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PROFESSOR: Hi.
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00:00:34,590 --> 00:00:36,470
Our lecture today
is on the one hand,
10
00:00:36,470 --> 00:00:40,790
deceptively simple, and on the
other hand, deceptively hard.
11
00:00:40,790 --> 00:00:43,110
That from a certain
point of view,
12
00:00:43,110 --> 00:00:46,620
to talk about the equations
of lines and planes
13
00:00:46,620 --> 00:00:50,200
is, first of all, a
topic that many people
14
00:00:50,200 --> 00:00:52,130
say, "I'm not interested
in studying geometry.
15
00:00:52,130 --> 00:00:54,230
When are we going to
get back to calculus?"
16
00:00:54,230 --> 00:00:58,120
And this is very analogous
to our problem of part one
17
00:00:58,120 --> 00:01:00,270
in calculus, if
you recall, where
18
00:01:00,270 --> 00:01:02,490
we started with
analytic geometry,
19
00:01:02,490 --> 00:01:04,470
and the question
came up, why do we
20
00:01:04,470 --> 00:01:07,610
need geometry when what
we really wanted to study
21
00:01:07,610 --> 00:01:08,470
was calculus?
22
00:01:08,470 --> 00:01:11,400
And the idea of graphs
played a very important role
23
00:01:11,400 --> 00:01:12,060
in calculus.
24
00:01:12,060 --> 00:01:16,070
We found out, for example, also,
that the relatively harmless
25
00:01:16,070 --> 00:01:19,090
straight line was a
rather crucial curve.
26
00:01:19,090 --> 00:01:23,400
Namely, given any curve, no
matter how random in the plane,
27
00:01:23,400 --> 00:01:25,640
provided only that
it was smooth,
28
00:01:25,640 --> 00:01:28,140
we were able to
approximate that curve
29
00:01:28,140 --> 00:01:32,170
by a sequence of tangent
lines to various points.
30
00:01:32,170 --> 00:01:35,530
And in a similar way,
one finds that planes
31
00:01:35,530 --> 00:01:39,830
will do for functions of
two real variables what
32
00:01:39,830 --> 00:01:42,620
the equations of lines
did for functions
33
00:01:42,620 --> 00:01:44,990
of a single real variable.
34
00:01:44,990 --> 00:01:47,940
As I say, we'll talk about that
in more detail as we go along.
35
00:01:47,940 --> 00:01:50,630
The other nice part
about this lesson
36
00:01:50,630 --> 00:01:54,350
is the fact that we can now take
some of our vector properties
37
00:01:54,350 --> 00:01:56,350
that we've learned
from the past lessons
38
00:01:56,350 --> 00:01:59,190
and hit what I call
the home run ball, once
39
00:01:59,190 --> 00:02:02,400
and for all now, finish
off what we've started.
40
00:02:02,400 --> 00:02:06,090
And that is, in the study of
three-dimensional geometry,
41
00:02:06,090 --> 00:02:09,360
the crucial characteristics
are those building blocks
42
00:02:09,360 --> 00:02:11,220
called lines and planes.
43
00:02:11,220 --> 00:02:13,610
And by finding
convenient recipes
44
00:02:13,610 --> 00:02:15,810
for expressing lines
and planes, we'll
45
00:02:15,810 --> 00:02:18,190
be most of the way
home as far as using
46
00:02:18,190 --> 00:02:19,640
this material is concerned.
47
00:02:19,640 --> 00:02:22,690
Again, we'll hit mainly
the highlights today.
48
00:02:22,690 --> 00:02:25,940
The remainder of the
material will be covered,
49
00:02:25,940 --> 00:02:29,280
I hope, adequately between
the text and the exercises.
50
00:02:29,280 --> 00:02:32,740
At any rate then,
the lesson today
51
00:02:32,740 --> 00:02:36,050
is "Equations of
Lines and Planes."
52
00:02:36,050 --> 00:02:39,880
And to refresh what I just
said before, the little ratio--
53
00:02:39,880 --> 00:02:45,510
planes are to surfaces what
lines are to curves-- that we
54
00:02:45,510 --> 00:02:47,990
can approximate curves
by tangent lines,
55
00:02:47,990 --> 00:02:52,240
we can approximate smooth
surfaces by tangent planes.
56
00:02:52,240 --> 00:02:54,630
Now what we would
like to do is go back
57
00:02:54,630 --> 00:02:58,322
to Cartesian coordinates and
find the equation of a plane.
58
00:02:58,322 --> 00:03:00,280
The first question that
we asked before we even
59
00:03:00,280 --> 00:03:03,920
use any coordinate system is
just how much information do
60
00:03:03,920 --> 00:03:06,950
you need before you
can determine a plane?
61
00:03:06,950 --> 00:03:10,880
Well one way we know of is to
know three points in the plane.
62
00:03:10,880 --> 00:03:13,560
Another way is analogous
to the line case,
63
00:03:13,560 --> 00:03:16,980
where to determine a line, you
needed a point and a slope.
64
00:03:16,980 --> 00:03:20,410
One way of determining
a plane is to know what?
65
00:03:20,410 --> 00:03:24,510
A point in the plane and
the direction of the plane.
66
00:03:24,510 --> 00:03:26,950
And one way of getting
the direction of the plane
67
00:03:26,950 --> 00:03:30,644
is to fix a normal, a
perpendicular to the plane.
68
00:03:30,644 --> 00:03:32,810
In other words, the approach
that we're going to use
69
00:03:32,810 --> 00:03:35,220
is, let's suppose that
we know a point that we
70
00:03:35,220 --> 00:03:36,710
want to be in our plane.
71
00:03:36,710 --> 00:03:38,890
And we'll call that point P_0.
72
00:03:38,890 --> 00:03:43,120
We'll denote it generically
by (x_0, y_0, z_0).
73
00:03:43,120 --> 00:03:48,100
And let's let capital N denote
the vector A*i plus B*j plus
74
00:03:48,100 --> 00:03:51,280
C*k, which is
perpendicular to our plane.
75
00:03:51,280 --> 00:03:51,780
OK?
76
00:03:51,780 --> 00:03:54,850
So we're given a vector
perpendicular to the plane,
77
00:03:54,850 --> 00:03:56,930
we're given a
point in the plane,
78
00:03:56,930 --> 00:03:58,710
and now in Cartesian
coordinates,
79
00:03:58,710 --> 00:04:01,090
we would like to know the
equation of the plane.
80
00:04:01,090 --> 00:04:04,460
And as we do so often, we simply
come back to our little diagram
81
00:04:04,460 --> 00:04:07,200
here that will utilize
vector geometry.
82
00:04:07,200 --> 00:04:10,510
What we say is, OK, here's
a little diagram here.
83
00:04:10,510 --> 00:04:12,660
Here's P_0 in the plane.
84
00:04:12,660 --> 00:04:15,030
Here's N, normal to the plane.
85
00:04:15,030 --> 00:04:17,120
Now we pick any
point P whatsoever,
86
00:04:17,120 --> 00:04:21,279
any point in the whole
world whatsoever, as long
87
00:04:21,279 --> 00:04:23,770
as it's in space,
three-dimensional space.
88
00:04:23,770 --> 00:04:26,740
And what we say is-- what
does it mean for the point P
89
00:04:26,740 --> 00:04:27,980
to be in the plane?
90
00:04:27,980 --> 00:04:33,880
Well, in vector language, for
the point P to be in the plane,
91
00:04:33,880 --> 00:04:38,590
I think it's rather obvious
that P_0 P had better
92
00:04:38,590 --> 00:04:42,800
be perpendicular to
N. On the other hand,
93
00:04:42,800 --> 00:04:45,010
in dot product
language, what does it
94
00:04:45,010 --> 00:04:48,380
mean to say that N is
perpendicular to P_0 P?
95
00:04:48,380 --> 00:04:54,410
That says that the dot product
of N and P_0 P must be 0.
96
00:04:54,410 --> 00:04:57,900
Now just to economize space,
let me utilize the notation
97
00:04:57,900 --> 00:04:59,640
that we've talked
about in the notes
98
00:04:59,640 --> 00:05:02,970
and in the exercises where
in Cartesian coordinates
99
00:05:02,970 --> 00:05:07,180
I'll abbreviate a vector
in i, j, and k components
100
00:05:07,180 --> 00:05:09,095
just by writing
down its components.
101
00:05:09,095 --> 00:05:12,450
In other words, let me
abbreviate N by A comma B comma
102
00:05:12,450 --> 00:05:16,840
C, which stands for A*i plus
B*j plus C*k, et cetera.
103
00:05:16,840 --> 00:05:19,940
Notice the beauty, now, of our
Cartesian coordinate system.
104
00:05:19,940 --> 00:05:23,510
N is the vector whose components
are A, B, and C. What about P_0
105
00:05:23,510 --> 00:05:24,470
P?
106
00:05:24,470 --> 00:05:27,000
Well, that's the
vector whose components
107
00:05:27,000 --> 00:05:32,000
are x minus x_0, y minus
y_0, and z minus z_0,
108
00:05:32,000 --> 00:05:34,590
that beauty of Cartesian
coordinates again.
109
00:05:34,590 --> 00:05:37,350
Consequently, I want the dot
product of these two vectors
110
00:05:37,350 --> 00:05:38,550
to be 0.
111
00:05:38,550 --> 00:05:40,770
But in Cartesian
coordinates, we know
112
00:05:40,770 --> 00:05:43,300
that to dot two
vectors, we simply
113
00:05:43,300 --> 00:05:46,490
multiply corresponding
components and add,
114
00:05:46,490 --> 00:05:48,190
and that results in what?
115
00:05:48,190 --> 00:05:54,360
A times (x minus x_0) plus
B times (y minus y_0),
116
00:05:54,360 --> 00:05:59,100
plus C times (z
minus z_0) equals 0.
117
00:05:59,100 --> 00:06:02,700
By the way, each of these
steps is reversible,
118
00:06:02,700 --> 00:06:07,770
meaning that under these given
conditions, the point (x, y, z)
119
00:06:07,770 --> 00:06:10,700
is in the given
plane if and only
120
00:06:10,700 --> 00:06:14,180
if this equation
here is satisfied.
121
00:06:14,180 --> 00:06:18,000
Another way of
saying that is what?
122
00:06:18,000 --> 00:06:26,140
If A, B, C, x_0, y_0, and
z_0 are given constants,
123
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
124
00:06:32,320 --> 00:06:37,790
z minus z_0 equals 0 is the
equation of the plane which
125
00:06:37,790 --> 00:06:42,550
passes through the
point x_0, y_0, z_0,
126
00:06:42,550 --> 00:06:48,880
and which has the vector A*i
plus B*j plus C*k as its normal
127
00:06:48,880 --> 00:06:49,910
vector.
128
00:06:49,910 --> 00:06:52,130
And perhaps the easiest
way to illustrate this
129
00:06:52,130 --> 00:06:54,660
is again, by means
of an example.
130
00:06:54,660 --> 00:06:57,730
Let me simply write down
a linear expression.
131
00:06:57,730 --> 00:07:02,860
I'll write down 2 times x
minus 1 plus 3 times y plus 2
132
00:07:02,860 --> 00:07:05,680
plus 4 times z minus 5.
133
00:07:05,680 --> 00:07:08,730
My claim is that this is
the special case where
134
00:07:08,730 --> 00:07:14,070
A, B, and C are played by
2, 3, and 4, respectively;
135
00:07:14,070 --> 00:07:21,020
x_0, y_0, and z_0 are played by
1, negative 2, and negative 5--
136
00:07:21,020 --> 00:07:24,230
the same thing as in
our previous study
137
00:07:24,230 --> 00:07:27,610
of ordinary
two-dimensional geometry.
138
00:07:27,610 --> 00:07:30,040
Remember that the standard
form of our equation
139
00:07:30,040 --> 00:07:31,640
uses a minus sign.
140
00:07:31,640 --> 00:07:33,830
Consequently, to
use the equation,
141
00:07:33,830 --> 00:07:37,320
where we see y plus 2,
we should rewrite that
142
00:07:37,320 --> 00:07:40,220
as y minus minus 2.
143
00:07:40,220 --> 00:07:43,520
And what we're saying is that
this equation is then what?
144
00:07:43,520 --> 00:07:50,130
It passes through
the point (1, -2, 5).
145
00:07:50,130 --> 00:07:57,250
And has as its normal the
vector 2*i plus 3*j plus 4*k.
146
00:07:57,250 --> 00:08:00,170
Actually, this is not a
very difficult concept
147
00:08:00,170 --> 00:08:05,070
once you try a few examples and
see what's happening over here.
148
00:08:05,070 --> 00:08:06,840
By the way, before
I go on, I just
149
00:08:06,840 --> 00:08:09,820
want to make a note that I'm
going to return to later on.
150
00:08:09,820 --> 00:08:11,710
Notice, by the way,
there is nothing
151
00:08:11,710 --> 00:08:15,160
sacred about the right-hand
side of this equation being 0.
152
00:08:15,160 --> 00:08:18,890
Notice that somehow or other,
the really important factor
153
00:08:18,890 --> 00:08:23,060
was 2x plus 3y plus 4z.
154
00:08:23,060 --> 00:08:27,380
In other words, notice that the
other terms led to a constant
155
00:08:27,380 --> 00:08:29,960
which could have been transposed
onto the right-hand side
156
00:08:29,960 --> 00:08:33,000
of the equation, and
that somehow or other,
157
00:08:33,000 --> 00:08:34,830
I can change the constant.
158
00:08:34,830 --> 00:08:38,190
But if these multipliers in
front-- the 2, 3, and 4--
159
00:08:38,190 --> 00:08:39,567
stay the same, see?
160
00:08:39,567 --> 00:08:40,900
Notice what I'm driving at here.
161
00:08:40,900 --> 00:08:43,940
No matter how I-- let me go
back up here for a second.
162
00:08:43,940 --> 00:08:49,860
No matter how I change x_0, y_0,
z_0, whatever plane this is,
163
00:08:49,860 --> 00:08:56,010
it still has A*i plus B*j
plus C*k as a normal vector.
164
00:08:56,010 --> 00:09:00,320
What I can change is what
point the plane passes through.
165
00:09:00,320 --> 00:09:04,290
In other words, somehow or other
if I leave A, B, and C fixed,
166
00:09:04,290 --> 00:09:07,900
but I vary x_0,
y_0, z_0, I generate
167
00:09:07,900 --> 00:09:10,430
a family of parallel planes.
168
00:09:10,430 --> 00:09:14,012
And that can be restated
somewhat differently.
169
00:09:14,012 --> 00:09:15,970
Well I guess if it wasn't
somewhat differently,
170
00:09:15,970 --> 00:09:18,430
that wouldn't be called
"restated," would it?
171
00:09:18,430 --> 00:09:19,750
Let's just note this way.
172
00:09:19,750 --> 00:09:25,840
For fixed A, B, and C, the
equation A*x plus B*y plus C*z
173
00:09:25,840 --> 00:09:29,180
equals D. See, forget about
the 0 on the right-hand side.
174
00:09:29,180 --> 00:09:31,280
Let D be an arbitrary constant.
175
00:09:31,280 --> 00:09:33,470
What I'm saying is
that this equation
176
00:09:33,470 --> 00:09:37,750
is a family of parallel planes.
177
00:09:37,750 --> 00:09:40,560
And why is it a family
of parallel planes?
178
00:09:40,560 --> 00:09:46,200
Because every plane in this
family has, as a normal vector,
179
00:09:46,200 --> 00:09:50,120
A*i plus B*j plus C*k.
180
00:09:50,120 --> 00:09:54,826
A*i plus B*j plus C*k.
181
00:09:54,826 --> 00:09:55,690
OK?
182
00:09:55,690 --> 00:09:57,690
The second point I
would like to emphasize
183
00:09:57,690 --> 00:09:59,780
about the equation
of our plane is
184
00:09:59,780 --> 00:10:03,616
that it's called a linear
equation, meaning--
185
00:10:03,616 --> 00:10:06,240
and I don't know why I suddenly
switched to small letters here,
186
00:10:06,240 --> 00:10:08,198
but that certainly doesn't
make any difference.
187
00:10:08,198 --> 00:10:10,900
With a, b, c, and
d as constants,
188
00:10:10,900 --> 00:10:15,640
observe that a*x plus b*y plus
c*z equals d is what we call
189
00:10:15,640 --> 00:10:19,720
a linear algebraic equation
in the variables x, y, and z.
190
00:10:19,720 --> 00:10:24,690
Namely, each variable appears
multiplied only by a constant.
191
00:10:24,690 --> 00:10:26,990
And we add these things up.
192
00:10:26,990 --> 00:10:31,830
And notice that in a way,
a plane should be linear,
193
00:10:31,830 --> 00:10:34,310
meaning there's no
curvature to a plane
194
00:10:34,310 --> 00:10:36,000
once it's fixed in space.
195
00:10:36,000 --> 00:10:40,220
And this sort of generalizes
the idea of the line.
196
00:10:40,220 --> 00:10:44,250
Remember, the general definition
of a line was of the form what?
197
00:10:44,250 --> 00:10:47,350
a*x plus b*y equals a constant.
198
00:10:47,350 --> 00:10:50,280
That was a two-dimensional
linear equation.
199
00:10:50,280 --> 00:10:53,354
The plane is a three-dimensional
linear equation.
200
00:10:53,354 --> 00:10:54,770
And one of the
subjects that we'll
201
00:10:54,770 --> 00:10:57,700
return to later in the
course, but I just mention it
202
00:10:57,700 --> 00:10:59,990
in passing now, is
that even though you
203
00:10:59,990 --> 00:11:03,040
can't draw in more than
three-dimensional space,
204
00:11:03,040 --> 00:11:05,620
if you have 15 variables,
you can certainly
205
00:11:05,620 --> 00:11:09,220
have a linear equation
in 15 unknowns.
206
00:11:09,220 --> 00:11:12,010
And the interesting point in
calculus of several variables
207
00:11:12,010 --> 00:11:15,980
is that even when you run out of
pictures-- when you can't draw
208
00:11:15,980 --> 00:11:18,690
the situation--
the linear equation
209
00:11:18,690 --> 00:11:22,960
plays a very, very special role
in the development of calculus
210
00:11:22,960 --> 00:11:26,050
of several variables,
analogous to what a line does
211
00:11:26,050 --> 00:11:28,870
for a curve in the
case of one variable
212
00:11:28,870 --> 00:11:30,920
and what a plane
does for a surface
213
00:11:30,920 --> 00:11:33,235
in the case of two variables.
214
00:11:33,235 --> 00:11:35,580
What I wanted to
emphasize though, also--
215
00:11:35,580 --> 00:11:38,350
and we'll come back to
this in very short order--
216
00:11:38,350 --> 00:11:42,980
is that a plane has
two degrees of freedom.
217
00:11:42,980 --> 00:11:44,660
Meaning what?
218
00:11:44,660 --> 00:11:48,977
That in a plane, observe that
given the linear equation,
219
00:11:48,977 --> 00:11:49,560
you have what?
220
00:11:49,560 --> 00:11:51,750
One linear equation
and three unknowns.
221
00:11:51,750 --> 00:11:56,310
It requires that you
pick two of the variables
222
00:11:56,310 --> 00:11:58,570
before the rest of the
equation is determined.
223
00:11:58,570 --> 00:12:03,770
In other words, if given
x + 2y + 3z = 6, if I say,
224
00:12:03,770 --> 00:12:06,850
let x be 15, notice
that I have what?
225
00:12:06,850 --> 00:12:13,210
15 + 2y + 3z = 6, which gives
me one equation and the two
226
00:12:13,210 --> 00:12:14,910
unknowns y and z.
227
00:12:14,910 --> 00:12:17,680
That to actually
uniquely fix anything,
228
00:12:17,680 --> 00:12:21,350
I must specify what two of
the three variables are.
229
00:12:21,350 --> 00:12:23,040
In other words,
in this equation,
230
00:12:23,040 --> 00:12:26,490
I can choose any two of the
three variables at random,
231
00:12:26,490 --> 00:12:28,350
and solve for the third.
232
00:12:28,350 --> 00:12:31,900
By the way, if anybody
is having difficulty
233
00:12:31,900 --> 00:12:34,090
understanding the
difference between the 6
234
00:12:34,090 --> 00:12:36,850
being on the right-hand
side as we have it now
235
00:12:36,850 --> 00:12:40,310
and the 0 as we
used it originally,
236
00:12:40,310 --> 00:12:45,110
notice that we can very
quickly find a point which
237
00:12:45,110 --> 00:12:47,200
this plane passes through.
238
00:12:47,200 --> 00:12:49,870
For example, among
other things, just set y
239
00:12:49,870 --> 00:12:53,350
and z equal to 0, in
which case x is 6.
240
00:12:53,350 --> 00:12:58,690
So certainly one point in
this plane is (6, 0, 0).
241
00:12:58,690 --> 00:13:01,920
What is a vector
perpendicular to this plane?
242
00:13:01,920 --> 00:13:06,700
It's the vector 1i
plus 2j plus 3k.
243
00:13:06,700 --> 00:13:09,790
So using the standard
form, we could write what?
244
00:13:09,790 --> 00:13:20,490
x minus 6 plus 2 times y
minus 0 plus 3 times z minus 0
245
00:13:20,490 --> 00:13:21,620
equals 0.
246
00:13:21,620 --> 00:13:24,450
Notice, by the way,
that algebraically,
247
00:13:24,450 --> 00:13:27,650
these two equations
are equivalent.
248
00:13:27,650 --> 00:13:30,230
But that in this form,
this tells me what?
249
00:13:30,230 --> 00:13:33,770
This specifies a point that
the plane passes through,
250
00:13:33,770 --> 00:13:34,300
namely what?
251
00:13:34,300 --> 00:13:37,680
This is the plane that
passes through (6, 0, 0),
252
00:13:37,680 --> 00:13:43,250
and has the vector i plus
2j plus 3k as a normal.
253
00:13:43,250 --> 00:13:45,740
By the way, you could do
this in different ways.
254
00:13:45,740 --> 00:13:48,050
Some person might
say, why couldn't you
255
00:13:48,050 --> 00:13:54,210
have transposed the 6 over
here, then taken 3z minus 6?
256
00:13:54,210 --> 00:13:54,710
You see?
257
00:13:54,710 --> 00:13:58,510
Why don't you let x and y be
0 and solve this equation,
258
00:13:58,510 --> 00:14:00,380
and get that z equals 2?
259
00:14:00,380 --> 00:14:05,580
In other words, isn't (0, 0,
2) also a point in the plane?
260
00:14:05,580 --> 00:14:07,650
And the answer is, yes it is.
261
00:14:07,650 --> 00:14:10,080
And you could have written
the equation now as what?
262
00:14:10,080 --> 00:14:17,110
x minus 0 plus twice y minus
0 plus 3 times z minus 2
263
00:14:17,110 --> 00:14:18,310
equals 0.
264
00:14:18,310 --> 00:14:19,360
That would be what?
265
00:14:19,360 --> 00:14:23,760
The equation of the plane
that passed through (0, 0, 2),
266
00:14:23,760 --> 00:14:28,250
and had as its normal
i plus 2j plus 3k.
267
00:14:28,250 --> 00:14:33,520
Of course what happens is
that (6, 0, 0) and (0, 0, 2)
268
00:14:33,520 --> 00:14:35,270
belong to the same plane.
269
00:14:35,270 --> 00:14:37,410
I mean, that's another
thing to keep in mind here,
270
00:14:37,410 --> 00:14:39,580
that the plane that
we're talking about
271
00:14:39,580 --> 00:14:41,620
passes through more
than one point.
272
00:14:41,620 --> 00:14:47,820
So x_0, y_0, z_0 can be played
by an infinity of choices.
273
00:14:47,820 --> 00:14:50,030
At any rate, let's
let that go now
274
00:14:50,030 --> 00:14:53,870
as the equation of our
plane, and let's talk now
275
00:14:53,870 --> 00:14:55,480
about the equation of a line.
276
00:14:55,480 --> 00:14:56,600
That's a plane.
277
00:14:56,600 --> 00:14:59,260
Let's talk about the
equation of a line.
278
00:14:59,260 --> 00:15:01,070
How do we determine a line?
279
00:15:01,070 --> 00:15:02,830
In two-dimensional
space, we said
280
00:15:02,830 --> 00:15:06,150
we needed to know a point
on the line and the slope.
281
00:15:06,150 --> 00:15:08,010
And another way
of saying that is,
282
00:15:08,010 --> 00:15:10,430
we need to know a
point on the line
283
00:15:10,430 --> 00:15:13,990
and we would like to know a
line parallel to the given line.
284
00:15:13,990 --> 00:15:16,790
In vector language,
what we say is, OK,
285
00:15:16,790 --> 00:15:19,200
let's suppose we're
given the line l.
286
00:15:19,200 --> 00:15:21,685
And we know that the
vector V, whose components
287
00:15:21,685 --> 00:15:25,800
are A, B, and C, that that
vector is parallel to l
288
00:15:25,800 --> 00:15:28,550
and that the point P_0
whose coordinates are
289
00:15:28,550 --> 00:15:32,940
(x_0, y_0, z_0), that that
point is on the line l.
290
00:15:32,940 --> 00:15:36,320
Then the question is, how do we
find the equation of the line
291
00:15:36,320 --> 00:15:37,030
l?
292
00:15:37,030 --> 00:15:41,150
And again, vector methods
come to our aid very nicely.
293
00:15:41,150 --> 00:15:46,330
What we say is, let's pick
any other point P in space.
294
00:15:46,330 --> 00:15:47,500
All right?
295
00:15:47,500 --> 00:15:52,390
What does it mean if the
point P is on the line l?
296
00:15:52,390 --> 00:15:55,060
If the point P is on
the line l, since we
297
00:15:55,060 --> 00:15:57,120
want to use vector
methods, let's
298
00:15:57,120 --> 00:16:01,750
simply observe that
since l is parallel to V,
299
00:16:01,750 --> 00:16:06,300
the vector P_0 P,
being parallel to V,
300
00:16:06,300 --> 00:16:09,120
must be a scalar
multiple of V. That's
301
00:16:09,120 --> 00:16:12,860
what parallel means for
vectors, scalar multiple.
302
00:16:12,860 --> 00:16:17,110
So P_0 p is equal to
some constant times V.
303
00:16:17,110 --> 00:16:19,470
And let me pause
here for a moment
304
00:16:19,470 --> 00:16:24,120
to point out that this
constant is really a variable.
305
00:16:24,120 --> 00:16:24,870
That sounds awful.
306
00:16:24,870 --> 00:16:26,470
How can a constant
be a variable?
307
00:16:26,470 --> 00:16:30,510
What I mean of course, is that
P was any point in this line.
308
00:16:30,510 --> 00:16:34,700
Notice that t determines
the length of P_0 P,
309
00:16:34,700 --> 00:16:38,370
and how long P_0 P is, is going
to depend on where I choose
310
00:16:38,370 --> 00:16:41,340
P. In other words, for
different choices of P,
311
00:16:41,340 --> 00:16:43,260
I get a different
scalar multiple.
312
00:16:43,260 --> 00:16:47,900
And by the way, if I choose
P on the wrong side of P_0,
313
00:16:47,900 --> 00:16:49,950
as I've deliberately
done over here,
314
00:16:49,950 --> 00:16:53,805
notice that P_0 P has
the opposite sense of V.
315
00:16:53,805 --> 00:16:56,070
So that t can even be negative.
316
00:16:56,070 --> 00:16:59,430
In other words, not only is t a
variable, but if it's negative,
317
00:16:59,430 --> 00:17:02,570
it means that P_0 P has
the opposite sense of V.
318
00:17:02,570 --> 00:17:04,819
If it's positive, they
have the same sense.
319
00:17:04,819 --> 00:17:06,589
But I'm not going to
belabor that point.
320
00:17:06,589 --> 00:17:09,660
What I'm now going to do is,
in Cartesian coordinates,
321
00:17:09,660 --> 00:17:11,700
see what this equation tells me.
322
00:17:11,700 --> 00:17:13,829
And right away,
it tells me what?
323
00:17:13,829 --> 00:17:19,250
That P_0 P is that vector whose
components are x minus x_0, y
324
00:17:19,250 --> 00:17:22,760
minus y_0, and z minus z_0.
325
00:17:22,760 --> 00:17:25,050
What vector is t times V?
326
00:17:25,050 --> 00:17:29,440
Well, V, we saw, had as
components, A, B, and C.
327
00:17:29,440 --> 00:17:32,350
And in Cartesian coordinates,
multiplying a vector
328
00:17:32,350 --> 00:17:37,540
by a scalar simply multiplies
each component by that scalar.
329
00:17:37,540 --> 00:17:41,050
So in other words, t times V is
the vector whose components are
330
00:17:41,050 --> 00:17:44,630
t*A, t*B, and t*C.
331
00:17:44,630 --> 00:17:47,270
We also know, in
Cartesian coordinates,
332
00:17:47,270 --> 00:17:49,800
that the only way that
two vectors can be equal
333
00:17:49,800 --> 00:17:51,840
is component by component.
334
00:17:51,840 --> 00:17:53,380
And that tells us what?
335
00:17:53,380 --> 00:17:56,510
That x minus x_0
must equal t times
336
00:17:56,510 --> 00:18:01,750
A. y minus y_0 must
equal t times B.
337
00:18:01,750 --> 00:18:06,180
And z minus z_0 must
equal t times C. That's
338
00:18:06,180 --> 00:18:08,790
these three equations here.
339
00:18:08,790 --> 00:18:10,550
What do all of these
three equations
340
00:18:10,550 --> 00:18:13,020
have in common numerically?
341
00:18:13,020 --> 00:18:15,370
They all have the factor t.
342
00:18:15,370 --> 00:18:18,640
And consequently, I can solve
each of these three equations
343
00:18:18,640 --> 00:18:20,620
for t.
344
00:18:20,620 --> 00:18:21,460
Namely, what?
345
00:18:21,460 --> 00:18:23,950
Divide both sides of
this equation by A,
346
00:18:23,950 --> 00:18:27,550
both sides of this equation by
B, both sides of this equation
347
00:18:27,550 --> 00:18:32,140
by C, being very careful that
neither A, B, nor C are 0.
348
00:18:32,140 --> 00:18:35,930
By the way, if they are 0,
straightforward ramifications
349
00:18:35,930 --> 00:18:38,920
take place that we'll leave
for the textbook to explain.
350
00:18:38,920 --> 00:18:40,570
Don't worry about
that part right now.
351
00:18:40,570 --> 00:18:42,550
We don't want to get
bogged down in that.
352
00:18:42,550 --> 00:18:46,050
But at any rate, if
we now go from here
353
00:18:46,050 --> 00:18:48,800
to see what that
says, we now wind up
354
00:18:48,800 --> 00:18:51,720
with the standard equation
of the straight line.
355
00:18:51,720 --> 00:18:59,250
Namely, if you have x minus
x_0 over A equals y minus y_0
356
00:18:59,250 --> 00:19:05,080
over B equals z minus z_0
over C equals some constant t,
357
00:19:05,080 --> 00:19:09,110
that particular form is
called the standard equation
358
00:19:09,110 --> 00:19:10,140
for a straight line.
359
00:19:10,140 --> 00:19:11,700
What straight line is it?
360
00:19:11,700 --> 00:19:14,880
It's the line which passes
through the point (x_0, y_0,
361
00:19:14,880 --> 00:19:21,910
z_0) and is parallel to the
vector A*i plus B*j plus C*k.
362
00:19:21,910 --> 00:19:26,200
By means of an example,
x minus 1 over 4
363
00:19:26,200 --> 00:19:31,590
equals y minus 5 over 3
equals z minus 6 over 7
364
00:19:31,590 --> 00:19:35,620
is the equation-- it's
one equation, really.
365
00:19:35,620 --> 00:19:38,557
It's the equation of a line
which has what property?
366
00:19:38,557 --> 00:19:40,140
It passes through
the point (1, 5, 6).
367
00:19:42,680 --> 00:19:48,670
And it's parallel to the
vector 4i plus 3j plus 7k.
368
00:19:48,670 --> 00:19:53,887
And by the way, I have to be
very, very on my guard here.
369
00:19:53,887 --> 00:19:55,470
There's something
very deceptive here.
370
00:19:55,470 --> 00:19:57,580
The equation of a
plane and a line
371
00:19:57,580 --> 00:19:59,400
are very, very much different.
372
00:19:59,400 --> 00:20:02,420
But they look enough alike
so it may confuse you.
373
00:20:02,420 --> 00:20:04,360
You know, it reminds
me of my daughter,
374
00:20:04,360 --> 00:20:07,069
who I get a lot of stories from,
was eating a sandwich one day.
375
00:20:07,069 --> 00:20:09,360
And I asked her what kind of
a sandwich she was eating.
376
00:20:09,360 --> 00:20:11,443
And she said it was like
a peanut butter and jelly
377
00:20:11,443 --> 00:20:12,139
sandwich.
378
00:20:12,139 --> 00:20:14,680
And I never heard of a sandwich
that was like a peanut butter
379
00:20:14,680 --> 00:20:15,471
and jelly sandwich.
380
00:20:15,471 --> 00:20:18,614
So I looked at it to see what it
was, and it was ham and cheese.
381
00:20:18,614 --> 00:20:20,780
And I say, why did you say
it was like peanut butter
382
00:20:20,780 --> 00:20:21,279
and jelly?
383
00:20:21,279 --> 00:20:23,140
And she says well, it
was two things in it.
384
00:20:23,140 --> 00:20:23,470
All right?
385
00:20:23,470 --> 00:20:23,969
Lookit.
386
00:20:23,969 --> 00:20:25,850
The equation of a
line and the plane
387
00:20:25,850 --> 00:20:29,450
have three things
in it-- x, y, and z.
388
00:20:29,450 --> 00:20:32,470
But to juxtaposition these,
let me write down the two
389
00:20:32,470 --> 00:20:35,810
things that may look confusing.
390
00:20:35,810 --> 00:20:37,806
Let's suppose I write this down.
391
00:20:37,806 --> 00:20:38,930
You see what I'm doing now?
392
00:20:38,930 --> 00:20:42,660
What I'm doing now is I'm
changing the equal signs here
393
00:20:42,660 --> 00:20:45,940
to plus signs and bringing
up the denominators here.
394
00:20:45,940 --> 00:20:47,350
See, this is a line.
395
00:20:47,350 --> 00:20:48,910
This is a plane.
396
00:20:48,910 --> 00:20:50,910
What plane is this?
397
00:20:50,910 --> 00:20:55,640
This is the plane which passes
through the point 1 comma
398
00:20:55,640 --> 00:21:02,700
5, comma 6, and has the line of
the vector 4i plus 3j plus 7k
399
00:21:02,700 --> 00:21:04,890
as its normal.
400
00:21:04,890 --> 00:21:06,460
How can I best
explain this to you
401
00:21:06,460 --> 00:21:08,210
to keep this straight
in your mind?
402
00:21:08,210 --> 00:21:10,390
Well, I think the
easiest way-- and again,
403
00:21:10,390 --> 00:21:12,040
notice what I'm
saying, see the x,
404
00:21:12,040 --> 00:21:14,540
y's, and z's here, the
x, y's and z's here.
405
00:21:14,540 --> 00:21:15,550
Which is which?
406
00:21:15,550 --> 00:21:18,620
The easiest way is to keep
track of degrees of freedom.
407
00:21:18,620 --> 00:21:20,870
Remember in the
plane, we said lookit.
408
00:21:20,870 --> 00:21:23,470
You can pick two of
the variables at random
409
00:21:23,470 --> 00:21:25,310
and solve for the third.
410
00:21:25,310 --> 00:21:28,970
I claim in this system--
in this system here,
411
00:21:28,970 --> 00:21:31,330
there is only one
degree of freedom.
412
00:21:31,330 --> 00:21:35,160
The line has one
degree of freedom.
413
00:21:35,160 --> 00:21:36,610
Namely, let's
repeat this example,
414
00:21:36,610 --> 00:21:39,130
so we don't have to keep
looking back to the board here.
415
00:21:39,130 --> 00:21:43,270
Let's take x minus 1 over
4 equals y minus 5 over 3
416
00:21:43,270 --> 00:21:45,662
equals z minus 6 over 7.
417
00:21:45,662 --> 00:21:47,620
And since we don't like
to work with fractions,
418
00:21:47,620 --> 00:21:51,310
I'll pick a number
that works out nicely.
419
00:21:51,310 --> 00:21:54,570
I say, OK, let's see
what happens when x is 9.
420
00:21:54,570 --> 00:21:56,070
Now here's the whole point.
421
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,
422
00:22:01,210 --> 00:22:04,140
this is fixed.
423
00:22:04,140 --> 00:22:04,640
Right?
424
00:22:04,640 --> 00:22:07,480
In fact, what does it become
fixed as soon as I do this?
425
00:22:07,480 --> 00:22:11,480
As soon as x equals 9,
x minus 1 over 4 is 2.
426
00:22:11,480 --> 00:22:16,627
Now notice that y minus
5 over 3 has to equal 2.
427
00:22:16,627 --> 00:22:17,960
Well I have no more choice then.
428
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
429
00:22:23,510 --> 00:22:26,950
has to equal 2-- you see
what I'm saying here?
430
00:22:26,950 --> 00:22:31,740
This fixes the fact that y must
be 11, and that z must be 20.
431
00:22:31,740 --> 00:22:33,980
In other words, the
choice of x equals
432
00:22:33,980 --> 00:22:39,080
9 forces me to make y
equal 11 and z equal 20.
433
00:22:39,080 --> 00:22:41,010
One degree of freedom.
434
00:22:41,010 --> 00:22:43,030
And by the way, if you
want to see this thing
435
00:22:43,030 --> 00:22:45,440
from a geometrical point
of view, what we're saying
436
00:22:45,440 --> 00:22:50,800
is, visualize this line cutting
through space, all right?
437
00:22:50,800 --> 00:22:53,120
Notice that directly
on that line,
438
00:22:53,120 --> 00:22:56,430
only one point will have
its x-coordinate equal to 9.
439
00:22:56,430 --> 00:22:58,110
And what we're
saying is, the point
440
00:22:58,110 --> 00:23:00,870
on that line whose
x-coordinate is 9
441
00:23:00,870 --> 00:23:05,080
is the point 9
comma 11 comma 20.
442
00:23:05,080 --> 00:23:05,710
OK?
443
00:23:05,710 --> 00:23:09,650
One degree of freedom again.
444
00:23:09,650 --> 00:23:11,450
That's very, very
crucial for you to see.
445
00:23:11,450 --> 00:23:14,500
By the way, I guess one thing
that bothers a lot of students
446
00:23:14,500 --> 00:23:17,220
is the fact that they read
this as two separate equations.
447
00:23:17,220 --> 00:23:20,820
They say, you know, why
isn't this x minus 1 over 4
448
00:23:20,820 --> 00:23:22,780
equals y minus 5 over 3?
449
00:23:22,780 --> 00:23:25,100
Why can't I treat
that as one equation?
450
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?
451
00:23:30,240 --> 00:23:33,180
Or why couldn't I
take x minus 1 over 4
452
00:23:33,180 --> 00:23:35,870
and say that equals
z minus 6 over 7?
453
00:23:35,870 --> 00:23:38,890
And the answer is, that
by itself isn't enough.
454
00:23:38,890 --> 00:23:41,210
But rather than give
you a negative answer,
455
00:23:41,210 --> 00:23:44,030
let me give you a positive one.
456
00:23:44,030 --> 00:23:47,560
Let me close today's lesson with
this particular illustration.
457
00:23:47,560 --> 00:23:53,510
Suppose we had solved x minus 1
over 4 equals y minus 5 over 3.
458
00:23:53,510 --> 00:23:56,880
What we would have
obtained is the equation
459
00:23:56,880 --> 00:23:59,380
4y minus 3x equals 17.
460
00:23:59,380 --> 00:24:01,070
Now this is very dangerous.
461
00:24:01,070 --> 00:24:04,870
When you look at the equation
4y minus 3x equals 17,
462
00:24:04,870 --> 00:24:06,450
I'll bet you
dollars to doughnuts
463
00:24:06,450 --> 00:24:12,370
you tend to think of this as
a line rather than as a plane.
464
00:24:12,370 --> 00:24:14,840
But the interesting
thing is, notice
465
00:24:14,840 --> 00:24:17,240
that the way we
got this equation
466
00:24:17,240 --> 00:24:20,840
was ignoring the
z-coordinate of our points.
467
00:24:20,840 --> 00:24:22,800
And what we're really
saying is, let's forget
468
00:24:22,800 --> 00:24:24,410
about the z-coordinate.
469
00:24:24,410 --> 00:24:27,680
In other words,
4y minus 3x equals
470
00:24:27,680 --> 00:24:32,900
17 may be viewed as a line,
but in this particular case,
471
00:24:32,900 --> 00:24:33,740
it's a plane.
472
00:24:33,740 --> 00:24:35,270
In fact, what plane is it?
473
00:24:35,270 --> 00:24:39,470
It's the plane that goes
through the line 4y minus 3x
474
00:24:39,470 --> 00:24:42,360
equals 17, which
lies on the xy-plane.
475
00:24:42,360 --> 00:24:45,060
It's the plane that goes
through that line perpendicular
476
00:24:45,060 --> 00:24:46,310
to the xy-plane.
477
00:24:46,310 --> 00:24:50,350
By the way, again, if you go
back to part one of this course
478
00:24:50,350 --> 00:24:53,540
where we stress sets,
the language of sets
479
00:24:53,540 --> 00:24:55,900
comes to our rescue very nicely.
480
00:24:55,900 --> 00:24:59,870
The difference between
whether 4y minus 3x equals
481
00:24:59,870 --> 00:25:03,010
17 is a plane or
whether it's a line
482
00:25:03,010 --> 00:25:08,100
hinges on whether we're
talking about the set of pairs
483
00:25:08,100 --> 00:25:12,640
x comma y such that
4y minus 3x equals 17,
484
00:25:12,640 --> 00:25:16,560
or whether we're talking about
the set of triplets (x, y, z)
485
00:25:16,560 --> 00:25:20,330
such that 4y minus 3x equals 17.
486
00:25:20,330 --> 00:25:22,520
In this particular
example, we're
487
00:25:22,520 --> 00:25:24,230
talking about points in space.
488
00:25:24,230 --> 00:25:26,290
In other words, our
universe of discourse
489
00:25:26,290 --> 00:25:31,680
are the points x comma y comma
z, not the points x comma y.
490
00:25:31,680 --> 00:25:34,650
Well anyway, rather than
to belabor this point, what
491
00:25:34,650 --> 00:25:39,050
I'm saying is, in the same way
that this equation represents
492
00:25:39,050 --> 00:25:47,780
a plane, in a similar way, had
we equated y minus 5 over 3
493
00:25:47,780 --> 00:25:50,380
equals z minus 6
over 7, we would've
494
00:25:50,380 --> 00:25:57,120
obtained the plane 7y
minus 3z equals 17.
495
00:25:57,120 --> 00:26:00,430
So if you don't like to look
at our set of three equations,
496
00:26:00,430 --> 00:26:03,280
if you'd like to look
at these three equations
497
00:26:03,280 --> 00:26:06,780
in pairs-- you see, if you
want to look at these three
498
00:26:06,780 --> 00:26:09,220
equations in pairs,
another way of saying
499
00:26:09,220 --> 00:26:15,220
it is this, that the triple
equality-- x minus 1 over 4
500
00:26:15,220 --> 00:26:19,620
equals y minus 5 over 3
equals z minus 6 over 7--
501
00:26:19,620 --> 00:26:23,430
that that may be viewed as
the intersection of the two
502
00:26:23,430 --> 00:26:27,880
planes-- namely the plane
determined by this equation
503
00:26:27,880 --> 00:26:30,540
and the plane determined
by this equation.
504
00:26:30,540 --> 00:26:33,690
Of course, someone can also
say, isn't there a plane
505
00:26:33,690 --> 00:26:37,610
determined by this
one and this one?
506
00:26:37,610 --> 00:26:39,430
And the answer is yes, there is.
507
00:26:39,430 --> 00:26:42,410
Notice that whereas you
have three equalities here,
508
00:26:42,410 --> 00:26:44,470
only two of them
are independent.
509
00:26:44,470 --> 00:26:46,590
Namely, as soon as
the first equals
510
00:26:46,590 --> 00:26:49,260
the second and the
second equals the third,
511
00:26:49,260 --> 00:26:52,080
the first must equal the third.
512
00:26:52,080 --> 00:26:52,880
OK?
513
00:26:52,880 --> 00:26:56,160
But the whole idea is, can
you now see the difference?
514
00:26:56,160 --> 00:26:58,780
The easiest way I
know of to distinguish
515
00:26:58,780 --> 00:27:02,860
the difference between the
equation of a line and a plane.
516
00:27:02,860 --> 00:27:05,790
The plane has two
degrees of freedom.
517
00:27:05,790 --> 00:27:08,320
The line has but one
degree of freedom.
518
00:27:08,320 --> 00:27:11,240
And that triple equality
says as soon as you've
519
00:27:11,240 --> 00:27:13,220
picked one of the
unknowns, you've
520
00:27:13,220 --> 00:27:15,050
determined all of the others.
521
00:27:15,050 --> 00:27:16,770
Whereas that string
of plus signs
522
00:27:16,770 --> 00:27:19,220
says that once you've
determined one,
523
00:27:19,220 --> 00:27:21,760
you still have
some freedom left.
524
00:27:21,760 --> 00:27:25,140
Now what we're going
to do is next time
525
00:27:25,140 --> 00:27:27,510
start a new phase of vectors.
526
00:27:27,510 --> 00:27:29,980
For the time being,
what we have now done
527
00:27:29,980 --> 00:27:32,040
is finished, at
least for the moment,
528
00:27:32,040 --> 00:27:35,950
our preliminary investigation
of three-dimensional space
529
00:27:35,950 --> 00:27:39,590
as seen through the eyes
of Cartesian coordinates.
530
00:27:39,590 --> 00:27:41,860
At any rate, until
next time, goodbye.
531
00:27:44,420 --> 00:27:46,790
Funding for the
publication of this video
532
00:27:46,790 --> 00:27:51,670
was provided by the Gabriella
and Paul Rosenbaum Foundation.
533
00:27:51,670 --> 00:27:55,850
Help OCW continue to provide
free and open access to MIT
534
00:27:55,850 --> 00:28:00,260
courses by making a donation
at ocw.mit.edu/donate.