1
00:00:01,000 --> 00:00:03,000
The following content is
provided under a Creative
2
00:00:03,000 --> 00:00:05,000
Commons license.
Your support will help MIT
3
00:00:05,000 --> 00:00:08,000
OpenCourseWare continue to offer
high quality educational
4
00:00:08,000 --> 00:00:13,000
resources for free.
To make a donation or to view
5
00:00:13,000 --> 00:00:18,000
additional materials from
hundreds of MIT courses,
6
00:00:18,000 --> 00:00:23,000
visit MIT OpenCourseWare at
ocw.mit.edu.
7
00:00:23,000 --> 00:00:30,000
The topic for today is going to
be equations of planes,
8
00:00:30,000 --> 00:00:39,000
and how they relate to linear
systems and matrices as we have
9
00:00:39,000 --> 00:00:49,000
seen during Tuesday's lecture.
So, let's start again with
10
00:00:49,000 --> 00:00:57,000
equations of planes.
Remember, we've seen briefly
11
00:00:57,000 --> 00:01:08,000
that an equation for a plane is
of the form ax by cz = d,
12
00:01:08,000 --> 00:01:16,000
where a, b, c,
and d are just numbers.
13
00:01:16,000 --> 00:01:21,000
This expresses the condition
for a point at coordinates x,
14
00:01:21,000 --> 00:01:29,000
y, z, to be in the plane.
An equation of this form
15
00:01:29,000 --> 00:01:38,000
defines a plane.
Let's see how that works, again.
16
00:01:38,000 --> 00:01:45,000
Let's start with an example.
Let's say that we want to find
17
00:01:45,000 --> 00:01:56,000
the equation of a plane through
the origin with normal vector --
18
00:01:56,000 --> 00:02:04,000
-- let's say vector N equals the
vector <1,5,
19
00:02:04,000 --> 00:02:09,000
10>.
How do we find an equation of
20
00:02:09,000 --> 00:02:16,000
this plane?
Remember that we can get an
21
00:02:16,000 --> 00:02:23,000
equation by thinking
geometrically.
22
00:02:23,000 --> 00:02:27,000
So, what's our thinking going
to be?
23
00:02:27,000 --> 00:02:41,000
Well, we have the x, y, z axes.
And, we have this vector N:
24
00:02:41,000 --> 00:02:46,000
.
It's supposed to be
25
00:02:46,000 --> 00:02:49,000
perpendicular to our plane.
And, our plane passes through
26
00:02:49,000 --> 00:02:54,000
the origin here.
So, we want to think of the
27
00:02:54,000 --> 00:03:00,000
plane that's perpendicular to
this vector.
28
00:03:00,000 --> 00:03:03,000
Well, when is a point in that
plane?
29
00:03:03,000 --> 00:03:11,000
Let's say we have a point,
P -- -- at coordinates x,
30
00:03:11,000 --> 00:03:15,000
y, z.
Well, the condition for P to be
31
00:03:15,000 --> 00:03:20,000
in the plane should be that we
have a right angle here.
32
00:03:20,000 --> 00:03:34,000
OK, so P is in the plane
whenever OP dot N is 0.
33
00:03:34,000 --> 00:03:38,000
And, if we write that
explicitly, the vector OP has
34
00:03:38,000 --> 00:03:40,000
components x,
y, z;
35
00:03:40,000 --> 00:03:48,000
N has components 1,5, 10.
So that will give us x 5y 10z =
36
00:03:48,000 --> 00:03:55,000
0.
That's the equation of our
37
00:03:55,000 --> 00:04:00,000
plane.
Now, let's think about a
38
00:04:00,000 --> 00:04:04,000
slightly different problem.
So, let's do another problem.
39
00:04:04,000 --> 00:04:11,000
Let's try to find the equation
of the plane through the point
40
00:04:11,000 --> 00:04:17,000
P0 with coordinates,
say, (2,1,-1),
41
00:04:17,000 --> 00:04:26,000
with normal vector,
again, the same N = <1,5,
42
00:04:26,000 --> 00:04:34,000
10>.
How do we find an equation of
43
00:04:34,000 --> 00:04:39,000
this thing?
Well, we're going to use the
44
00:04:39,000 --> 00:04:44,000
same method.
In fact, let's think for a
45
00:04:44,000 --> 00:04:47,000
second.
I said we have our normal
46
00:04:47,000 --> 00:04:52,000
vector, N, and it's going to be
perpendicular to both planes at
47
00:04:52,000 --> 00:04:53,000
the same time.
So, in fact,
48
00:04:53,000 --> 00:04:56,000
our two planes will be parallel
to each other.
49
00:04:56,000 --> 00:04:58,000
The difference is,
well, before,
50
00:04:58,000 --> 00:05:00,000
we had a plane that was
perpendicular to N,
51
00:05:00,000 --> 00:05:05,000
and passing through the origin.
And now, we have a new plane
52
00:05:05,000 --> 00:05:10,000
that's going to pass not through
the origin but through this
53
00:05:10,000 --> 00:05:13,000
point, P0.
I don't really know where it
54
00:05:13,000 --> 00:05:15,000
is, but let's say,
for example,
55
00:05:15,000 --> 00:05:21,000
that P0 is here.
Then, I will just have to shift
56
00:05:21,000 --> 00:05:26,000
my plane so that,
instead of passing through the
57
00:05:26,000 --> 00:05:32,000
origin, it passes through this
new point.
58
00:05:32,000 --> 00:05:36,000
How am I going to do that?
Well, now, for a point P to be
59
00:05:36,000 --> 00:05:41,000
in our new plane,
we need the vector no longer OP
60
00:05:41,000 --> 00:05:44,000
but P0P to be perpendicular to
N.
61
00:05:44,000 --> 00:05:57,000
So P is in this new plane if
the vector P0P is perpendicular
62
00:05:57,000 --> 00:06:01,000
to N.
And now, let's think,
63
00:06:01,000 --> 00:06:05,000
what's the vector P0P?
Well, we take the coordinates
64
00:06:05,000 --> 00:06:08,000
of P, and we subtract those of
P0.
65
00:06:08,000 --> 00:06:15,000
So, that should be x-2,
y-1, and z 1,
66
00:06:15,000 --> 00:06:23,000
dot product with <1,5,
10> equals 0.
67
00:06:23,000 --> 00:06:41,000
Let's expand this.
We get (x-2) 5(y-1) 10(z 1) = 0.
68
00:06:41,000 --> 00:06:45,000
Let's put the constants on the
other side.
69
00:06:45,000 --> 00:06:50,000
We get: x 5y 10z equals -- here
minus two becomes two,
70
00:06:50,000 --> 00:06:56,000
minus five becomes five,
ten becomes minus ten.
71
00:06:56,000 --> 00:07:01,000
I think we end up with negative
three.
72
00:07:01,000 --> 00:07:05,000
So, the only thing that changes
between these two equations is
73
00:07:05,000 --> 00:07:07,000
the constant term on the
right-hand side,
74
00:07:07,000 --> 00:07:11,000
the thing that I called d.
The other common feature is
75
00:07:11,000 --> 00:07:13,000
that the coefficients of x,
y, and z: one,
76
00:07:13,000 --> 00:07:15,000
five, and ten,
correspond exactly to the
77
00:07:15,000 --> 00:07:19,000
normal vector.
That's something you should
78
00:07:19,000 --> 00:07:22,000
remember about planes.
These coefficients here
79
00:07:22,000 --> 00:07:27,000
correspond exactly to a normal
vector and, well,
80
00:07:27,000 --> 00:07:33,000
this constant term here roughly
measures how far you move
81
00:07:33,000 --> 00:07:35,000
from...
I f you have a plane through
82
00:07:35,000 --> 00:07:37,000
the origin, the right-hand side
will be zero.
83
00:07:37,000 --> 00:07:41,000
And, if you move to a parallel
plane, then this number will
84
00:07:41,000 --> 00:07:44,000
become something else.
Actually, how could we have
85
00:07:44,000 --> 00:07:48,000
found that -3 more quickly?
Well, we know that the first
86
00:07:48,000 --> 00:07:51,000
part of the equation is like
this.
87
00:07:51,000 --> 00:07:55,000
And we know something else.
We know that the point P0 is in
88
00:07:55,000 --> 00:08:00,000
the plane.
So, if we plug the coordinates
89
00:08:00,000 --> 00:08:05,000
of P0 into this,
well, x is 2 5 times 1 10 times
90
00:08:05,000 --> 00:08:08,000
-1.
We get -3.
91
00:08:08,000 --> 00:08:12,000
So, in fact,
the number we should have here
92
00:08:12,000 --> 00:08:16,000
should be minus three so that P0
is a solution.
93
00:08:16,000 --> 00:08:25,000
Let me point out -- (I'll put a
1 here again) -- these three
94
00:08:25,000 --> 00:08:31,000
numbers: 1,5,
10, are exactly the normal
95
00:08:31,000 --> 00:08:37,000
vector.
And one way that we can get
96
00:08:37,000 --> 00:08:45,000
this number here is by computing
the value of the left-hand side
97
00:08:45,000 --> 00:08:51,000
at the point P0.
We plug in the point P0 into
98
00:08:51,000 --> 00:08:56,000
the left hand side.
OK, any questions about that?
99
00:09:07,000 --> 00:09:10,000
By the way, of course,
a plane doesn't have just one
100
00:09:10,000 --> 00:09:12,000
equation.
It has infinitely many
101
00:09:12,000 --> 00:09:18,000
equations because if instead,
say, I multiply everything by
102
00:09:18,000 --> 00:09:23,000
two, 2x 10y 20z = -6 is also an
equation for this plane.
103
00:09:23,000 --> 00:09:32,000
That's because we have normal
vectors of all sizes -- we can
104
00:09:32,000 --> 00:09:40,000
choose how big we make it.
Again, the single most
105
00:09:40,000 --> 00:09:49,000
important thing here:
in the equation ax by cz = d,
106
00:09:49,000 --> 00:09:57,000
the coefficients,
a, b, c, give us a normal
107
00:09:57,000 --> 00:10:03,000
vector to the plane.
So, that's why,
108
00:10:03,000 --> 00:10:07,000
in fact, what matters to us the
most is finding the normal
109
00:10:07,000 --> 00:10:08,000
vector.
In particular,
110
00:10:08,000 --> 00:10:11,000
if you remember,
last time I explained something
111
00:10:11,000 --> 00:10:14,000
about how we can find a normal
vector to a plane if we know
112
00:10:14,000 --> 00:10:17,000
points in the plane.
Namely, we can take the cross
113
00:10:17,000 --> 00:10:20,000
product of two vectors contained
in the plane.
114
00:10:48,000 --> 00:10:54,000
Let's just do an example to see
if we completely understand
115
00:10:54,000 --> 00:10:59,000
what's going on.
Let's say that I give you the
116
00:10:59,000 --> 00:11:03,000
vector with components
,
117
00:11:03,000 --> 00:11:08,000
and I give you the plane x y 3z
= 5.
118
00:11:08,000 --> 00:11:12,000
So, do you think that this
vector is parallel to the plane,
119
00:11:12,000 --> 00:11:15,000
perpendicular to it,
neither?
120
00:11:24,000 --> 00:11:43,000
I'm starting to see a few votes.
OK, I see that most of you are
121
00:11:43,000 --> 00:11:48,000
answering number two:
this vector is perpendicular to
122
00:11:48,000 --> 00:11:51,000
the plane.
There are some other answers
123
00:11:51,000 --> 00:11:59,000
too.
Well, let's try to figure it
124
00:11:59,000 --> 00:12:02,000
out.
Let's do the example.
125
00:12:02,000 --> 00:12:12,000
Say v is <1,2,
-1> and the plane is x y 3z
126
00:12:12,000 --> 00:12:15,000
= 5.
Let's just draw that plane
127
00:12:15,000 --> 00:12:18,000
anywhere -- it doesn't really
matter.
128
00:12:18,000 --> 00:12:21,000
Let's first get a normal vector
out of it.
129
00:12:21,000 --> 00:12:28,000
Well, to get a normal vector to
the plane, what I will do is
130
00:12:28,000 --> 00:12:33,000
take the coefficients of x,
y, and z.
131
00:12:33,000 --> 00:12:36,000
So, that's .
So
132
00:12:36,000 --> 00:12:40,000
is perpendicular to the plane.
How do we get all the other
133
00:12:40,000 --> 00:12:43,000
vectors that are perpendicular
to the plane?
134
00:12:43,000 --> 00:12:47,000
Well, all the perpendicular
vectors are parallel to each
135
00:12:47,000 --> 00:12:50,000
other.
That means that they are just
136
00:12:50,000 --> 00:12:54,000
obtained by multiplying this guy
by some number.
137
00:12:55,000 --> 00:12:59,000
for example,
would still be perpendicular to
138
00:12:59,000 --> 00:13:00,000
the plane.
139
00:13:01,000 --> 00:13:04,000
is also perpendicular to the
plane.
140
00:13:04,000 --> 00:13:07,000
But now, see,
these guys are not proportional
141
00:13:07,000 --> 00:13:18,000
to each other.
So, V is not perpendicular to
142
00:13:18,000 --> 00:13:28,000
the plane.
So it's not perpendicular to
143
00:13:28,000 --> 00:13:33,000
the plane.
Being perpendicular to the
144
00:13:33,000 --> 00:13:37,000
plane is the same as being
parallel to its normal vector.
145
00:13:37,000 --> 00:13:41,000
Now, what about testing if v
is, instead, parallel to the
146
00:13:41,000 --> 00:13:43,000
plane?
Well, it's parallel to the
147
00:13:43,000 --> 00:13:46,000
plane if it's perpendicular to
N.
148
00:13:46,000 --> 00:13:46,000
Let's check.
149
00:13:56,000 --> 00:14:04,000
So, let's try to see if v is
perpendicular to N.
150
00:14:04,000 --> 00:14:11,000
Well, let's do v.N.
That's <1,2,
151
00:14:11,000 --> 00:14:15,000
- 1> dot <1,1,
3>.
152
00:14:15,000 --> 00:14:25,000
You get 1 2 - 3=0.
So, yes.
153
00:14:25,000 --> 00:14:37,000
If it's perpendicular to N,
it means -- It's actually going
154
00:14:37,000 --> 00:14:43,000
to be parallel to the plane.
155
00:14:56,000 --> 00:15:00,000
OK, any questions?
Yes?
156
00:15:00,000 --> 00:15:03,000
[QUESTION FROM STUDENT:]
When you plug the vector into
157
00:15:03,000 --> 00:15:05,000
the plane equation,
you get zero.
158
00:15:05,000 --> 00:15:13,000
What does that mean?
Let's see.
159
00:15:13,000 --> 00:15:18,000
If I plug the vector into the
plane equation:
160
00:15:18,000 --> 00:15:23,000
1 2-3, well,
the left hand side becomes
161
00:15:23,000 --> 00:15:30,000
zero.
So, it's not a solution of the
162
00:15:30,000 --> 00:15:34,000
plane equation.
There's two different things
163
00:15:34,000 --> 00:15:38,000
here.
One is that the point with
164
00:15:38,000 --> 00:15:44,000
coordinates (1,2,- 1) is not in
the plane.
165
00:15:44,000 --> 00:15:52,000
What that tells us is that,
if I put my vector V at the
166
00:15:52,000 --> 00:16:01,000
origin, then its head is not
going to be in the plane.
167
00:16:01,000 --> 00:16:03,000
On the other hand,
you're right,
168
00:16:03,000 --> 00:16:06,000
the left hand side evaluates to
zero.
169
00:16:06,000 --> 00:16:09,000
What that means is that,
if instead I had taken the
170
00:16:09,000 --> 00:16:12,000
plane x y 3z = 0,
then it would be inside.
171
00:16:12,000 --> 00:16:21,000
The plane is x y 3z = 5,
so x y 3z = 0 would be a plane
172
00:16:21,000 --> 00:16:27,000
parallel to it,
but through the origin.
173
00:16:27,000 --> 00:16:30,000
So, that would be another way
to see that the vector is
174
00:16:30,000 --> 00:16:33,000
parallel to the plane.
If we move the plane to a
175
00:16:33,000 --> 00:16:37,000
parallel plane through the
origin, then the endpoint of the
176
00:16:37,000 --> 00:16:44,000
vector is in the plane.
OK, that's another way to
177
00:16:44,000 --> 00:16:56,000
convince ourselves.
Any other questions?
178
00:16:56,000 --> 00:17:04,000
OK, let's move on.
So, last time we learned about
179
00:17:04,000 --> 00:17:08,000
matrices and linear systems.
So, let's try to think,
180
00:17:08,000 --> 00:17:12,000
now, about linear systems in
terms of equations of planes and
181
00:17:12,000 --> 00:17:15,000
intersections of planes.
Remember that a linear system
182
00:17:15,000 --> 00:17:19,000
is a bunch of equations -- say,
a 3x3 linear system is three
183
00:17:19,000 --> 00:17:23,000
different equations.
Each of them is the equation of
184
00:17:23,000 --> 00:17:25,000
a plane.
So, in fact,
185
00:17:25,000 --> 00:17:29,000
if we try to solve a system of
equations, that means actually
186
00:17:29,000 --> 00:17:33,000
we are trying to find a point
that is on several planes at the
187
00:17:33,000 --> 00:17:46,000
same time.
So...
188
00:17:46,000 --> 00:17:52,000
Let's say that we have a 3x3
linear system.
189
00:17:52,000 --> 00:18:04,000
Just to take an example -- it
doesn't really matter what I
190
00:18:04,000 --> 00:18:14,000
give you, but let's say I give
you x z = 1, x y = 2,
191
00:18:14,000 --> 00:18:21,000
x 2y 3z = 3.
What does it mean to solve this?
192
00:18:21,000 --> 00:18:27,000
It means we want to find x,
y, z which satisfy all of these
193
00:18:27,000 --> 00:18:30,000
conditions.
Let's just look at the first
194
00:18:30,000 --> 00:18:33,000
equation, first.
Well, the first equation says
195
00:18:33,000 --> 00:18:37,000
our point should be on the plane
which has this equation.
196
00:18:37,000 --> 00:18:42,000
Then, the second equation says
that our point should also be on
197
00:18:42,000 --> 00:18:46,000
that plane.
So, if you just look at the
198
00:18:46,000 --> 00:18:50,000
first two equations,
you have two planes.
199
00:18:50,000 --> 00:19:08,000
And the solutions -- these two
equations determine for you two
200
00:19:08,000 --> 00:19:22,000
planes, and two planes intersect
in a line.
201
00:19:22,000 --> 00:19:27,000
Now, what happens with the
third equation?
202
00:19:27,000 --> 00:19:30,000
That's actually going to be a
third plane.
203
00:19:30,000 --> 00:19:33,000
So, if we want to solve the
first two equations,
204
00:19:33,000 --> 00:19:37,000
we have to be on this line.
And if we want to solve the
205
00:19:37,000 --> 00:19:41,000
third one, we also need to be on
another plane.
206
00:19:41,000 --> 00:19:52,000
And, in general,
the three planes intersect in a
207
00:19:52,000 --> 00:20:02,000
point because this line of
intersection...
208
00:20:02,000 --> 00:20:04,000
Three planes intersect in a
point,
209
00:20:04,000 --> 00:20:09,000
and one way to think about it
is that the line where the first
210
00:20:09,000 --> 00:20:14,000
two planes intersect meets the
third plane in a point.
211
00:20:14,000 --> 00:20:21,000
And, that point is the solution
to the linear system.
212
00:20:21,000 --> 00:20:28,000
The line -- this is
mathematical notation for the
213
00:20:28,000 --> 00:20:36,000
intersection between the first
two planes -- intersects the
214
00:20:36,000 --> 00:20:46,000
third plane in a point,
which is going to be the
215
00:20:46,000 --> 00:20:53,000
solution.
So, how do we find the solution?
216
00:20:53,000 --> 00:20:58,000
One way is to draw pictures and
try to figure out where the
217
00:20:58,000 --> 00:21:03,000
solution is, but that's not how
we do it in practice if we are
218
00:21:03,000 --> 00:21:07,000
given the equations.
Let me use matrix notation.
219
00:21:07,000 --> 00:21:17,000
Remember, we saw on Tuesday
that the solution to AX = B is
220
00:21:17,000 --> 00:21:23,000
given by X = A inverse B.
We got from here to there by
221
00:21:23,000 --> 00:21:26,000
multiplying on the left by A
inverse.
222
00:21:26,000 --> 00:21:32,000
A inverse AX simplifies to X
equals A inverse B.
223
00:21:32,000 --> 00:21:35,000
And, once again,
it's A inverse B and not BA
224
00:21:35,000 --> 00:21:37,000
inverse.
If you try to set up the
225
00:21:37,000 --> 00:21:39,000
multiplication,
BA inverse doesn't work.
226
00:21:39,000 --> 00:21:47,000
The sizes are not compatible,
you can't multiply the other
227
00:21:47,000 --> 00:21:54,000
way around.
OK, that's pretty good --
228
00:21:54,000 --> 00:22:03,000
unless it doesn't work that way.
What could go wrong?
229
00:22:03,000 --> 00:22:07,000
Well, let's say that our first
two planes do intersect nicely
230
00:22:07,000 --> 00:22:10,000
in a line, but let's think about
the third plane.
231
00:22:10,000 --> 00:22:13,000
Maybe the third plane does not
intersect that line nicely in a
232
00:22:13,000 --> 00:22:19,000
point.
Maybe it's actually parallel to
233
00:22:19,000 --> 00:22:26,000
that line.
Let's try to think about this
234
00:22:26,000 --> 00:22:33,000
question for a second.
Let's say that the set of
235
00:22:33,000 --> 00:22:39,000
solutions to a 3x3 linear system
is not just one point.
236
00:22:39,000 --> 00:22:43,000
So, we don't have a unique
solution that we can get this
237
00:22:43,000 --> 00:22:53,000
way.
What do you think could happen?
238
00:22:53,000 --> 00:22:58,000
OK, I see that answers number
three and five seem to be
239
00:22:58,000 --> 00:23:03,000
dominating.
There's also a bit of answer
240
00:23:03,000 --> 00:23:06,000
number one.
In fact, these are pretty good
241
00:23:06,000 --> 00:23:08,000
answers.
I see that some of you figured
242
00:23:08,000 --> 00:23:12,000
out that you can answer one and
three at the same time,
243
00:23:12,000 --> 00:23:15,000
or three and five at the same
time.
244
00:23:15,000 --> 00:23:18,000
I yet have to see somebody with
three hands answer all three
245
00:23:18,000 --> 00:23:20,000
numbers at the same time.
OK.
246
00:23:20,000 --> 00:23:26,000
Indeed, we'll see very soon
that we could have either no
247
00:23:26,000 --> 00:23:29,000
solution, a line,
or a plane.
248
00:23:29,000 --> 00:23:33,000
The other answers:
"two points"
249
00:23:33,000 --> 00:23:35,000
(two solutions),
we will see,
250
00:23:35,000 --> 00:23:37,000
is actually not a possibility
because if you have two
251
00:23:37,000 --> 00:23:40,000
different solutions,
then the entire line through
252
00:23:40,000 --> 00:23:44,000
these two points is also going
to be made of solutions.
253
00:23:44,000 --> 00:23:47,000
"A tetrahedron"
is just there to amuse you,
254
00:23:47,000 --> 00:23:51,000
it's actually not a good answer
to the question.
255
00:23:51,000 --> 00:23:54,000
It's not very likely that you
will get a tetrahedron out of
256
00:23:54,000 --> 00:23:56,000
intersecting planes.
"A plane"
257
00:23:56,000 --> 00:23:58,000
is indeed possible,
and "I don't know"
258
00:23:58,000 --> 00:24:00,000
is still OK for a few more
minutes,
259
00:24:00,000 --> 00:24:04,000
but we're going to get to the
bottom of this,
260
00:24:04,000 --> 00:24:09,000
and then we will know.
OK, let's try to figure out
261
00:24:09,000 --> 00:24:16,000
what can happen.
Let me go back to my picture.
262
00:24:16,000 --> 00:24:20,000
I had my first two planes;
they determine a line.
263
00:24:20,000 --> 00:24:23,000
And now I have my third plane.
Maybe my third plane is
264
00:24:23,000 --> 00:24:29,000
actually parallel to the line
but doesn't pass through it.
265
00:24:29,000 --> 00:24:32,000
Well, then, there's no
solutions because,
266
00:24:32,000 --> 00:24:37,000
to solve the system of
equations, I need to be in the
267
00:24:37,000 --> 00:24:40,000
first two planes.
So, that means I need to be in
268
00:24:40,000 --> 00:24:43,000
that vertical line.
(That line was supposed to be
269
00:24:43,000 --> 00:24:47,000
red, but I guess it doesn't
really show up as red).
270
00:24:47,000 --> 00:24:49,000
And it also needs to be in the
third plane.
271
00:24:49,000 --> 00:24:52,000
But the line and the plane are
parallel to each other.
272
00:24:52,000 --> 00:24:55,000
There's just no place where
they intersect.
273
00:24:55,000 --> 00:24:59,000
So there's no way to solve all
the equations.
274
00:24:59,000 --> 00:25:03,000
On the other hand,
the other thing that could
275
00:25:03,000 --> 00:25:07,000
happen is that actually the line
is contained in the plane.
276
00:25:07,000 --> 00:25:13,000
And then, any point on that
line will automatically solve
277
00:25:13,000 --> 00:25:19,000
the third equation.
So if you try solving a system
278
00:25:19,000 --> 00:25:23,000
that looks like this by hand,
if you do substitutions,
279
00:25:23,000 --> 00:25:25,000
eliminations,
and so on,
280
00:25:25,000 --> 00:25:28,000
what you will notice is that,
after you have dealt with two
281
00:25:28,000 --> 00:25:31,000
of the equations,
the third one would actually
282
00:25:31,000 --> 00:25:35,000
turn out to be the same as what
you got out of the first two.
283
00:25:35,000 --> 00:25:36,000
It doesn't give you any
additional information.
284
00:25:36,000 --> 00:25:41,000
It's as if you had only two
equations.
285
00:25:41,000 --> 00:25:45,000
The previous case would be when
actually the third equation
286
00:25:45,000 --> 00:25:49,000
contradicts something that you
can get out of the first two.
287
00:25:49,000 --> 00:25:51,000
For example,
maybe out of the first two,
288
00:25:51,000 --> 00:25:54,000
you got that x plus z equals
one, and the third equation is x
289
00:25:54,000 --> 00:25:57,000
plus z equals two.
Well, it can't be one and two
290
00:25:57,000 --> 00:26:00,000
at the same time.
Another way to say it is that
291
00:26:00,000 --> 00:26:04,000
this picture is one where you
can get out of the equations
292
00:26:04,000 --> 00:26:07,000
that a number equals a different
number.
293
00:26:07,000 --> 00:26:10,000
That's impossible.
And, that picture is one where
294
00:26:10,000 --> 00:26:12,000
out of the equations you get
zero equals zero,
295
00:26:12,000 --> 00:26:15,000
which is certainly true,
but isn't a very useful
296
00:26:15,000 --> 00:26:19,000
equation.
So, you can't actually finish
297
00:26:19,000 --> 00:26:27,000
solving.
OK, let me write that down.
298
00:26:27,000 --> 00:26:48,000
unless the third plane is
parallel to the line where P1
299
00:26:48,000 --> 00:26:58,000
and P2 intersect.
Then there's two subcases.
300
00:26:58,000 --> 00:27:11,000
If the line of intersections of
P1 and P2 is actually contained
301
00:27:11,000 --> 00:27:22,000
in P3 (the third plane),
then we have infinitely many
302
00:27:22,000 --> 00:27:26,000
solutions.
Namely, any point on the line
303
00:27:26,000 --> 00:27:29,000
will automatically solve the
third equation.
304
00:27:49,000 --> 00:28:05,000
The other subcase is if the
line of the intersection of P1
305
00:28:05,000 --> 00:28:19,000
and P2 is parallel to P3 and not
contained in it.
306
00:28:19,000 --> 00:28:35,000
Then we get no solutions.
Just to show you the pictures
307
00:28:35,000 --> 00:28:38,000
once again: when we have the
first two planes,
308
00:28:38,000 --> 00:28:42,000
they give us a line.
And now, depending on what
309
00:28:42,000 --> 00:28:45,000
happens to that line in relation
to the third plane,
310
00:28:45,000 --> 00:28:50,000
various situations can happen.
If the line hits the third
311
00:28:50,000 --> 00:28:55,000
plane in a point,
then that's going to be our
312
00:28:55,000 --> 00:28:58,000
solution.
If that line,
313
00:28:58,000 --> 00:29:01,000
instead, is parallel to the
third plane, well,
314
00:29:01,000 --> 00:29:05,000
if it's parallel and outside of
it, then we have no solution.
315
00:29:05,000 --> 00:29:16,000
If it's parallel and contained
in it, then we have infinitely
316
00:29:16,000 --> 00:29:23,000
many solutions.
So, going back to our list of
317
00:29:23,000 --> 00:29:29,000
possibilities,
let's see what can happen.
318
00:29:29,000 --> 00:29:32,000
No solution:
we've seen that it happens when
319
00:29:32,000 --> 00:29:37,000
the line where the first two
planes intersect is parallel to
320
00:29:37,000 --> 00:29:40,000
the third one.
Two points: well,
321
00:29:40,000 --> 00:29:45,000
that didn't come up.
As I said, the problem is that,
322
00:29:45,000 --> 00:29:49,000
if the line of intersections of
the first two planes has two
323
00:29:49,000 --> 00:29:52,000
points that are in the third
plane,
324
00:29:52,000 --> 00:29:55,000
then that means the entire line
must actually be in the third
325
00:29:55,000 --> 00:29:58,000
plane.
So, if you have two solutions,
326
00:29:58,000 --> 00:30:03,000
then you have more than two.
In fact, you have infinitely
327
00:30:03,000 --> 00:30:05,000
many, and we've seen that can
happen.
328
00:30:05,000 --> 00:30:10,000
A tetrahedron:
still doesn't look very
329
00:30:10,000 --> 00:30:13,000
promising.
What about a plane?
330
00:30:13,000 --> 00:30:17,000
Well, that's a case that I
didn't explain because I've been
331
00:30:17,000 --> 00:30:20,000
assuming that P1 and P2 are
different planes and they
332
00:30:20,000 --> 00:30:23,000
intersect in a line.
But, in fact,
333
00:30:23,000 --> 00:30:26,000
they could be parallel,
in which case we already have
334
00:30:26,000 --> 00:30:28,000
no solution to the first two
equations;
335
00:30:28,000 --> 00:30:32,000
or they could be the same plane.
And now, if the third plane is
336
00:30:32,000 --> 00:30:36,000
also the same plane -- if all
three planes are the same plane,
337
00:30:36,000 --> 00:30:38,000
then you have a plane of
solutions.
338
00:30:38,000 --> 00:30:40,000
If I give you three times the
same equation,
339
00:30:40,000 --> 00:30:44,000
that is a linear system.
It's not a very interesting
340
00:30:44,000 --> 00:30:50,000
one, but it's a linear system.
And "I don't know"
341
00:30:50,000 --> 00:30:58,000
is no longer a solution either.
OK, any questions?
342
00:30:58,000 --> 00:31:01,000
[STUDENT QUESTION:]
What's the geometric
343
00:31:01,000 --> 00:31:04,000
significance of the plane x y z
equals 1, as opposed to 2,
344
00:31:04,000 --> 00:31:07,000
or 3?
That's a very good question.
345
00:31:07,000 --> 00:31:10,000
The question is,
what is the geometric
346
00:31:10,000 --> 00:31:14,000
significance of an equation like
x y z equals to 1,2,
347
00:31:14,000 --> 00:31:19,000
3, or something else?
Well, if the equation is x y z
348
00:31:19,000 --> 00:31:23,000
equals zero, it means that our
plane is passing through the
349
00:31:23,000 --> 00:31:25,000
origin.
And then, if we change the
350
00:31:25,000 --> 00:31:28,000
constant, it means we move to a
parallel plane.
351
00:31:28,000 --> 00:31:31,000
So, the first guess that you
might have is that this number
352
00:31:31,000 --> 00:31:35,000
on the right-hand side is the
distance between the origin and
353
00:31:35,000 --> 00:31:37,000
the plane.
It tells us how far from the
354
00:31:37,000 --> 00:31:42,000
origin we are.
That is not quite true.
355
00:31:42,000 --> 00:31:47,000
In fact, that would be true if
the coefficients here formed a
356
00:31:47,000 --> 00:31:50,000
unit vector.
Then this would just be the
357
00:31:50,000 --> 00:31:55,000
distance to the origin.
Otherwise, you have to actually
358
00:31:55,000 --> 00:31:57,000
scale by the length of this
normal vector.
359
00:31:57,000 --> 00:32:01,000
And, I think there's a problem
in the Notes that will show you
360
00:32:01,000 --> 00:32:05,000
exactly how this works.
You should think of it roughly
361
00:32:05,000 --> 00:32:09,000
as how much we have moved the
plane away from the origin.
362
00:32:09,000 --> 00:32:13,000
That's the meaning of the last
term, D, in the right-hand side
363
00:32:13,000 --> 00:32:14,000
of the equation.
364
00:32:29,000 --> 00:32:34,000
So, let's try to think about
what exactly these cases are --
365
00:32:34,000 --> 00:32:38,000
how do we detect in which
situation we are?
366
00:32:38,000 --> 00:32:43,000
It's all very nice in the
picture, but it's difficult to
367
00:32:43,000 --> 00:32:46,000
draw planes.
In fact, when I draw these
368
00:32:46,000 --> 00:32:48,000
pictures, I'm always very
careful not to actually pretend
369
00:32:48,000 --> 00:32:51,000
to draw an actual plane given by
an equation.
370
00:32:51,000 --> 00:32:56,000
When I do, then it's blatantly
false -- it's difficult to draw
371
00:32:56,000 --> 00:32:58,000
a plane correctly.
So, instead,
372
00:32:58,000 --> 00:33:02,000
let's try to think about it in
terms of matrices.
373
00:33:02,000 --> 00:33:04,000
In particular,
what's wrong with this?
374
00:33:04,000 --> 00:33:09,000
Why can't we always say the
solution is X = A inverse B?
375
00:33:09,000 --> 00:33:19,000
Well, the point is that,
actually, you cannot always
376
00:33:19,000 --> 00:33:26,000
invert a matrix.
Recall we've seen this formula:
377
00:33:26,000 --> 00:33:32,000
A inverse is one over
determinant of A times the
378
00:33:32,000 --> 00:33:36,000
adjoint matrix.
And we've learned how to
379
00:33:36,000 --> 00:33:39,000
compute this thing:
remember, we had to take
380
00:33:39,000 --> 00:33:43,000
minors, then flip some signs,
and then transpose.
381
00:33:43,000 --> 00:33:46,000
That step we can always do.
We can always do these
382
00:33:46,000 --> 00:33:48,000
calculations.
But then, at the end,
383
00:33:48,000 --> 00:33:51,000
we have to divide by the
determinant.
384
00:33:51,000 --> 00:33:53,000
That's fine if the determinant
is not zero.
385
00:33:53,000 --> 00:34:00,000
But, if the determinant is
zero, then certainly we cannot
386
00:34:00,000 --> 00:34:05,000
do that.
What I didn't mention last time
387
00:34:05,000 --> 00:34:11,000
is that the matrix is invertible
-- that means it has an inverse
388
00:34:11,000 --> 00:34:16,000
-- exactly when its determinant
is not zero.
389
00:34:16,000 --> 00:34:20,000
That's something we should
remember.
390
00:34:20,000 --> 00:34:24,000
So, if the determinant is not
zero, then we can use our method
391
00:34:24,000 --> 00:34:28,000
to find the inverse.
And then we can solve using
392
00:34:28,000 --> 00:34:31,000
this method.
If not, then not.
393
00:34:31,000 --> 00:34:33,000
Yes?
[STUDENT QUESTION:]
394
00:34:33,000 --> 00:34:36,000
Sorry, can you reexplain that?
You can invert A if the
395
00:34:36,000 --> 00:34:38,000
determinant of A is not equal to
zero?
396
00:34:38,000 --> 00:34:41,000
That's correct.
We can invert the matrix A if
397
00:34:41,000 --> 00:34:46,000
the determinant is not zero.
If you look again at the method
398
00:34:46,000 --> 00:34:49,000
that we saw last time:
first we had to compute the
399
00:34:49,000 --> 00:34:52,000
adjoint matrix.
And, these are operations we
400
00:34:52,000 --> 00:34:54,000
can always do.
If we are given a 3x3 matrix,
401
00:34:54,000 --> 00:34:56,000
we can always compute the
adjoint.
402
00:34:56,000 --> 00:34:59,000
And then, the last step to find
the inverse was to divide by the
403
00:34:59,000 --> 00:35:02,000
determinant.
And that we can only do if the
404
00:35:02,000 --> 00:35:06,000
determinant is not zero.
So, if we have a matrix whose
405
00:35:06,000 --> 00:35:09,000
determinant is not zero,
then we know how to find the
406
00:35:09,000 --> 00:35:11,000
inverse.
If the determinant is zero,
407
00:35:11,000 --> 00:35:14,000
then of course this method
doesn't work.
408
00:35:14,000 --> 00:35:17,000
I'm actually saying even more:
there isn't an inverse at all.
409
00:35:17,000 --> 00:35:19,000
It's not just that our method
fails.
410
00:35:19,000 --> 00:35:27,000
I cannot take the inverse of a
matrix with determinant zero.
411
00:35:27,000 --> 00:35:30,000
Geometrically,
the situation where the
412
00:35:30,000 --> 00:35:34,000
determinant is not zero is
exactly this nice usual
413
00:35:34,000 --> 00:35:39,000
situation where the three planes
intersect in a point,
414
00:35:39,000 --> 00:35:45,000
while the situation where the
determinant is zero is this
415
00:35:45,000 --> 00:35:52,000
situation here where the line
determined by the first two
416
00:35:52,000 --> 00:35:56,000
planes is parallel to the third
plane.
417
00:35:56,000 --> 00:36:06,000
Let me emphasize this again,
and let's see again what
418
00:36:06,000 --> 00:36:19,000
happens.
Let's start with an easier case.
419
00:36:19,000 --> 00:36:21,000
It's called the case of a
homogeneous system.
420
00:36:21,000 --> 00:36:27,000
It's called homogeneous because
it's the situation where the
421
00:36:27,000 --> 00:36:31,000
equations are invariant under
scaling.
422
00:36:31,000 --> 00:36:35,000
So, a homogeneous system is one
where the right hand side is
423
00:36:35,000 --> 00:36:38,000
zero -- there's no B.
If you want,
424
00:36:38,000 --> 00:36:42,000
the constant terms here are all
zero: 0,0, 0.
425
00:36:42,000 --> 00:36:46,000
OK, so this one is not
homogenous.
426
00:36:46,000 --> 00:36:57,000
So, let's see what happens
there.
427
00:36:57,000 --> 00:37:02,000
Let's take an example.
Instead of this system,
428
00:37:02,000 --> 00:37:10,000
we could take x z = 0,
x y = 0, and x 2y 3z also
429
00:37:10,000 --> 00:37:16,000
equals zero.
Can we solve these equations?
430
00:37:16,000 --> 00:37:20,000
I think actually you already
know a very simple solution to
431
00:37:20,000 --> 00:37:23,000
these equations.
Yeah, you can just take x,
432
00:37:23,000 --> 00:37:34,000
y, and z all to be zero.
So, there's always an obvious
433
00:37:34,000 --> 00:37:44,000
solution -- -- namely,
(0,0, 0).
434
00:37:44,000 --> 00:37:53,000
And, in mathematical jargon,
this is called the trivial
435
00:37:53,000 --> 00:37:57,000
solution.
There's always this trivial
436
00:37:57,000 --> 00:37:59,000
solution.
What's the geometric
437
00:37:59,000 --> 00:38:01,000
interpretation?
Well, having zeros here means
438
00:38:01,000 --> 00:38:04,000
that all three planes pass
through the origin.
439
00:38:04,000 --> 00:38:07,000
So, certainly the origin is
always a solution.
440
00:38:21,000 --> 00:38:35,000
The origin is always a solution
because the three planes -- --
441
00:38:35,000 --> 00:38:45,000
pass through the origin.
Now there's two subcases.
442
00:38:45,000 --> 00:38:52,000
One case is if the determinant
of the matrix A is nonzero.
443
00:38:52,000 --> 00:39:01,000
That means that we can invert A.
So, if we can invert A,
444
00:39:01,000 --> 00:39:07,000
then we can solve the system by
multiplying by A inverse.
445
00:39:07,000 --> 00:39:13,000
If we multiply by A inverse,
we'll get X equals A inverse
446
00:39:13,000 --> 00:39:21,000
times zero, which is zero.
That's the only solution
447
00:39:21,000 --> 00:39:24,000
because,
if AX is zero,
448
00:39:24,000 --> 00:39:27,000
then let's multiply by A
inverse: we get that A inverse
449
00:39:27,000 --> 00:39:29,000
AX, which is X,
equals A inverse zero,
450
00:39:29,000 --> 00:39:32,000
which is zero.
We get that X equals zero.
451
00:39:32,000 --> 00:39:42,000
We've solved it,
there's no other solution.
452
00:39:42,000 --> 00:39:55,000
To go back to these pictures
that we all enjoy,
453
00:39:55,000 --> 00:40:03,000
it's this case.
Now the other case,
454
00:40:03,000 --> 00:40:13,000
if the determinant of A equals
zero, then this method doesn't
455
00:40:13,000 --> 00:40:18,000
quite work.
What does it mean that the
456
00:40:18,000 --> 00:40:22,000
determinant of A is zero?
Remember, the entries in A are
457
00:40:22,000 --> 00:40:25,000
the coefficients in the
equations.
458
00:40:25,000 --> 00:40:29,000
But now, the coefficients in
the equations are exactly the
459
00:40:29,000 --> 00:40:36,000
normal vectors to the planes.
So, that's the same thing as
460
00:40:36,000 --> 00:40:47,000
saying that the determinant of
the three normal vectors to our
461
00:40:47,000 --> 00:40:54,000
three planes is 0.
That means that N1,
462
00:40:54,000 --> 00:41:02,000
N2, and N3 are actually in a
same plane -- they're coplanar.
463
00:41:02,000 --> 00:41:06,000
These three vectors are
coplanar.
464
00:41:06,000 --> 00:41:14,000
So, let's see what happens.
I claim it will correspond to
465
00:41:14,000 --> 00:41:20,000
this situation here.
Let's draw the normal vectors
466
00:41:20,000 --> 00:41:27,000
to these three planes.
(Well, it's not very easy to
467
00:41:27,000 --> 00:41:33,000
see, but I've tried to draw the
normal vectors to my planes.)
468
00:41:33,000 --> 00:41:37,000
They are all in the direction
that's perpendicular to the line
469
00:41:37,000 --> 00:41:40,000
of intersection.
They are all in the same plane.
470
00:41:40,000 --> 00:41:44,000
So, if I try to form a
parallelepiped with these three
471
00:41:44,000 --> 00:41:47,000
normal vectors,
well, I will get something
472
00:41:47,000 --> 00:41:50,000
that's completely flat,
and has no volume,
473
00:41:50,000 --> 00:42:04,000
has volume zero.
So the parallelepiped -- -- has
474
00:42:04,000 --> 00:42:11,000
volume 0.
And the fact that the normal
475
00:42:11,000 --> 00:42:19,000
vectors are coplanar tells us
that, in fact -- (well,
476
00:42:19,000 --> 00:42:25,000
let me start a new blackboard).
Let's say that our normal
477
00:42:25,000 --> 00:42:28,000
vectors, N1, N2,
N3, are all in the same plane.
478
00:42:28,000 --> 00:42:32,000
And let's think about the
direction that's perpendicular
479
00:42:32,000 --> 00:42:35,000
to N1, N2, and N3 at the same
time.
480
00:42:35,000 --> 00:42:37,000
I claim that it will be the
line of intersection.
481
00:43:08,000 --> 00:43:12,000
So, let me try to draw that
picture again.
482
00:43:12,000 --> 00:43:26,000
We have three planes -- (now
you see why I prepared a picture
483
00:43:26,000 --> 00:43:31,000
in advance.
It's easier to draw it
484
00:43:31,000 --> 00:43:37,000
beforehand).
And I said their normal vectors
485
00:43:37,000 --> 00:43:41,000
are all in the same plane.
What else do I know?
486
00:43:41,000 --> 00:43:45,000
I know that all these planes
pass through the origin.
487
00:43:45,000 --> 00:43:50,000
So the origin is somewhere in
the intersection of the three
488
00:43:50,000 --> 00:43:59,000
planes.
Now, I said that the normal
489
00:43:59,000 --> 00:44:13,000
vectors to my three planes are
all actually coplanar.
490
00:44:13,000 --> 00:44:23,000
So N1, N2, N3 determine a plane.
Now, if I look at the line
491
00:44:23,000 --> 00:44:27,000
through the origin that's
perpendicular to N1,
492
00:44:27,000 --> 00:44:33,000
N2, and N3,
so, perpendicular to this red
493
00:44:33,000 --> 00:44:39,000
plane here,
it's supposed to be in all the
494
00:44:39,000 --> 00:44:44,000
planes.
(You can see that better on the
495
00:44:44,000 --> 00:44:47,000
side screens).
And why is that?
496
00:44:47,000 --> 00:44:51,000
Well, that's because my line is
perpendicular to the normal
497
00:44:51,000 --> 00:44:54,000
vectors, so it's parallel to the
planes.
498
00:44:54,000 --> 00:44:58,000
It's parallel to all the planes.
Now, why is it in the planes
499
00:44:58,000 --> 00:45:01,000
instead of parallel to them?
Well, that's because my line
500
00:45:01,000 --> 00:45:03,000
goes through the origin,
and the origin is on the
501
00:45:03,000 --> 00:45:07,000
planes.
So, certainly my line has to be
502
00:45:07,000 --> 00:45:11,000
contained in the planes,
not parallel to them.
503
00:45:11,000 --> 00:45:26,000
So the line through the origin
and perpendicular to the plane
504
00:45:26,000 --> 00:45:39,000
of N1, N2, N3 -- -- is parallel
to all three planes.
505
00:45:39,000 --> 00:45:47,000
And, because the planes go
through the origin,
506
00:45:47,000 --> 00:45:58,000
it's contained in them.
So what happens here is I have,
507
00:45:58,000 --> 00:46:06,000
in fact, infinitely many
solutions.
508
00:46:06,000 --> 00:46:09,000
How do I find these solutions?
Well, if I want to find
509
00:46:09,000 --> 00:46:13,000
something that's perpendicular
to N1, N2, and N3 -- if I just
510
00:46:13,000 --> 00:46:16,000
want to be perpendicular to N1
and N2,
511
00:46:16,000 --> 00:46:29,000
I can take their cross product.
So, for example,
512
00:46:29,000 --> 00:46:38,000
N1 cross N2 is perpendicular to
N1 and to N2,
513
00:46:38,000 --> 00:46:43,000
and also to N3,
because N3 is in the same plane
514
00:46:43,000 --> 00:46:46,000
as N1 and N2,
so, if you're perpendicular to
515
00:46:46,000 --> 00:46:49,000
N1 and N2, you are also
perpendicular to N3.
516
00:46:49,000 --> 00:47:03,000
It's automatic.
So, it's a nontrivial solution.
517
00:47:03,000 --> 00:47:09,000
This vector goes along the line
of intersections.
518
00:47:09,000 --> 00:47:13,000
OK, that's the case of
homogeneous systems.
519
00:47:13,000 --> 00:47:24,000
And then, let's finish with the
other case, the general case.
520
00:47:24,000 --> 00:47:32,000
If we look at a system,
AX = B, with B now anything,
521
00:47:32,000 --> 00:47:41,000
there's two cases.
If the determinant of A is not
522
00:47:41,000 --> 00:47:51,000
zero, then there is a unique
solution -- -- namely,
523
00:47:51,000 --> 00:47:59,000
X equals A inverse B.
If the determinant of A is
524
00:47:59,000 --> 00:48:02,000
zero,
then it means we have the
525
00:48:02,000 --> 00:48:06,000
situation with planes that are
all parallel to a same line,
526
00:48:06,000 --> 00:48:18,000
and then we have either no
solution or infinitely many
527
00:48:18,000 --> 00:48:23,000
solutions.
It cannot be a single solution.
528
00:48:23,000 --> 00:48:26,000
Now, whether you have no
solutions or infinitely many
529
00:48:26,000 --> 00:48:30,000
solutions, we haven't actually
developed the tools to answer
530
00:48:30,000 --> 00:48:32,000
that.
But, if you try solving the
531
00:48:32,000 --> 00:48:34,000
system by hand,
by elimination,
532
00:48:34,000 --> 00:48:37,000
you will see that you end up
maybe with something that says
533
00:48:37,000 --> 00:48:40,000
zero equals zero,
and you have infinitely many
534
00:48:40,000 --> 00:48:42,000
solutions.
Actually, if you can find one
535
00:48:42,000 --> 00:48:45,000
solution, then you know that
there's infinitely many.
536
00:48:45,000 --> 00:48:48,000
On the other hand,
if you end up with something
537
00:48:48,000 --> 00:48:51,000
that's a contradiction,
like one equals two,
538
00:48:51,000 --> 00:48:54,000
then you know there's no
solutions.
539
00:48:54,000 --> 00:48:58,000
That's the end for today.
Tomorrow, we will learn about
540
00:48:58,000 --> 00:49:01,000
parametric equations for lines
and curves.