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PROFESSOR: Hi.
00:00:34.590 --> 00:00:36.470
Our lecture today
is on the one hand,
00:00:36.470 --> 00:00:40.790
deceptively simple, and on the
other hand, deceptively hard.
00:00:40.790 --> 00:00:43.110
That from a certain
point of view,
00:00:43.110 --> 00:00:46.620
to talk about the equations
of lines and planes
00:00:46.620 --> 00:00:50.200
is, first of all, a
topic that many people
00:00:50.200 --> 00:00:52.130
say, "I'm not interested
in studying geometry.
00:00:52.130 --> 00:00:54.230
When are we going to
get back to calculus?"
00:00:54.230 --> 00:00:58.120
And this is very analogous
to our problem of part one
00:00:58.120 --> 00:01:00.270
in calculus, if
you recall, where
00:01:00.270 --> 00:01:02.490
we started with
analytic geometry,
00:01:02.490 --> 00:01:04.470
and the question
came up, why do we
00:01:04.470 --> 00:01:07.610
need geometry when what
we really wanted to study
00:01:07.610 --> 00:01:08.470
was calculus?
00:01:08.470 --> 00:01:11.400
And the idea of graphs
played a very important role
00:01:11.400 --> 00:01:12.060
in calculus.
00:01:12.060 --> 00:01:16.070
We found out, for example, also,
that the relatively harmless
00:01:16.070 --> 00:01:19.090
straight line was a
rather crucial curve.
00:01:19.090 --> 00:01:23.400
Namely, given any curve, no
matter how random in the plane,
00:01:23.400 --> 00:01:25.640
provided only that
it was smooth,
00:01:25.640 --> 00:01:28.140
we were able to
approximate that curve
00:01:28.140 --> 00:01:32.170
by a sequence of tangent
lines to various points.
00:01:32.170 --> 00:01:35.530
And in a similar way,
one finds that planes
00:01:35.530 --> 00:01:39.830
will do for functions of
two real variables what
00:01:39.830 --> 00:01:42.620
the equations of lines
did for functions
00:01:42.620 --> 00:01:44.990
of a single real variable.
00:01:44.990 --> 00:01:47.940
As I say, we'll talk about that
in more detail as we go along.
00:01:47.940 --> 00:01:50.630
The other nice part
about this lesson
00:01:50.630 --> 00:01:54.350
is the fact that we can now take
some of our vector properties
00:01:54.350 --> 00:01:56.350
that we've learned
from the past lessons
00:01:56.350 --> 00:01:59.190
and hit what I call
the home run ball, once
00:01:59.190 --> 00:02:02.400
and for all now, finish
off what we've started.
00:02:02.400 --> 00:02:06.090
And that is, in the study of
three-dimensional geometry,
00:02:06.090 --> 00:02:09.360
the crucial characteristics
are those building blocks
00:02:09.360 --> 00:02:11.220
called lines and planes.
00:02:11.220 --> 00:02:13.610
And by finding
convenient recipes
00:02:13.610 --> 00:02:15.810
for expressing lines
and planes, we'll
00:02:15.810 --> 00:02:18.190
be most of the way
home as far as using
00:02:18.190 --> 00:02:19.640
this material is concerned.
00:02:19.640 --> 00:02:22.690
Again, we'll hit mainly
the highlights today.
00:02:22.690 --> 00:02:25.940
The remainder of the
material will be covered,
00:02:25.940 --> 00:02:29.280
I hope, adequately between
the text and the exercises.
00:02:29.280 --> 00:02:32.740
At any rate then,
the lesson today
00:02:32.740 --> 00:02:36.050
is "Equations of
Lines and Planes."
00:02:36.050 --> 00:02:39.880
And to refresh what I just
said before, the little ratio--
00:02:39.880 --> 00:02:45.510
planes are to surfaces what
lines are to curves-- that we
00:02:45.510 --> 00:02:47.990
can approximate curves
by tangent lines,
00:02:47.990 --> 00:02:52.240
we can approximate smooth
surfaces by tangent planes.
00:02:52.240 --> 00:02:54.630
Now what we would
like to do is go back
00:02:54.630 --> 00:02:58.322
to Cartesian coordinates and
find the equation of a plane.
00:02:58.322 --> 00:03:00.280
The first question that
we asked before we even
00:03:00.280 --> 00:03:03.920
use any coordinate system is
just how much information do
00:03:03.920 --> 00:03:06.950
you need before you
can determine a plane?
00:03:06.950 --> 00:03:10.880
Well one way we know of is to
know three points in the plane.
00:03:10.880 --> 00:03:13.560
Another way is analogous
to the line case,
00:03:13.560 --> 00:03:16.980
where to determine a line, you
needed a point and a slope.
00:03:16.980 --> 00:03:20.410
One way of determining
a plane is to know what?
00:03:20.410 --> 00:03:24.510
A point in the plane and
the direction of the plane.
00:03:24.510 --> 00:03:26.950
And one way of getting
the direction of the plane
00:03:26.950 --> 00:03:30.644
is to fix a normal, a
perpendicular to the plane.
00:03:30.644 --> 00:03:32.810
In other words, the approach
that we're going to use
00:03:32.810 --> 00:03:35.220
is, let's suppose that
we know a point that we
00:03:35.220 --> 00:03:36.710
want to be in our plane.
00:03:36.710 --> 00:03:38.890
And we'll call that point P_0.
00:03:38.890 --> 00:03:43.120
We'll denote it generically
by (x_0, y_0, z_0).
00:03:43.120 --> 00:03:48.100
And let's let capital N denote
the vector A*i plus B*j plus
00:03:48.100 --> 00:03:51.280
C*k, which is
perpendicular to our plane.
00:03:51.280 --> 00:03:51.780
OK?
00:03:51.780 --> 00:03:54.850
So we're given a vector
perpendicular to the plane,
00:03:54.850 --> 00:03:56.930
we're given a
point in the plane,
00:03:56.930 --> 00:03:58.710
and now in Cartesian
coordinates,
00:03:58.710 --> 00:04:01.090
we would like to know the
equation of the plane.
00:04:01.090 --> 00:04:04.460
And as we do so often, we simply
come back to our little diagram
00:04:04.460 --> 00:04:07.200
here that will utilize
vector geometry.
00:04:07.200 --> 00:04:10.510
What we say is, OK, here's
a little diagram here.
00:04:10.510 --> 00:04:12.660
Here's P_0 in the plane.
00:04:12.660 --> 00:04:15.030
Here's N, normal to the plane.
00:04:15.030 --> 00:04:17.120
Now we pick any
point P whatsoever,
00:04:17.120 --> 00:04:21.279
any point in the whole
world whatsoever, as long
00:04:21.279 --> 00:04:23.770
as it's in space,
three-dimensional space.
00:04:23.770 --> 00:04:26.740
And what we say is-- what
does it mean for the point P
00:04:26.740 --> 00:04:27.980
to be in the plane?
00:04:27.980 --> 00:04:33.880
Well, in vector language, for
the point P to be in the plane,
00:04:33.880 --> 00:04:38.590
I think it's rather obvious
that P_0 P had better
00:04:38.590 --> 00:04:42.800
be perpendicular to
N. On the other hand,
00:04:42.800 --> 00:04:45.010
in dot product
language, what does it
00:04:45.010 --> 00:04:48.380
mean to say that N is
perpendicular to P_0 P?
00:04:48.380 --> 00:04:54.410
That says that the dot product
of N and P_0 P must be 0.
00:04:54.410 --> 00:04:57.900
Now just to economize space,
let me utilize the notation
00:04:57.900 --> 00:04:59.640
that we've talked
about in the notes
00:04:59.640 --> 00:05:02.970
and in the exercises where
in Cartesian coordinates
00:05:02.970 --> 00:05:07.180
I'll abbreviate a vector
in i, j, and k components
00:05:07.180 --> 00:05:09.095
just by writing
down its components.
00:05:09.095 --> 00:05:12.450
In other words, let me
abbreviate N by A comma B comma
00:05:12.450 --> 00:05:16.840
C, which stands for A*i plus
B*j plus C*k, et cetera.
00:05:16.840 --> 00:05:19.940
Notice the beauty, now, of our
Cartesian coordinate system.
00:05:19.940 --> 00:05:23.510
N is the vector whose components
are A, B, and C. What about P_0
00:05:23.510 --> 00:05:24.470
P?
00:05:24.470 --> 00:05:27.000
Well, that's the
vector whose components
00:05:27.000 --> 00:05:32.000
are x minus x_0, y minus
y_0, and z minus z_0,
00:05:32.000 --> 00:05:34.590
that beauty of Cartesian
coordinates again.
00:05:34.590 --> 00:05:37.350
Consequently, I want the dot
product of these two vectors
00:05:37.350 --> 00:05:38.550
to be 0.
00:05:38.550 --> 00:05:40.770
But in Cartesian
coordinates, we know
00:05:40.770 --> 00:05:43.300
that to dot two
vectors, we simply
00:05:43.300 --> 00:05:46.490
multiply corresponding
components and add,
00:05:46.490 --> 00:05:48.190
and that results in what?
00:05:48.190 --> 00:05:54.360
A times (x minus x_0) plus
B times (y minus y_0),
00:05:54.360 --> 00:05:59.100
plus C times (z
minus z_0) equals 0.
00:05:59.100 --> 00:06:02.700
By the way, each of these
steps is reversible,
00:06:02.700 --> 00:06:07.770
meaning that under these given
conditions, the point (x, y, z)
00:06:07.770 --> 00:06:10.700
is in the given
plane if and only
00:06:10.700 --> 00:06:14.180
if this equation
here is satisfied.
00:06:14.180 --> 00:06:18.000
Another way of
saying that is what?
00:06:18.000 --> 00:06:26.140
If A, B, C, x_0, y_0, and
z_0 are given constants,
00:06:26.140 --> 00:06:32.320
then A times x minus x_0 plus B
times y minus y_0 plus C times
00:06:32.320 --> 00:06:37.790
z minus z_0 equals 0 is the
equation of the plane which
00:06:37.790 --> 00:06:42.550
passes through the
point x_0, y_0, z_0,
00:06:42.550 --> 00:06:48.880
and which has the vector A*i
plus B*j plus C*k as its normal
00:06:48.880 --> 00:06:49.910
vector.
00:06:49.910 --> 00:06:52.130
And perhaps the easiest
way to illustrate this
00:06:52.130 --> 00:06:54.660
is again, by means
of an example.
00:06:54.660 --> 00:06:57.730
Let me simply write down
a linear expression.
00:06:57.730 --> 00:07:02.860
I'll write down 2 times x
minus 1 plus 3 times y plus 2
00:07:02.860 --> 00:07:05.680
plus 4 times z minus 5.
00:07:05.680 --> 00:07:08.730
My claim is that this is
the special case where
00:07:08.730 --> 00:07:14.070
A, B, and C are played by
2, 3, and 4, respectively;
00:07:14.070 --> 00:07:21.020
x_0, y_0, and z_0 are played by
1, negative 2, and negative 5--
00:07:21.020 --> 00:07:24.230
the same thing as in
our previous study
00:07:24.230 --> 00:07:27.610
of ordinary
two-dimensional geometry.
00:07:27.610 --> 00:07:30.040
Remember that the standard
form of our equation
00:07:30.040 --> 00:07:31.640
uses a minus sign.
00:07:31.640 --> 00:07:33.830
Consequently, to
use the equation,
00:07:33.830 --> 00:07:37.320
where we see y plus 2,
we should rewrite that
00:07:37.320 --> 00:07:40.220
as y minus minus 2.
00:07:40.220 --> 00:07:43.520
And what we're saying is that
this equation is then what?
00:07:43.520 --> 00:07:50.130
It passes through
the point (1, -2, 5).
00:07:50.130 --> 00:07:57.250
And has as its normal the
vector 2*i plus 3*j plus 4*k.
00:07:57.250 --> 00:08:00.170
Actually, this is not a
very difficult concept
00:08:00.170 --> 00:08:05.070
once you try a few examples and
see what's happening over here.
00:08:05.070 --> 00:08:06.840
By the way, before
I go on, I just
00:08:06.840 --> 00:08:09.820
want to make a note that I'm
going to return to later on.
00:08:09.820 --> 00:08:11.710
Notice, by the way,
there is nothing
00:08:11.710 --> 00:08:15.160
sacred about the right-hand
side of this equation being 0.
00:08:15.160 --> 00:08:18.890
Notice that somehow or other,
the really important factor
00:08:18.890 --> 00:08:23.060
was 2x plus 3y plus 4z.
00:08:23.060 --> 00:08:27.380
In other words, notice that the
other terms led to a constant
00:08:27.380 --> 00:08:29.960
which could have been transposed
onto the right-hand side
00:08:29.960 --> 00:08:33.000
of the equation, and
that somehow or other,
00:08:33.000 --> 00:08:34.830
I can change the constant.
00:08:34.830 --> 00:08:38.190
But if these multipliers in
front-- the 2, 3, and 4--
00:08:38.190 --> 00:08:39.567
stay the same, see?
00:08:39.567 --> 00:08:40.900
Notice what I'm driving at here.
00:08:40.900 --> 00:08:43.940
No matter how I-- let me go
back up here for a second.
00:08:43.940 --> 00:08:49.860
No matter how I change x_0, y_0,
z_0, whatever plane this is,
00:08:49.860 --> 00:08:56.010
it still has A*i plus B*j
plus C*k as a normal vector.
00:08:56.010 --> 00:09:00.320
What I can change is what
point the plane passes through.
00:09:00.320 --> 00:09:04.290
In other words, somehow or other
if I leave A, B, and C fixed,
00:09:04.290 --> 00:09:07.900
but I vary x_0,
y_0, z_0, I generate
00:09:07.900 --> 00:09:10.430
a family of parallel planes.
00:09:10.430 --> 00:09:14.012
And that can be restated
somewhat differently.
00:09:14.012 --> 00:09:15.970
Well I guess if it wasn't
somewhat differently,
00:09:15.970 --> 00:09:18.430
that wouldn't be called
"restated," would it?
00:09:18.430 --> 00:09:19.750
Let's just note this way.
00:09:19.750 --> 00:09:25.840
For fixed A, B, and C, the
equation A*x plus B*y plus C*z
00:09:25.840 --> 00:09:29.180
equals D. See, forget about
the 0 on the right-hand side.
00:09:29.180 --> 00:09:31.280
Let D be an arbitrary constant.
00:09:31.280 --> 00:09:33.470
What I'm saying is
that this equation
00:09:33.470 --> 00:09:37.750
is a family of parallel planes.
00:09:37.750 --> 00:09:40.560
And why is it a family
of parallel planes?
00:09:40.560 --> 00:09:46.200
Because every plane in this
family has, as a normal vector,
00:09:46.200 --> 00:09:50.120
A*i plus B*j plus C*k.
00:09:50.120 --> 00:09:54.826
A*i plus B*j plus C*k.
00:09:54.826 --> 00:09:55.690
OK?
00:09:55.690 --> 00:09:57.690
The second point I
would like to emphasize
00:09:57.690 --> 00:09:59.780
about the equation
of our plane is
00:09:59.780 --> 00:10:03.616
that it's called a linear
equation, meaning--
00:10:03.616 --> 00:10:06.240
and I don't know why I suddenly
switched to small letters here,
00:10:06.240 --> 00:10:08.198
but that certainly doesn't
make any difference.
00:10:08.198 --> 00:10:10.900
With a, b, c, and
d as constants,
00:10:10.900 --> 00:10:15.640
observe that a*x plus b*y plus
c*z equals d is what we call
00:10:15.640 --> 00:10:19.720
a linear algebraic equation
in the variables x, y, and z.
00:10:19.720 --> 00:10:24.690
Namely, each variable appears
multiplied only by a constant.
00:10:24.690 --> 00:10:26.990
And we add these things up.
00:10:26.990 --> 00:10:31.830
And notice that in a way,
a plane should be linear,
00:10:31.830 --> 00:10:34.310
meaning there's no
curvature to a plane
00:10:34.310 --> 00:10:36.000
once it's fixed in space.
00:10:36.000 --> 00:10:40.220
And this sort of generalizes
the idea of the line.
00:10:40.220 --> 00:10:44.250
Remember, the general definition
of a line was of the form what?
00:10:44.250 --> 00:10:47.350
a*x plus b*y equals a constant.
00:10:47.350 --> 00:10:50.280
That was a two-dimensional
linear equation.
00:10:50.280 --> 00:10:53.354
The plane is a three-dimensional
linear equation.
00:10:53.354 --> 00:10:54.770
And one of the
subjects that we'll
00:10:54.770 --> 00:10:57.700
return to later in the
course, but I just mention it
00:10:57.700 --> 00:10:59.990
in passing now, is
that even though you
00:10:59.990 --> 00:11:03.040
can't draw in more than
three-dimensional space,
00:11:03.040 --> 00:11:05.620
if you have 15 variables,
you can certainly
00:11:05.620 --> 00:11:09.220
have a linear equation
in 15 unknowns.
00:11:09.220 --> 00:11:12.010
And the interesting point in
calculus of several variables
00:11:12.010 --> 00:11:15.980
is that even when you run out of
pictures-- when you can't draw
00:11:15.980 --> 00:11:18.690
the situation--
the linear equation
00:11:18.690 --> 00:11:22.960
plays a very, very special role
in the development of calculus
00:11:22.960 --> 00:11:26.050
of several variables,
analogous to what a line does
00:11:26.050 --> 00:11:28.870
for a curve in the
case of one variable
00:11:28.870 --> 00:11:30.920
and what a plane
does for a surface
00:11:30.920 --> 00:11:33.235
in the case of two variables.
00:11:33.235 --> 00:11:35.580
What I wanted to
emphasize though, also--
00:11:35.580 --> 00:11:38.350
and we'll come back to
this in very short order--
00:11:38.350 --> 00:11:42.980
is that a plane has
two degrees of freedom.
00:11:42.980 --> 00:11:44.660
Meaning what?
00:11:44.660 --> 00:11:48.977
That in a plane, observe that
given the linear equation,
00:11:48.977 --> 00:11:49.560
you have what?
00:11:49.560 --> 00:11:51.750
One linear equation
and three unknowns.
00:11:51.750 --> 00:11:56.310
It requires that you
pick two of the variables
00:11:56.310 --> 00:11:58.570
before the rest of the
equation is determined.
00:11:58.570 --> 00:12:03.770
In other words, if given
x + 2y + 3z = 6, if I say,
00:12:03.770 --> 00:12:06.850
let x be 15, notice
that I have what?
00:12:06.850 --> 00:12:13.210
15 + 2y + 3z = 6, which gives
me one equation and the two
00:12:13.210 --> 00:12:14.910
unknowns y and z.
00:12:14.910 --> 00:12:17.680
That to actually
uniquely fix anything,
00:12:17.680 --> 00:12:21.350
I must specify what two of
the three variables are.
00:12:21.350 --> 00:12:23.040
In other words,
in this equation,
00:12:23.040 --> 00:12:26.490
I can choose any two of the
three variables at random,
00:12:26.490 --> 00:12:28.350
and solve for the third.
00:12:28.350 --> 00:12:31.900
By the way, if anybody
is having difficulty
00:12:31.900 --> 00:12:34.090
understanding the
difference between the 6
00:12:34.090 --> 00:12:36.850
being on the right-hand
side as we have it now
00:12:36.850 --> 00:12:40.310
and the 0 as we
used it originally,
00:12:40.310 --> 00:12:45.110
notice that we can very
quickly find a point which
00:12:45.110 --> 00:12:47.200
this plane passes through.
00:12:47.200 --> 00:12:49.870
For example, among
other things, just set y
00:12:49.870 --> 00:12:53.350
and z equal to 0, in
which case x is 6.
00:12:53.350 --> 00:12:58.690
So certainly one point in
this plane is (6, 0, 0).
00:12:58.690 --> 00:13:01.920
What is a vector
perpendicular to this plane?
00:13:01.920 --> 00:13:06.700
It's the vector 1i
plus 2j plus 3k.
00:13:06.700 --> 00:13:09.790
So using the standard
form, we could write what?
00:13:09.790 --> 00:13:20.490
x minus 6 plus 2 times y
minus 0 plus 3 times z minus 0
00:13:20.490 --> 00:13:21.620
equals 0.
00:13:21.620 --> 00:13:24.450
Notice, by the way,
that algebraically,
00:13:24.450 --> 00:13:27.650
these two equations
are equivalent.
00:13:27.650 --> 00:13:30.230
But that in this form,
this tells me what?
00:13:30.230 --> 00:13:33.770
This specifies a point that
the plane passes through,
00:13:33.770 --> 00:13:34.300
namely what?
00:13:34.300 --> 00:13:37.680
This is the plane that
passes through (6, 0, 0),
00:13:37.680 --> 00:13:43.250
and has the vector i plus
2j plus 3k as a normal.
00:13:43.250 --> 00:13:45.740
By the way, you could do
this in different ways.
00:13:45.740 --> 00:13:48.050
Some person might
say, why couldn't you
00:13:48.050 --> 00:13:54.210
have transposed the 6 over
here, then taken 3z minus 6?
00:13:54.210 --> 00:13:54.710
You see?
00:13:54.710 --> 00:13:58.510
Why don't you let x and y be
0 and solve this equation,
00:13:58.510 --> 00:14:00.380
and get that z equals 2?
00:14:00.380 --> 00:14:05.580
In other words, isn't (0, 0,
2) also a point in the plane?
00:14:05.580 --> 00:14:07.650
And the answer is, yes it is.
00:14:07.650 --> 00:14:10.080
And you could have written
the equation now as what?
00:14:10.080 --> 00:14:17.110
x minus 0 plus twice y minus
0 plus 3 times z minus 2
00:14:17.110 --> 00:14:18.310
equals 0.
00:14:18.310 --> 00:14:19.360
That would be what?
00:14:19.360 --> 00:14:23.760
The equation of the plane
that passed through (0, 0, 2),
00:14:23.760 --> 00:14:28.250
and had as its normal
i plus 2j plus 3k.
00:14:28.250 --> 00:14:33.520
Of course what happens is
that (6, 0, 0) and (0, 0, 2)
00:14:33.520 --> 00:14:35.270
belong to the same plane.
00:14:35.270 --> 00:14:37.410
I mean, that's another
thing to keep in mind here,
00:14:37.410 --> 00:14:39.580
that the plane that
we're talking about
00:14:39.580 --> 00:14:41.620
passes through more
than one point.
00:14:41.620 --> 00:14:47.820
So x_0, y_0, z_0 can be played
by an infinity of choices.
00:14:47.820 --> 00:14:50.030
At any rate, let's
let that go now
00:14:50.030 --> 00:14:53.870
as the equation of our
plane, and let's talk now
00:14:53.870 --> 00:14:55.480
about the equation of a line.
00:14:55.480 --> 00:14:56.600
That's a plane.
00:14:56.600 --> 00:14:59.260
Let's talk about the
equation of a line.
00:14:59.260 --> 00:15:01.070
How do we determine a line?
00:15:01.070 --> 00:15:02.830
In two-dimensional
space, we said
00:15:02.830 --> 00:15:06.150
we needed to know a point
on the line and the slope.
00:15:06.150 --> 00:15:08.010
And another way
of saying that is,
00:15:08.010 --> 00:15:10.430
we need to know a
point on the line
00:15:10.430 --> 00:15:13.990
and we would like to know a
line parallel to the given line.
00:15:13.990 --> 00:15:16.790
In vector language,
what we say is, OK,
00:15:16.790 --> 00:15:19.200
let's suppose we're
given the line l.
00:15:19.200 --> 00:15:21.685
And we know that the
vector V, whose components
00:15:21.685 --> 00:15:25.800
are A, B, and C, that that
vector is parallel to l
00:15:25.800 --> 00:15:28.550
and that the point P_0
whose coordinates are
00:15:28.550 --> 00:15:32.940
(x_0, y_0, z_0), that that
point is on the line l.
00:15:32.940 --> 00:15:36.320
Then the question is, how do we
find the equation of the line
00:15:36.320 --> 00:15:37.030
l?
00:15:37.030 --> 00:15:41.150
And again, vector methods
come to our aid very nicely.
00:15:41.150 --> 00:15:46.330
What we say is, let's pick
any other point P in space.
00:15:46.330 --> 00:15:47.500
All right?
00:15:47.500 --> 00:15:52.390
What does it mean if the
point P is on the line l?
00:15:52.390 --> 00:15:55.060
If the point P is on
the line l, since we
00:15:55.060 --> 00:15:57.120
want to use vector
methods, let's
00:15:57.120 --> 00:16:01.750
simply observe that
since l is parallel to V,
00:16:01.750 --> 00:16:06.300
the vector P_0 P,
being parallel to V,
00:16:06.300 --> 00:16:09.120
must be a scalar
multiple of V. That's
00:16:09.120 --> 00:16:12.860
what parallel means for
vectors, scalar multiple.
00:16:12.860 --> 00:16:17.110
So P_0 p is equal to
some constant times V.
00:16:17.110 --> 00:16:19.470
And let me pause
here for a moment
00:16:19.470 --> 00:16:24.120
to point out that this
constant is really a variable.
00:16:24.120 --> 00:16:24.870
That sounds awful.
00:16:24.870 --> 00:16:26.470
How can a constant
be a variable?
00:16:26.470 --> 00:16:30.510
What I mean of course, is that
P was any point in this line.
00:16:30.510 --> 00:16:34.700
Notice that t determines
the length of P_0 P,
00:16:34.700 --> 00:16:38.370
and how long P_0 P is, is going
to depend on where I choose
00:16:38.370 --> 00:16:41.340
P. In other words, for
different choices of P,
00:16:41.340 --> 00:16:43.260
I get a different
scalar multiple.
00:16:43.260 --> 00:16:47.900
And by the way, if I choose
P on the wrong side of P_0,
00:16:47.900 --> 00:16:49.950
as I've deliberately
done over here,
00:16:49.950 --> 00:16:53.805
notice that P_0 P has
the opposite sense of V.
00:16:53.805 --> 00:16:56.070
So that t can even be negative.
00:16:56.070 --> 00:16:59.430
In other words, not only is t a
variable, but if it's negative,
00:16:59.430 --> 00:17:02.570
it means that P_0 P has
the opposite sense of V.
00:17:02.570 --> 00:17:04.819
If it's positive, they
have the same sense.
00:17:04.819 --> 00:17:06.589
But I'm not going to
belabor that point.
00:17:06.589 --> 00:17:09.660
What I'm now going to do is,
in Cartesian coordinates,
00:17:09.660 --> 00:17:11.700
see what this equation tells me.
00:17:11.700 --> 00:17:13.829
And right away,
it tells me what?
00:17:13.829 --> 00:17:19.250
That P_0 P is that vector whose
components are x minus x_0, y
00:17:19.250 --> 00:17:22.760
minus y_0, and z minus z_0.
00:17:22.760 --> 00:17:25.050
What vector is t times V?
00:17:25.050 --> 00:17:29.440
Well, V, we saw, had as
components, A, B, and C.
00:17:29.440 --> 00:17:32.350
And in Cartesian coordinates,
multiplying a vector
00:17:32.350 --> 00:17:37.540
by a scalar simply multiplies
each component by that scalar.
00:17:37.540 --> 00:17:41.050
So in other words, t times V is
the vector whose components are
00:17:41.050 --> 00:17:44.630
t*A, t*B, and t*C.
00:17:44.630 --> 00:17:47.270
We also know, in
Cartesian coordinates,
00:17:47.270 --> 00:17:49.800
that the only way that
two vectors can be equal
00:17:49.800 --> 00:17:51.840
is component by component.
00:17:51.840 --> 00:17:53.380
And that tells us what?
00:17:53.380 --> 00:17:56.510
That x minus x_0
must equal t times
00:17:56.510 --> 00:18:01.750
A. y minus y_0 must
equal t times B.
00:18:01.750 --> 00:18:06.180
And z minus z_0 must
equal t times C. That's
00:18:06.180 --> 00:18:08.790
these three equations here.
00:18:08.790 --> 00:18:10.550
What do all of these
three equations
00:18:10.550 --> 00:18:13.020
have in common numerically?
00:18:13.020 --> 00:18:15.370
They all have the factor t.
00:18:15.370 --> 00:18:18.640
And consequently, I can solve
each of these three equations
00:18:18.640 --> 00:18:20.620
for t.
00:18:20.620 --> 00:18:21.460
Namely, what?
00:18:21.460 --> 00:18:23.950
Divide both sides of
this equation by A,
00:18:23.950 --> 00:18:27.550
both sides of this equation by
B, both sides of this equation
00:18:27.550 --> 00:18:32.140
by C, being very careful that
neither A, B, nor C are 0.
00:18:32.140 --> 00:18:35.930
By the way, if they are 0,
straightforward ramifications
00:18:35.930 --> 00:18:38.920
take place that we'll leave
for the textbook to explain.
00:18:38.920 --> 00:18:40.570
Don't worry about
that part right now.
00:18:40.570 --> 00:18:42.550
We don't want to get
bogged down in that.
00:18:42.550 --> 00:18:46.050
But at any rate, if
we now go from here
00:18:46.050 --> 00:18:48.800
to see what that
says, we now wind up
00:18:48.800 --> 00:18:51.720
with the standard equation
of the straight line.
00:18:51.720 --> 00:18:59.250
Namely, if you have x minus
x_0 over A equals y minus y_0
00:18:59.250 --> 00:19:05.080
over B equals z minus z_0
over C equals some constant t,
00:19:05.080 --> 00:19:09.110
that particular form is
called the standard equation
00:19:09.110 --> 00:19:10.140
for a straight line.
00:19:10.140 --> 00:19:11.700
What straight line is it?
00:19:11.700 --> 00:19:14.880
It's the line which passes
through the point (x_0, y_0,
00:19:14.880 --> 00:19:21.910
z_0) and is parallel to the
vector A*i plus B*j plus C*k.
00:19:21.910 --> 00:19:26.200
By means of an example,
x minus 1 over 4
00:19:26.200 --> 00:19:31.590
equals y minus 5 over 3
equals z minus 6 over 7
00:19:31.590 --> 00:19:35.620
is the equation-- it's
one equation, really.
00:19:35.620 --> 00:19:38.557
It's the equation of a line
which has what property?
00:19:38.557 --> 00:19:40.140
It passes through
the point (1, 5, 6).
00:19:42.680 --> 00:19:48.670
And it's parallel to the
vector 4i plus 3j plus 7k.
00:19:48.670 --> 00:19:53.887
And by the way, I have to be
very, very on my guard here.
00:19:53.887 --> 00:19:55.470
There's something
very deceptive here.
00:19:55.470 --> 00:19:57.580
The equation of a
plane and a line
00:19:57.580 --> 00:19:59.400
are very, very much different.
00:19:59.400 --> 00:20:02.420
But they look enough alike
so it may confuse you.
00:20:02.420 --> 00:20:04.360
You know, it reminds
me of my daughter,
00:20:04.360 --> 00:20:07.069
who I get a lot of stories from,
was eating a sandwich one day.
00:20:07.069 --> 00:20:09.360
And I asked her what kind of
a sandwich she was eating.
00:20:09.360 --> 00:20:11.443
And she said it was like
a peanut butter and jelly
00:20:11.443 --> 00:20:12.139
sandwich.
00:20:12.139 --> 00:20:14.680
And I never heard of a sandwich
that was like a peanut butter
00:20:14.680 --> 00:20:15.471
and jelly sandwich.
00:20:15.471 --> 00:20:18.614
So I looked at it to see what it
was, and it was ham and cheese.
00:20:18.614 --> 00:20:20.780
And I say, why did you say
it was like peanut butter
00:20:20.780 --> 00:20:21.279
and jelly?
00:20:21.279 --> 00:20:23.140
And she says well, it
was two things in it.
00:20:23.140 --> 00:20:23.470
All right?
00:20:23.470 --> 00:20:23.969
Lookit.
00:20:23.969 --> 00:20:25.850
The equation of a
line and the plane
00:20:25.850 --> 00:20:29.450
have three things
in it-- x, y, and z.
00:20:29.450 --> 00:20:32.470
But to juxtaposition these,
let me write down the two
00:20:32.470 --> 00:20:35.810
things that may look confusing.
00:20:35.810 --> 00:20:37.806
Let's suppose I write this down.
00:20:37.806 --> 00:20:38.930
You see what I'm doing now?
00:20:38.930 --> 00:20:42.660
What I'm doing now is I'm
changing the equal signs here
00:20:42.660 --> 00:20:45.940
to plus signs and bringing
up the denominators here.
00:20:45.940 --> 00:20:47.350
See, this is a line.
00:20:47.350 --> 00:20:48.910
This is a plane.
00:20:48.910 --> 00:20:50.910
What plane is this?
00:20:50.910 --> 00:20:55.640
This is the plane which passes
through the point 1 comma
00:20:55.640 --> 00:21:02.700
5, comma 6, and has the line of
the vector 4i plus 3j plus 7k
00:21:02.700 --> 00:21:04.890
as its normal.
00:21:04.890 --> 00:21:06.460
How can I best
explain this to you
00:21:06.460 --> 00:21:08.210
to keep this straight
in your mind?
00:21:08.210 --> 00:21:10.390
Well, I think the
easiest way-- and again,
00:21:10.390 --> 00:21:12.040
notice what I'm
saying, see the x,
00:21:12.040 --> 00:21:14.540
y's, and z's here, the
x, y's and z's here.
00:21:14.540 --> 00:21:15.550
Which is which?
00:21:15.550 --> 00:21:18.620
The easiest way is to keep
track of degrees of freedom.
00:21:18.620 --> 00:21:20.870
Remember in the
plane, we said lookit.
00:21:20.870 --> 00:21:23.470
You can pick two of
the variables at random
00:21:23.470 --> 00:21:25.310
and solve for the third.
00:21:25.310 --> 00:21:28.970
I claim in this system--
in this system here,
00:21:28.970 --> 00:21:31.330
there is only one
degree of freedom.
00:21:31.330 --> 00:21:35.160
The line has one
degree of freedom.
00:21:35.160 --> 00:21:36.610
Namely, let's
repeat this example,
00:21:36.610 --> 00:21:39.130
so we don't have to keep
looking back to the board here.
00:21:39.130 --> 00:21:43.270
Let's take x minus 1 over
4 equals y minus 5 over 3
00:21:43.270 --> 00:21:45.662
equals z minus 6 over 7.
00:21:45.662 --> 00:21:47.620
And since we don't like
to work with fractions,
00:21:47.620 --> 00:21:51.310
I'll pick a number
that works out nicely.
00:21:51.310 --> 00:21:54.570
I say, OK, let's see
what happens when x is 9.
00:21:54.570 --> 00:21:56.070
Now here's the whole point.
00:21:56.070 --> 00:22:01.210
As soon as I say that x equals
9, as soon as I let x equal 9,
00:22:01.210 --> 00:22:04.140
this is fixed.
00:22:04.140 --> 00:22:04.640
Right?
00:22:04.640 --> 00:22:07.480
In fact, what does it become
fixed as soon as I do this?
00:22:07.480 --> 00:22:11.480
As soon as x equals 9,
x minus 1 over 4 is 2.
00:22:11.480 --> 00:22:16.627
Now notice that y minus
5 over 3 has to equal 2.
00:22:16.627 --> 00:22:17.960
Well I have no more choice then.
00:22:17.960 --> 00:22:23.510
If y minus 5 over 3 has to equal
2, and also z minus 6 over 7
00:22:23.510 --> 00:22:26.950
has to equal 2-- you see
what I'm saying here?
00:22:26.950 --> 00:22:31.740
This fixes the fact that y must
be 11, and that z must be 20.
00:22:31.740 --> 00:22:33.980
In other words, the
choice of x equals
00:22:33.980 --> 00:22:39.080
9 forces me to make y
equal 11 and z equal 20.
00:22:39.080 --> 00:22:41.010
One degree of freedom.
00:22:41.010 --> 00:22:43.030
And by the way, if you
want to see this thing
00:22:43.030 --> 00:22:45.440
from a geometrical point
of view, what we're saying
00:22:45.440 --> 00:22:50.800
is, visualize this line cutting
through space, all right?
00:22:50.800 --> 00:22:53.120
Notice that directly
on that line,
00:22:53.120 --> 00:22:56.430
only one point will have
its x-coordinate equal to 9.
00:22:56.430 --> 00:22:58.110
And what we're
saying is, the point
00:22:58.110 --> 00:23:00.870
on that line whose
x-coordinate is 9
00:23:00.870 --> 00:23:05.080
is the point 9
comma 11 comma 20.
00:23:05.080 --> 00:23:05.710
OK?
00:23:05.710 --> 00:23:09.650
One degree of freedom again.
00:23:09.650 --> 00:23:11.450
That's very, very
crucial for you to see.
00:23:11.450 --> 00:23:14.500
By the way, I guess one thing
that bothers a lot of students
00:23:14.500 --> 00:23:17.220
is the fact that they read
this as two separate equations.
00:23:17.220 --> 00:23:20.820
They say, you know, why
isn't this x minus 1 over 4
00:23:20.820 --> 00:23:22.780
equals y minus 5 over 3?
00:23:22.780 --> 00:23:25.100
Why can't I treat
that as one equation?
00:23:25.100 --> 00:23:30.240
Or why couldn't I take y minus 5
over 3 equals z minus 6 over 7?
00:23:30.240 --> 00:23:33.180
Or why couldn't I
take x minus 1 over 4
00:23:33.180 --> 00:23:35.870
and say that equals
z minus 6 over 7?
00:23:35.870 --> 00:23:38.890
And the answer is, that
by itself isn't enough.
00:23:38.890 --> 00:23:41.210
But rather than give
you a negative answer,
00:23:41.210 --> 00:23:44.030
let me give you a positive one.
00:23:44.030 --> 00:23:47.560
Let me close today's lesson with
this particular illustration.
00:23:47.560 --> 00:23:53.510
Suppose we had solved x minus 1
over 4 equals y minus 5 over 3.
00:23:53.510 --> 00:23:56.880
What we would have
obtained is the equation
00:23:56.880 --> 00:23:59.380
4y minus 3x equals 17.
00:23:59.380 --> 00:24:01.070
Now this is very dangerous.
00:24:01.070 --> 00:24:04.870
When you look at the equation
4y minus 3x equals 17,
00:24:04.870 --> 00:24:06.450
I'll bet you
dollars to doughnuts
00:24:06.450 --> 00:24:12.370
you tend to think of this as
a line rather than as a plane.
00:24:12.370 --> 00:24:14.840
But the interesting
thing is, notice
00:24:14.840 --> 00:24:17.240
that the way we
got this equation
00:24:17.240 --> 00:24:20.840
was ignoring the
z-coordinate of our points.
00:24:20.840 --> 00:24:22.800
And what we're really
saying is, let's forget
00:24:22.800 --> 00:24:24.410
about the z-coordinate.
00:24:24.410 --> 00:24:27.680
In other words,
4y minus 3x equals
00:24:27.680 --> 00:24:32.900
17 may be viewed as a line,
but in this particular case,
00:24:32.900 --> 00:24:33.740
it's a plane.
00:24:33.740 --> 00:24:35.270
In fact, what plane is it?
00:24:35.270 --> 00:24:39.470
It's the plane that goes
through the line 4y minus 3x
00:24:39.470 --> 00:24:42.360
equals 17, which
lies on the xy-plane.
00:24:42.360 --> 00:24:45.060
It's the plane that goes
through that line perpendicular
00:24:45.060 --> 00:24:46.310
to the xy-plane.
00:24:46.310 --> 00:24:50.350
By the way, again, if you go
back to part one of this course
00:24:50.350 --> 00:24:53.540
where we stress sets,
the language of sets
00:24:53.540 --> 00:24:55.900
comes to our rescue very nicely.
00:24:55.900 --> 00:24:59.870
The difference between
whether 4y minus 3x equals
00:24:59.870 --> 00:25:03.010
17 is a plane or
whether it's a line
00:25:03.010 --> 00:25:08.100
hinges on whether we're
talking about the set of pairs
00:25:08.100 --> 00:25:12.640
x comma y such that
4y minus 3x equals 17,
00:25:12.640 --> 00:25:16.560
or whether we're talking about
the set of triplets (x, y, z)
00:25:16.560 --> 00:25:20.330
such that 4y minus 3x equals 17.
00:25:20.330 --> 00:25:22.520
In this particular
example, we're
00:25:22.520 --> 00:25:24.230
talking about points in space.
00:25:24.230 --> 00:25:26.290
In other words, our
universe of discourse
00:25:26.290 --> 00:25:31.680
are the points x comma y comma
z, not the points x comma y.
00:25:31.680 --> 00:25:34.650
Well anyway, rather than
to belabor this point, what
00:25:34.650 --> 00:25:39.050
I'm saying is, in the same way
that this equation represents
00:25:39.050 --> 00:25:47.780
a plane, in a similar way, had
we equated y minus 5 over 3
00:25:47.780 --> 00:25:50.380
equals z minus 6
over 7, we would've
00:25:50.380 --> 00:25:57.120
obtained the plane 7y
minus 3z equals 17.
00:25:57.120 --> 00:26:00.430
So if you don't like to look
at our set of three equations,
00:26:00.430 --> 00:26:03.280
if you'd like to look
at these three equations
00:26:03.280 --> 00:26:06.780
in pairs-- you see, if you
want to look at these three
00:26:06.780 --> 00:26:09.220
equations in pairs,
another way of saying
00:26:09.220 --> 00:26:15.220
it is this, that the triple
equality-- x minus 1 over 4
00:26:15.220 --> 00:26:19.620
equals y minus 5 over 3
equals z minus 6 over 7--
00:26:19.620 --> 00:26:23.430
that that may be viewed as
the intersection of the two
00:26:23.430 --> 00:26:27.880
planes-- namely the plane
determined by this equation
00:26:27.880 --> 00:26:30.540
and the plane determined
by this equation.
00:26:30.540 --> 00:26:33.690
Of course, someone can also
say, isn't there a plane
00:26:33.690 --> 00:26:37.610
determined by this
one and this one?
00:26:37.610 --> 00:26:39.430
And the answer is yes, there is.
00:26:39.430 --> 00:26:42.410
Notice that whereas you
have three equalities here,
00:26:42.410 --> 00:26:44.470
only two of them
are independent.
00:26:44.470 --> 00:26:46.590
Namely, as soon as
the first equals
00:26:46.590 --> 00:26:49.260
the second and the
second equals the third,
00:26:49.260 --> 00:26:52.080
the first must equal the third.
00:26:52.080 --> 00:26:52.880
OK?
00:26:52.880 --> 00:26:56.160
But the whole idea is, can
you now see the difference?
00:26:56.160 --> 00:26:58.780
The easiest way I
know of to distinguish
00:26:58.780 --> 00:27:02.860
the difference between the
equation of a line and a plane.
00:27:02.860 --> 00:27:05.790
The plane has two
degrees of freedom.
00:27:05.790 --> 00:27:08.320
The line has but one
degree of freedom.
00:27:08.320 --> 00:27:11.240
And that triple equality
says as soon as you've
00:27:11.240 --> 00:27:13.220
picked one of the
unknowns, you've
00:27:13.220 --> 00:27:15.050
determined all of the others.
00:27:15.050 --> 00:27:16.770
Whereas that string
of plus signs
00:27:16.770 --> 00:27:19.220
says that once you've
determined one,
00:27:19.220 --> 00:27:21.760
you still have
some freedom left.
00:27:21.760 --> 00:27:25.140
Now what we're going
to do is next time
00:27:25.140 --> 00:27:27.510
start a new phase of vectors.
00:27:27.510 --> 00:27:29.980
For the time being,
what we have now done
00:27:29.980 --> 00:27:32.040
is finished, at
least for the moment,
00:27:32.040 --> 00:27:35.950
our preliminary investigation
of three-dimensional space
00:27:35.950 --> 00:27:39.590
as seen through the eyes
of Cartesian coordinates.
00:27:39.590 --> 00:27:41.860
At any rate, until
next time, goodbye.
00:27:44.420 --> 00:27:46.790
Funding for the
publication of this video
00:27:46.790 --> 00:27:51.670
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:27:51.670 --> 00:27:55.850
Help OCW continue to provide
free and open access to MIT
00:27:55.850 --> 00:28:00.260
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at ocw.mit.edu/donate.