.010
In other words, if I tell
you the range and bearing,
00:15:36.010 --> 00:15:38.750
I've certainly given
you as much information
00:15:38.750 --> 00:15:41.610
as if I told you the
i and j components.
00:15:41.610 --> 00:15:44.940
Now here's the important point,
the real kicker to this thing.
00:15:44.940 --> 00:15:49.180
And that is that the vector
A minus B-- and A minus B
00:15:49.180 --> 00:15:50.030
is what?
00:15:50.030 --> 00:15:51.070
It's this vector here.
00:15:55.400 --> 00:15:57.840
And by the way, notice how
vector arithmetic goes.
00:15:57.840 --> 00:15:59.950
How could you tell
quickly whether this
00:15:59.950 --> 00:16:02.400
is A minus B or B minus A?
00:16:02.400 --> 00:16:05.040
We talked about this
in the previous unit,
00:16:05.040 --> 00:16:07.130
and had exercises
and discussion on it.
00:16:07.130 --> 00:16:10.600
But notice again, the
structure of vector arithmetic.
00:16:10.600 --> 00:16:14.380
If you look at addition,
what is this vector here?
00:16:14.380 --> 00:16:17.830
This is the vector which
must be added on to B
00:16:17.830 --> 00:16:22.610
to give the vector A. And that,
by definition, is called what?
00:16:22.610 --> 00:16:25.430
A minus B. A minus
B is what vector?
00:16:25.430 --> 00:16:30.200
The vector you must add on
to B to yield A. Subtraction
00:16:30.200 --> 00:16:32.020
is still the
inverse of addition.
00:16:32.020 --> 00:16:36.260
And hopefully by now, you should
be becoming much less fearful
00:16:36.260 --> 00:16:39.300
of this vector notation,
that structurally, it's
00:16:39.300 --> 00:16:43.810
the same as our first
unit of arithmetic.
00:16:43.810 --> 00:16:47.780
Well, numerical arithmetic,
OK, the more regular type,
00:16:47.780 --> 00:16:49.470
ordinary type of arithmetic.
00:16:49.470 --> 00:16:51.130
But here's the important point.
00:16:51.130 --> 00:16:55.070
This vector is still
A minus B. But I
00:16:55.070 --> 00:17:00.400
claim that it's trivial to
see that this vector is not
00:17:00.400 --> 00:17:03.800
the vector whose
magnitude is r_2 minus r_1
00:17:03.800 --> 00:17:07.720
and whose angle of bearing
is theta_2 minus theta_1.
00:17:07.720 --> 00:17:10.940
Now the word "trivial" is a very
misleading word in mathematics.
00:17:10.940 --> 00:17:13.420
One of our famous
mathematical anecdotes
00:17:13.420 --> 00:17:16.276
is the professor who said
that a proof was trivial.
00:17:16.276 --> 00:17:18.359
The student says, "It
doesn't seem trivial to me."
00:17:18.359 --> 00:17:20.810
Their professor says,
"Well, it is trivial."
00:17:20.810 --> 00:17:22.490
He didn't quite
see it in a hurry.
00:17:22.490 --> 00:17:23.400
He says, "Wait here."
00:17:23.400 --> 00:17:25.030
He ran down to his office.
00:17:25.030 --> 00:17:28.260
Came up three hours later
with a ream of papers
00:17:28.260 --> 00:17:30.880
that he had written on,
bathed in perspiration,
00:17:30.880 --> 00:17:31.637
and a big smile.
00:17:31.637 --> 00:17:32.720
And he said, "I was right.
00:17:32.720 --> 00:17:34.020
It was trivial."
00:17:34.020 --> 00:17:36.170
So just because what
I think is trivial
00:17:36.170 --> 00:17:38.130
may not be what you
think is trivial,
00:17:38.130 --> 00:17:43.060
let me go through this statement
in more computational detail.
00:17:43.060 --> 00:17:46.610
What I'm saying is, let v
denote the vector that we
00:17:46.610 --> 00:17:49.120
called A minus B before.
00:17:49.120 --> 00:17:52.670
Notice that in terms of what
the vectors A and B are,
00:17:52.670 --> 00:17:54.840
this length is r_1.
00:17:54.840 --> 00:17:56.650
This length is r_2.
00:17:56.650 --> 00:17:59.670
And notice that since this
whole angle was theta_2,
00:17:59.670 --> 00:18:03.630
and the angle from here to here
was theta_1, this angle in here
00:18:03.630 --> 00:18:06.140
must be the difference between
those two angles, which
00:18:06.140 --> 00:18:08.380
is theta_2 minus theta_1.
00:18:08.380 --> 00:18:11.320
Now notice that this
is still a triangle.
00:18:11.320 --> 00:18:13.120
The length of this
triangle is still
00:18:13.120 --> 00:18:16.850
expressible in terms of these
sides and the included angle.
00:18:16.850 --> 00:18:20.110
But you may recall from plane
trigonometry and our refresher
00:18:20.110 --> 00:18:22.630
of that in the first
part of our course-- part
00:18:22.630 --> 00:18:25.660
one-- that to find the
third side of a triangle,
00:18:25.660 --> 00:18:27.850
given two sides and
the included angle,
00:18:27.850 --> 00:18:30.010
one must use the Law of Cosines.
00:18:30.010 --> 00:18:33.490
In other words, the magnitude
of v has what property?
00:18:33.490 --> 00:18:36.860
That its square is equal to the
sum of the squares of these two
00:18:36.860 --> 00:18:40.870
sides minus twice the
product of these two
00:18:40.870 --> 00:18:44.220
lengths times the cosine
of the included angle.
00:18:44.220 --> 00:18:49.170
In other words, this is just--
I hope this shows up all right--
00:18:49.170 --> 00:18:51.650
this is the Law of Cosines.
00:18:51.650 --> 00:18:54.350
And if it's difficult to
read, don't bother reading it.
00:18:54.350 --> 00:18:56.720
It's still the Law of
Cosines, and hopefully you
00:18:56.720 --> 00:18:58.070
recognize it as such.
00:18:58.070 --> 00:19:00.590
In other words, to find
the magnitude of v,
00:19:00.590 --> 00:19:03.880
it's not just r_1
or r_2 minus r_1.
00:19:03.880 --> 00:19:08.380
It's the square root of r_1
squared plus r_2 squared minus
00:19:08.380 --> 00:19:11.790
2r_1*r_2 cosine
theta_2 minus theta_1.
00:19:11.790 --> 00:19:14.080
Can you use polar
coordinates if you want?
00:19:14.080 --> 00:19:15.700
The answer is you bet you can.
00:19:15.700 --> 00:19:18.540
But when you do use it, make
sure that whenever you're
00:19:18.540 --> 00:19:21.150
going to find the magnitude,
that you don't just
00:19:21.150 --> 00:19:22.470
subtract the two magnitudes.
00:19:22.470 --> 00:19:23.870
You have to use this.
00:19:23.870 --> 00:19:26.770
And by the way, notice I haven't
even gone into this part,
00:19:26.770 --> 00:19:28.780
because it's irrelevant
from the point of view
00:19:28.780 --> 00:19:30.610
that I'm trying
to emphasize now.
00:19:30.610 --> 00:19:33.700
This mess just gives you
the magnitude of a vector.
00:19:33.700 --> 00:19:36.040
It doesn't even tell you
what direction it's in.
00:19:36.040 --> 00:19:38.830
I have to still do a
heck of a lot of geometry
00:19:38.830 --> 00:19:43.060
if I want to find out
what this angle here
00:19:43.060 --> 00:19:46.130
is-- a lot of work involved.
00:19:46.130 --> 00:19:47.990
I can use polar
coordinates, but I
00:19:47.990 --> 00:19:50.770
lose some of the luxury
of Cartesian coordinates.
00:19:50.770 --> 00:19:55.010
In fact, notice that the
vector whose magnitude
00:19:55.010 --> 00:19:58.040
is r_2 minus r_1 and
whose angle of bearing
00:19:58.040 --> 00:20:01.750
is theta_2 minus theta_1,
can also be computed.
00:20:01.750 --> 00:20:05.850
Namely, how do we find
theta_2 minus theta_1?
00:20:05.850 --> 00:20:08.980
Well up here, we have
the angle theta_2.
00:20:08.980 --> 00:20:10.580
We have the angle theta_1.
00:20:10.580 --> 00:20:12.960
We computed theta_2
minus theta_1.
00:20:12.960 --> 00:20:17.210
I can now mark off that angle
here, theta_2 minus theta_1.
00:20:17.210 --> 00:20:20.820
I can now take a
circle of radius r_1--
00:20:20.820 --> 00:20:23.410
well, why even say a
circle-- mark off the length
00:20:23.410 --> 00:20:27.590
r_1 onto r_2, assuming r_2 is
the greater of the two lengths.
00:20:27.590 --> 00:20:30.690
That difference will
be r_2 minus r_1.
00:20:30.690 --> 00:20:33.290
In other words, if I
swing an arc over here,
00:20:33.290 --> 00:20:35.360
this would r_1 also.
00:20:35.360 --> 00:20:39.100
This distance here
would be r_2 minus r_1.
00:20:39.100 --> 00:20:42.990
I take that distance, and
mark it off down here.
00:20:42.990 --> 00:20:44.570
And what vector is this?
00:20:44.570 --> 00:20:47.100
This is the vector, which
in polar coordinates
00:20:47.100 --> 00:20:51.690
would have its magnitude
equal to r_2 minus r_1,
00:20:51.690 --> 00:20:55.980
and have its angle equal
to theta_2 minus theta_1.
00:20:55.980 --> 00:20:58.340
And all I want you
to see is no matter
00:20:58.340 --> 00:21:00.780
how you slice it, just
from this picture alone,
00:21:00.780 --> 00:21:06.530
this vector is not the
same as this vector.
00:21:06.530 --> 00:21:08.590
And by the way, don't make
the mistake of saying,
00:21:08.590 --> 00:21:10.230
gee, the way you've
draw them, they
00:21:10.230 --> 00:21:12.250
look like they could
be the same length.
00:21:12.250 --> 00:21:15.560
Remember that even if by
coincidence these two vectors
00:21:15.560 --> 00:21:19.000
had the same length,
you must remember
00:21:19.000 --> 00:21:22.850
that vector equality requires
not just the same magnitude,
00:21:22.850 --> 00:21:24.480
but the same direction.
00:21:24.480 --> 00:21:27.090
And even with the same
direction, the same sense.
00:21:27.090 --> 00:21:29.940
What should be
obvious is, at least
00:21:29.940 --> 00:21:33.420
based on this one picture, that
the direction of this vector
00:21:33.420 --> 00:21:37.450
certainly is not the same as
the direction of this vector.
00:21:37.450 --> 00:21:39.920
In other words, I guess I've
said this many times in part
00:21:39.920 --> 00:21:43.790
one, and even though it comes
out like phony facetiousness,
00:21:43.790 --> 00:21:45.430
I mean it quite sincerely.
00:21:45.430 --> 00:21:48.010
Given two vectors in
polar coordinates,
00:21:48.010 --> 00:21:52.690
one certainly has
the right to invent
00:21:52.690 --> 00:21:57.910
the vector r_2 minus r_1
comma theta_2 minus theta_1.
00:21:57.910 --> 00:22:00.230
You certainly have
the right to do that,
00:22:00.230 --> 00:22:03.100
but you don't have the right
to call that A minus B. What
00:22:03.100 --> 00:22:05.680
I meant by being facetious
is, you have the right
00:22:05.680 --> 00:22:08.060
to call it that, but
you're going to be wrong.
00:22:08.060 --> 00:22:10.920
Because A minus B has
already been defined.
00:22:10.920 --> 00:22:13.180
And all we have now
is a choice of what
00:22:13.180 --> 00:22:17.650
the answer looks like in a
particular coordinate system.
00:22:17.650 --> 00:22:19.880
And by the way, just
as a quick review,
00:22:19.880 --> 00:22:22.430
assuming that you've studied
different number bases at one
00:22:22.430 --> 00:22:25.220
time in your careers, this
has come up many times
00:22:25.220 --> 00:22:27.420
in the mathematics
curriculum under the heading
00:22:27.420 --> 00:22:30.680
of "Number-versus-Numeral,"
or more generally,
00:22:30.680 --> 00:22:34.330
I call it the heading of
"Name-versus-Concept."
00:22:34.330 --> 00:22:36.610
For example, consider
the number 6.
00:22:36.610 --> 00:22:40.450
6 is this many, and no matter
how you slice it again,
00:22:40.450 --> 00:22:41.890
this many is an even number.
00:22:41.890 --> 00:22:46.950
It breaks up into bundles
of 2 with none left over.
00:22:46.950 --> 00:22:47.960
OK?
00:22:47.960 --> 00:22:52.130
If you were to write 6 as a
base 5 numeral, it's what?
00:22:52.130 --> 00:22:55.260
One bundle of 5
with 1 left over.
00:22:55.260 --> 00:22:58.380
In other words, notice
that the number 6
00:22:58.380 --> 00:23:01.200
ends in an odd digit in base 5.
00:23:01.200 --> 00:23:03.800
In other words, the
test for evenness
00:23:03.800 --> 00:23:06.890
by looking at the last digit
depends on the number base.
00:23:06.890 --> 00:23:09.110
In other words, in an
even base, a number
00:23:09.110 --> 00:23:11.710
is even if its
units digit is even.
00:23:11.710 --> 00:23:14.800
In an odd base, a
number may or may not
00:23:14.800 --> 00:23:18.589
be even, depending on whether
its units digit is even.
00:23:18.589 --> 00:23:20.380
In other words, the
actual test that you're
00:23:20.380 --> 00:23:24.430
using-- the actual convenient
computational property--
00:23:24.430 --> 00:23:25.900
depends on the base.
00:23:25.900 --> 00:23:29.260
But the important point
is that 6 is always even,
00:23:29.260 --> 00:23:30.890
independently of a number base.
00:23:30.890 --> 00:23:34.050
In fact, the number
6 as six tally marks
00:23:34.050 --> 00:23:38.644
was invented long before
place-value enumeration.
00:23:38.644 --> 00:23:39.810
You see what I'm driving at?
00:23:39.810 --> 00:23:42.990
In other words, the number
concept stays the same.
00:23:42.990 --> 00:23:44.800
The concept stays the same.
00:23:44.800 --> 00:23:46.570
What the thing
looks like in terms
00:23:46.570 --> 00:23:48.670
of a particular set
of names, that's
00:23:48.670 --> 00:23:51.510
what depends on the
names that you choose.
00:23:51.510 --> 00:23:52.397
All right?
00:23:52.397 --> 00:23:54.230
I think that is very,
very important to see.
00:23:54.230 --> 00:23:56.760
The vector properties
do not change.
00:23:56.760 --> 00:23:59.450
The convenient
computational recipes
00:23:59.450 --> 00:24:02.320
do depend on what coordinate
system you're choosing.
00:24:02.320 --> 00:24:04.280
And that's one of the
reasons-- the same as
00:24:04.280 --> 00:24:07.640
in part one of this course--
why we find it very convenient,
00:24:07.640 --> 00:24:10.510
whenever possible, to use
Cartesian coordinates.
00:24:10.510 --> 00:24:11.305
OK?
00:24:11.305 --> 00:24:14.225
And let me just
summarize this some more.
00:24:17.240 --> 00:24:22.460
Structurally, this is
very important here.
00:24:22.460 --> 00:24:26.700
Structurally, arrow arithmetic
is the same for both two
00:24:26.700 --> 00:24:28.410
and three dimensions.
00:24:28.410 --> 00:24:30.890
For example, without
belaboring the point,
00:24:30.890 --> 00:24:33.040
whether we're dealing with
two-dimensional arrows
00:24:33.040 --> 00:24:36.260
or three-dimensional arrows,
A plus B is B plus A.
00:24:36.260 --> 00:24:41.230
A plus (B plus C) is equal
to (A plus B) plus C.
00:24:41.230 --> 00:24:44.520
and let me just say, et cetera,
and not belabor this particular
00:24:44.520 --> 00:24:48.180
point, that structurally, we
cannot tell the difference
00:24:48.180 --> 00:24:51.910
between two-dimensional vectors
and three-dimensional vectors.
00:24:51.910 --> 00:24:56.170
What we can tell the
difference between, I guess,
00:24:56.170 --> 00:24:57.860
is the geometry.
00:24:57.860 --> 00:25:01.070
The geometry may be more
difficult to visualize
00:25:01.070 --> 00:25:03.260
in three-space
than in two-space.
00:25:03.260 --> 00:25:07.710
But structurally, we
cannot tell the difference.
00:25:07.710 --> 00:25:09.840
And I guess this is what
this thing is all about,
00:25:09.840 --> 00:25:13.380
that as you read the textbook
and do your assignment,
00:25:13.380 --> 00:25:15.990
there is going to be a
tendency on your part
00:25:15.990 --> 00:25:18.370
to try to rely on diagrams.
00:25:18.370 --> 00:25:22.130
And boy, you have to be an
expert in descriptive geometry
00:25:22.130 --> 00:25:26.720
to be able to take a view of
something in three dimensions,
00:25:26.720 --> 00:25:31.430
and try to get its true lengths,
as you look along various axes
00:25:31.430 --> 00:25:33.080
and lines of sight.
00:25:33.080 --> 00:25:35.890
The beauty is going to
be that as we continue on
00:25:35.890 --> 00:25:40.160
with this course, we will
use geometry for motivation.
00:25:40.160 --> 00:25:42.780
But once we have motivated
things by geometry,
00:25:42.780 --> 00:25:47.010
we will extract those rules that
we can use even in cases where
00:25:47.010 --> 00:25:48.490
we can't see the picture.
00:25:48.490 --> 00:25:51.290
In much the same way, going
back to our first lecture
00:25:51.290 --> 00:25:54.380
when we talked about
why b to the 0 equals 1,
00:25:54.380 --> 00:25:57.910
we made up our
rules to make sure
00:25:57.910 --> 00:26:03.840
that they would conform to the
nice computational recipes that
00:26:03.840 --> 00:26:06.970
were prevalent in the
more simple cases.
00:26:06.970 --> 00:26:09.480
And we are going to play
this thing to the hilt.
00:26:09.480 --> 00:26:12.460
We're going to really
exploit this and explore it
00:26:12.460 --> 00:26:14.980
in future lectures, and in
fact, throughout the rest
00:26:14.980 --> 00:26:15.880
of this course.
00:26:15.880 --> 00:26:18.100
But more about that next time.
00:26:18.100 --> 00:26:22.140
And until next time, good bye.
00:26:22.140 --> 00:26:24.510
Funding for the
publication of this video
00:26:24.510 --> 00:26:29.390
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:26:29.390 --> 00:26:33.560
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free and open access to MIT
00:26:33.560 --> 00:26:37.978
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