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The topic for today is going to
be equations of planes,

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00:00:30,000 --> 00:00:39,000
and how they relate to linear
systems and matrices as we have

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seen during Tuesday's lecture.
So, let's start again with

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00:00:49,000 --> 00:00:57,000
equations of planes.
Remember, we've seen briefly

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that an equation for a plane is
of the form ax by cz = d,

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where a, b, c,
and d are just numbers.

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00:01:16,000 --> 00:01:21,000
This expresses the condition
for a point at coordinates x,

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00:01:21,000 --> 00:01:29,000
y, z, to be in the plane.
An equation of this form

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00:01:29,000 --> 00:01:38,000
defines a plane.
Let's see how that works, again.

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00:01:38,000 --> 00:01:45,000
Let's start with an example.
Let's say that we want to find

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00:01:45,000 --> 00:01:56,000
the equation of a plane through
the origin with normal vector --

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00:01:56,000 --> 00:02:04,000
-- let's say vector N equals the
vector <1,5,

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00:02:04,000 --> 00:02:09,000
10>.
How do we find an equation of

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00:02:09,000 --> 00:02:16,000
this plane?
Remember that we can get an

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00:02:16,000 --> 00:02:23,000
equation by thinking
geometrically.

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00:02:23,000 --> 00:02:27,000
So, what's our thinking going
to be?

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00:02:27,000 --> 00:02:41,000
Well, we have the x, y, z axes.
And, we have this vector N:

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00:02:41,000 --> 00:02:46,000
.
It's supposed to be

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00:02:46,000 --> 00:02:49,000
perpendicular to our plane.
And, our plane passes through

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00:02:49,000 --> 00:02:54,000
the origin here.
So, we want to think of the

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00:02:54,000 --> 00:03:00,000
plane that's perpendicular to
this vector.

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00:03:00,000 --> 00:03:03,000
Well, when is a point in that
plane?

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00:03:03,000 --> 00:03:11,000
Let's say we have a point,
P -- -- at coordinates x,

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00:03:11,000 --> 00:03:15,000
y, z.
Well, the condition for P to be

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00:03:15,000 --> 00:03:20,000
in the plane should be that we
have a right angle here.

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00:03:20,000 --> 00:03:34,000
OK, so P is in the plane
whenever OP dot N is 0.

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00:03:34,000 --> 00:03:38,000
And, if we write that
explicitly, the vector OP has

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00:03:38,000 --> 00:03:40,000
components x,
y, z;

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00:03:40,000 --> 00:03:48,000
N has components 1,5, 10.
So that will give us x 5y 10z =

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00:03:48,000 --> 00:03:55,000
0.
That's the equation of our

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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.

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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,

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00:04:26,000 --> 00:04:34,000
10>.
How do we find an equation of

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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

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00:04:44,000 --> 00:04:47,000
second.
I said we have our normal

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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,

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00:04:53,000 --> 00:04:56,000
our two planes will be parallel
to each other.

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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,

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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

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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,

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00:05:15,000 --> 00:05:21,000
that P0 is here.
Then, I will just have to shift

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00:05:21,000 --> 00:05:26,000
my plane so that,
instead of passing through the

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00:05:26,000 --> 00:05:32,000
origin, it passes through this
new point.

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00:05:32,000 --> 00:05:36,000
How am I going to do that?
Well, now, for a point P to be

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00:05:36,000 --> 00:05:41,000
in our new plane,
we need the vector no longer OP

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00:05:41,000 --> 00:05:44,000
but P0P to be perpendicular to
N.

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00:05:44,000 --> 00:05:57,000
So P is in this new plane if
the vector P0P is perpendicular

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00:05:57,000 --> 00:06:01,000
to N.
And now, let's think,

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00:06:01,000 --> 00:06:05,000
what's the vector P0P?
Well, we take the coordinates

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00:06:05,000 --> 00:06:08,000
of P, and we subtract those of
P0.

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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.

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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,

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00:06:50,000 --> 00:06:56,000
minus five becomes five,
ten becomes minus ten.

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00:06:56,000 --> 00:07:01,000
I think we end up with negative
three.

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00:07:01,000 --> 00:07:05,000
So, the only thing that changes
between these two equations is

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00:07:05,000 --> 00:07:07,000
the constant term on the
right-hand side,

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00:07:07,000 --> 00:07:11,000
the thing that I called d.
The other common feature is

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00:07:11,000 --> 00:07:13,000
that the coefficients of x,
y, and z: one,

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00:07:13,000 --> 00:07:15,000
five, and ten,
correspond exactly to the

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normal vector.
That's something you should

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00:07:19,000 --> 00:07:22,000
remember about planes.
These coefficients here

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correspond exactly to a normal
vector and, well,

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00:07:27,000 --> 00:07:33,000
this constant term here roughly
measures how far you move

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00:07:33,000 --> 00:07:35,000
from...
I f you have a plane through

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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

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00:07:41,000 --> 00:07:44,000
become something else.
Actually, how could we have

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00:07:44,000 --> 00:07:48,000
found that -3 more quickly?
Well, we know that the first

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00:07:48,000 --> 00:07:51,000
part of the equation is like
this.

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00:07:51,000 --> 00:07:55,000
And we know something else.
We know that the point P0 is in

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00:07:55,000 --> 00:08:00,000
the plane.
So, if we plug the coordinates

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00:08:00,000 --> 00:08:05,000
of P0 into this,
well, x is 2 5 times 1 10 times

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00:08:05,000 --> 00:08:08,000
-1.
We get -3.

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00:08:08,000 --> 00:08:12,000
So, in fact,
the number we should have here

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00:08:12,000 --> 00:08:16,000
should be minus three so that P0
is a solution.

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00:08:16,000 --> 00:08:25,000
Let me point out -- (I'll put a
1 here again) -- these three

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numbers: 1,5,
10, are exactly the normal

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00:08:31,000 --> 00:08:37,000
vector.
And one way that we can get

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00:08:37,000 --> 00:08:45,000
this number here is by computing
the value of the left-hand side

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00:08:45,000 --> 00:08:51,000
at the point P0.
We plug in the point P0 into

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00:08:51,000 --> 00:08:56,000
the left hand side.
OK, any questions about that?

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00:09:07,000 --> 00:09:10,000
By the way, of course,
a plane doesn't have just one

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00:09:10,000 --> 00:09:12,000
equation.
It has infinitely many

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00:09:12,000 --> 00:09:18,000
equations because if instead,
say, I multiply everything by

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00:09:18,000 --> 00:09:23,000
two, 2x 10y 20z = -6 is also an
equation for this plane.

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00:09:23,000 --> 00:09:32,000
That's because we have normal
vectors of all sizes -- we can

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00:09:32,000 --> 00:09:40,000
choose how big we make it.
Again, the single most

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00:09:40,000 --> 00:09:49,000
important thing here:
in the equation ax by cz = d,

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00:09:49,000 --> 00:09:57,000
the coefficients,
a, b, c, give us a normal

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vector to the plane.
So, that's why,

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00:10:03,000 --> 00:10:07,000
in fact, what matters to us the
most is finding the normal

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00:10:07,000 --> 00:10:08,000
vector.
In particular,

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00:10:08,000 --> 00:10:11,000
if you remember,
last time I explained something

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00:10:11,000 --> 00:10:14,000
about how we can find a normal
vector to a plane if we know

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00:10:14,000 --> 00:10:17,000
points in the plane.
Namely, we can take the cross

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00:10:17,000 --> 00:10:20,000
product of two vectors contained
in the plane.

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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

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00:10:59,000 --> 00:11:03,000
vector with components
,

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00:11:03,000 --> 00:11:08,000
and I give you the plane x y 3z
= 5.

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00:11:08,000 --> 00:11:12,000
So, do you think that this
vector is parallel to the plane,

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00:11:12,000 --> 00:11:15,000
perpendicular to it,
neither?

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00:11:24,000 --> 00:11:43,000
I'm starting to see a few votes.
OK, I see that most of you are

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answering number two:
this vector is perpendicular to

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00:11:48,000 --> 00:11:51,000
the plane.
There are some other answers

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00:11:51,000 --> 00:11:59,000
too.
Well, let's try to figure it

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00:11:59,000 --> 00:12:02,000
out.
Let's do the example.

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00:12:02,000 --> 00:12:12,000
Say v is <1,2,
-1> and the plane is x y 3z

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00:12:12,000 --> 00:12:15,000
= 5.
Let's just draw that plane

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00:12:15,000 --> 00:12:18,000
anywhere -- it doesn't really
matter.

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00:12:18,000 --> 00:12:21,000
Let's first get a normal vector
out of it.

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00:12:21,000 --> 00:12:28,000
Well, to get a normal vector to
the plane, what I will do is

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00:12:28,000 --> 00:12:33,000
take the coefficients of x,
y, and z.

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00:12:33,000 --> 00:12:36,000
So, that's .
So

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00:12:36,000 --> 00:12:40,000
is perpendicular to the plane.
How do we get all the other

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00:12:40,000 --> 00:12:43,000
vectors that are perpendicular
to the plane?

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00:12:43,000 --> 00:12:47,000
Well, all the perpendicular
vectors are parallel to each

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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

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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

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00:13:07,000 --> 00:13:18,000
to each other.
So, V is not perpendicular to

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00:13:18,000 --> 00:13:28,000
the plane.
So it's not perpendicular to

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00:13:28,000 --> 00:13:33,000
the plane.
Being perpendicular to the

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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.

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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.

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00:14:25,000 --> 00:14:37,000
If it's perpendicular to N,
it means -- It's actually going

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00:14:37,000 --> 00:14:43,000
to be parallel to the plane.

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00:14:56,000 --> 00:15:00,000
OK, any questions?
Yes?

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00:15:00,000 --> 00:15:03,000
[QUESTION FROM STUDENT:]
When you plug the vector into

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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

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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

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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

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00:17:04,000 --> 00:17:08,000
matrices and linear systems.
So, let's try to think,

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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.

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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.