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The topic for today is going to
be equations of planes,
00:00:30.000 --> 00:00:39.000
and how they relate to linear
systems and matrices as we have
00:00:39.000 --> 00:00:49.000
seen during Tuesday's lecture.
So, let's start again with
00:00:49.000 --> 00:00:57.000
equations of planes.
Remember, we've seen briefly
00:00:57.000 --> 00:01:08.000
that an equation for a plane is
of the form ax by cz = d,
00:01:08.000 --> 00:01:16.000
where a, b, c,
and d are just numbers.
00:01:16.000 --> 00:01:21.000
This expresses the condition
for a point at coordinates x,
00:01:21.000 --> 00:01:29.000
y, z, to be in the plane.
An equation of this form
00:01:29.000 --> 00:01:38.000
defines a plane.
Let's see how that works, again.
00:01:38.000 --> 00:01:45.000
Let's start with an example.
Let's say that we want to find
00:01:45.000 --> 00:01:56.000
the equation of a plane through
the origin with normal vector --
00:01:56.000 --> 00:02:04.000
-- let's say vector N equals the
vector <1,5,
00:02:04.000 --> 00:02:09.000
10>.
How do we find an equation of
00:02:09.000 --> 00:02:16.000
this plane?
Remember that we can get an
00:02:16.000 --> 00:02:23.000
equation by thinking
geometrically.
00:02:23.000 --> 00:02:27.000
So, what's our thinking going
to be?
00:02:27.000 --> 00:02:41.000
Well, we have the x, y, z axes.
And, we have this vector N:
00:02:41.000 --> 00:02:46.000
.
It's supposed to be
00:02:46.000 --> 00:02:49.000
perpendicular to our plane.
And, our plane passes through
00:02:49.000 --> 00:02:54.000
the origin here.
So, we want to think of the
00:02:54.000 --> 00:03:00.000
plane that's perpendicular to
this vector.
00:03:00.000 --> 00:03:03.000
Well, when is a point in that
plane?
00:03:03.000 --> 00:03:11.000
Let's say we have a point,
P -- -- at coordinates x,
00:03:11.000 --> 00:03:15.000
y, z.
Well, the condition for P to be
00:03:15.000 --> 00:03:20.000
in the plane should be that we
have a right angle here.
00:03:20.000 --> 00:03:34.000
OK, so P is in the plane
whenever OP dot N is 0.
00:03:34.000 --> 00:03:38.000
And, if we write that
explicitly, the vector OP has
00:03:38.000 --> 00:03:40.000
components x,
y, z;
00:03:40.000 --> 00:03:48.000
N has components 1,5, 10.
So that will give us x 5y 10z =
00:03:48.000 --> 00:03:55.000
0.
That's the equation of our
00:03:55.000 --> 00:04:00.000
plane.
Now, let's think about a
00:04:00.000 --> 00:04:04.000
slightly different problem.
So, let's do another problem.
00:04:04.000 --> 00:04:11.000
Let's try to find the equation
of the plane through the point
00:04:11.000 --> 00:04:17.000
P0 with coordinates,
say, (2,1,-1),
00:04:17.000 --> 00:04:26.000
with normal vector,
again, the same N = <1,5,
00:04:26.000 --> 00:04:34.000
10>.
How do we find an equation of
00:04:34.000 --> 00:04:39.000
this thing?
Well, we're going to use the
00:04:39.000 --> 00:04:44.000
same method.
In fact, let's think for a
00:04:44.000 --> 00:04:47.000
second.
I said we have our normal
00:04:47.000 --> 00:04:52.000
vector, N, and it's going to be
perpendicular to both planes at
00:04:52.000 --> 00:04:53.000
the same time.
So, in fact,
00:04:53.000 --> 00:04:56.000
our two planes will be parallel
to each other.
00:04:56.000 --> 00:04:58.000
The difference is,
well, before,
00:04:58.000 --> 00:05:00.000
we had a plane that was
perpendicular to N,
00:05:00.000 --> 00:05:05.000
and passing through the origin.
And now, we have a new plane
00:05:05.000 --> 00:05:10.000
that's going to pass not through
the origin but through this
00:05:10.000 --> 00:05:13.000
point, P0.
I don't really know where it
00:05:13.000 --> 00:05:15.000
is, but let's say,
for example,
00:05:15.000 --> 00:05:21.000
that P0 is here.
Then, I will just have to shift
00:05:21.000 --> 00:05:26.000
my plane so that,
instead of passing through the
00:05:26.000 --> 00:05:32.000
origin, it passes through this
new point.
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How am I going to do that?
Well, now, for a point P to be
00:05:36.000 --> 00:05:41.000
in our new plane,
we need the vector no longer OP
00:05:41.000 --> 00:05:44.000
but P0P to be perpendicular to
N.
00:05:44.000 --> 00:05:57.000
So P is in this new plane if
the vector P0P is perpendicular
00:05:57.000 --> 00:06:01.000
to N.
And now, let's think,
00:06:01.000 --> 00:06:05.000
what's the vector P0P?
Well, we take the coordinates
00:06:05.000 --> 00:06:08.000
of P, and we subtract those of
P0.
00:06:08.000 --> 00:06:15.000
So, that should be x-2,
y-1, and z 1,
00:06:15.000 --> 00:06:23.000
dot product with <1,5,
10> equals 0.
00:06:23.000 --> 00:06:41.000
Let's expand this.
We get (x-2) 5(y-1) 10(z 1) = 0.
00:06:41.000 --> 00:06:45.000
Let's put the constants on the
other side.
00:06:45.000 --> 00:06:50.000
We get: x 5y 10z equals -- here
minus two becomes two,
00:06:50.000 --> 00:06:56.000
minus five becomes five,
ten becomes minus ten.
00:06:56.000 --> 00:07:01.000
I think we end up with negative
three.
00:07:01.000 --> 00:07:05.000
So, the only thing that changes
between these two equations is
00:07:05.000 --> 00:07:07.000
the constant term on the
right-hand side,
00:07:07.000 --> 00:07:11.000
the thing that I called d.
The other common feature is
00:07:11.000 --> 00:07:13.000
that the coefficients of x,
y, and z: one,
00:07:13.000 --> 00:07:15.000
five, and ten,
correspond exactly to the
00:07:15.000 --> 00:07:19.000
normal vector.
That's something you should
00:07:19.000 --> 00:07:22.000
remember about planes.
These coefficients here
00:07:22.000 --> 00:07:27.000
correspond exactly to a normal
vector and, well,
00:07:27.000 --> 00:07:33.000
this constant term here roughly
measures how far you move
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from...
I f you have a plane through
00:07:35.000 --> 00:07:37.000
the origin, the right-hand side
will be zero.
00:07:37.000 --> 00:07:41.000
And, if you move to a parallel
plane, then this number will
00:07:41.000 --> 00:07:44.000
become something else.
Actually, how could we have
00:07:44.000 --> 00:07:48.000
found that -3 more quickly?
Well, we know that the first
00:07:48.000 --> 00:07:51.000
part of the equation is like
this.
00:07:51.000 --> 00:07:55.000
And we know something else.
We know that the point P0 is in
00:07:55.000 --> 00:08:00.000
the plane.
So, if we plug the coordinates
00:08:00.000 --> 00:08:05.000
of P0 into this,
well, x is 2 5 times 1 10 times
00:08:05.000 --> 00:08:08.000
-1.
We get -3.
00:08:08.000 --> 00:08:12.000
So, in fact,
the number we should have here
00:08:12.000 --> 00:08:16.000
should be minus three so that P0
is a solution.
00:08:16.000 --> 00:08:25.000
Let me point out -- (I'll put a
1 here again) -- these three
00:08:25.000 --> 00:08:31.000
numbers: 1,5,
10, are exactly the normal
00:08:31.000 --> 00:08:37.000
vector.
And one way that we can get
00:08:37.000 --> 00:08:45.000
this number here is by computing
the value of the left-hand side
00:08:45.000 --> 00:08:51.000
at the point P0.
We plug in the point P0 into
00:08:51.000 --> 00:08:56.000
the left hand side.
OK, any questions about that?
00:09:07.000 --> 00:09:10.000
By the way, of course,
a plane doesn't have just one
00:09:10.000 --> 00:09:12.000
equation.
It has infinitely many
00:09:12.000 --> 00:09:18.000
equations because if instead,
say, I multiply everything by
00:09:18.000 --> 00:09:23.000
two, 2x 10y 20z = -6 is also an
equation for this plane.
00:09:23.000 --> 00:09:32.000
That's because we have normal
vectors of all sizes -- we can
00:09:32.000 --> 00:09:40.000
choose how big we make it.
Again, the single most
00:09:40.000 --> 00:09:49.000
important thing here:
in the equation ax by cz = d,
00:09:49.000 --> 00:09:57.000
the coefficients,
a, b, c, give us a normal
00:09:57.000 --> 00:10:03.000
vector to the plane.
So, that's why,
00:10:03.000 --> 00:10:07.000
in fact, what matters to us the
most is finding the normal
00:10:07.000 --> 00:10:08.000
vector.
In particular,
00:10:08.000 --> 00:10:11.000
if you remember,
last time I explained something
00:10:11.000 --> 00:10:14.000
about how we can find a normal
vector to a plane if we know
00:10:14.000 --> 00:10:17.000
points in the plane.
Namely, we can take the cross
00:10:17.000 --> 00:10:20.000
product of two vectors contained
in the plane.
00:10:48.000 --> 00:10:54.000
Let's just do an example to see
if we completely understand
00:10:54.000 --> 00:10:59.000
what's going on.
Let's say that I give you the
00:10:59.000 --> 00:11:03.000
vector with components
,
00:11:03.000 --> 00:11:08.000
and I give you the plane x y 3z
= 5.
00:11:08.000 --> 00:11:12.000
So, do you think that this
vector is parallel to the plane,
00:11:12.000 --> 00:11:15.000
perpendicular to it,
neither?
00:11:24.000 --> 00:11:43.000
I'm starting to see a few votes.
OK, I see that most of you are
00:11:43.000 --> 00:11:48.000
answering number two:
this vector is perpendicular to
00:11:48.000 --> 00:11:51.000
the plane.
There are some other answers
00:11:51.000 --> 00:11:59.000
too.
Well, let's try to figure it
00:11:59.000 --> 00:12:02.000
out.
Let's do the example.
00:12:02.000 --> 00:12:12.000
Say v is <1,2,
-1> and the plane is x y 3z
00:12:12.000 --> 00:12:15.000
= 5.
Let's just draw that plane
00:12:15.000 --> 00:12:18.000
anywhere -- it doesn't really
matter.
00:12:18.000 --> 00:12:21.000
Let's first get a normal vector
out of it.
00:12:21.000 --> 00:12:28.000
Well, to get a normal vector to
the plane, what I will do is
00:12:28.000 --> 00:12:33.000
take the coefficients of x,
y, and z.
00:12:33.000 --> 00:12:36.000
So, that's .
So
00:12:36.000 --> 00:12:40.000
is perpendicular to the plane.
How do we get all the other
00:12:40.000 --> 00:12:43.000
vectors that are perpendicular
to the plane?
00:12:43.000 --> 00:12:47.000
Well, all the perpendicular
vectors are parallel to each
00:12:47.000 --> 00:12:50.000
other.
That means that they are just
00:12:50.000 --> 00:12:54.000
obtained by multiplying this guy
by some number.
00:12:55.000 --> 00:12:59.000
for example,
would still be perpendicular to
00:12:59.000 --> 00:13:00.000
the plane.
00:13:01.000 --> 00:13:04.000
is also perpendicular to the
plane.
00:13:04.000 --> 00:13:07.000
But now, see,
these guys are not proportional
00:13:07.000 --> 00:13:18.000
to each other.
So, V is not perpendicular to
00:13:18.000 --> 00:13:28.000
the plane.
So it's not perpendicular to
00:13:28.000 --> 00:13:33.000
the plane.
Being perpendicular to the
00:13:33.000 --> 00:13:37.000
plane is the same as being
parallel to its normal vector.
00:13:37.000 --> 00:13:41.000
Now, what about testing if v
is, instead, parallel to the
00:13:41.000 --> 00:13:43.000
plane?
Well, it's parallel to the
00:13:43.000 --> 00:13:46.000
plane if it's perpendicular to
N.
00:13:46.000 --> 00:13:46.000
Let's check.
00:13:56.000 --> 00:14:04.000
So, let's try to see if v is
perpendicular to N.
00:14:04.000 --> 00:14:11.000
Well, let's do v.N.
That's <1,2,
00:14:11.000 --> 00:14:15.000
- 1> dot <1,1,
3>.
00:14:15.000 --> 00:14:25.000
You get 1 2 - 3=0.
So, yes.
00:14:25.000 --> 00:14:37.000
If it's perpendicular to N,
it means -- It's actually going
00:14:37.000 --> 00:14:43.000
to be parallel to the plane.
00:14:56.000 --> 00:15:00.000
OK, any questions?
Yes?
00:15:00.000 --> 00:15:03.000
[QUESTION FROM STUDENT:]
When you plug the vector into
00:15:03.000 --> 00:15:05.000
the plane equation,
you get zero.
00:15:05.000 --> 00:15:13.000
What does that mean?
Let's see.
00:15:13.000 --> 00:15:18.000
If I plug the vector into the
plane equation:
00:15:18.000 --> 00:15:23.000
1 2-3, well,
the left hand side becomes
00:15:23.000 --> 00:15:30.000
zero.
So, it's not a solution of the
00:15:30.000 --> 00:15:34.000
plane equation.
There's two different things
00:15:34.000 --> 00:15:38.000
here.
One is that the point with
00:15:38.000 --> 00:15:44.000
coordinates (1,2,- 1) is not in
the plane.
00:15:44.000 --> 00:15:52.000
What that tells us is that,
if I put my vector V at the
00:15:52.000 --> 00:16:01.000
origin, then its head is not
going to be in the plane.
00:16:01.000 --> 00:16:03.000
On the other hand,
you're right,
00:16:03.000 --> 00:16:06.000
the left hand side evaluates to
zero.
00:16:06.000 --> 00:16:09.000
What that means is that,
if instead I had taken the
00:16:09.000 --> 00:16:12.000
plane x y 3z = 0,
then it would be inside.
00:16:12.000 --> 00:16:21.000
The plane is x y 3z = 5,
so x y 3z = 0 would be a plane
00:16:21.000 --> 00:16:27.000
parallel to it,
but through the origin.
00:16:27.000 --> 00:16:30.000
So, that would be another way
to see that the vector is
00:16:30.000 --> 00:16:33.000
parallel to the plane.
If we move the plane to a
00:16:33.000 --> 00:16:37.000
parallel plane through the
origin, then the endpoint of the
00:16:37.000 --> 00:16:44.000
vector is in the plane.
OK, that's another way to
00:16:44.000 --> 00:16:56.000
convince ourselves.
Any other questions?
00:16:56.000 --> 00:17:04.000
OK, let's move on.
So, last time we learned about
00:17:04.000 --> 00:17:08.000
matrices and linear systems.
So, let's try to think,
00:17:08.000 --> 00:17:12.000
now, about linear systems in
terms of equations of planes and
00:17:12.000 --> 00:17:15.000
intersections of planes.
Remember that a linear system
00:17:15.000 --> 00:17:19.000
is a bunch of equations -- say,
a 3x3 linear system is three
00:17:19.000 --> 00:17:23.000
different equations.
Each of them is the equation of
00:17:23.000 --> 00:17:25.000
a plane.
So, in fact,
00:17:25.000 --> 00:17:29.000
if we try to solve a system of
equations, that means actually
00:17:29.000 --> 00:17:33.000
we are trying to find a point
that is on several planes at the
00:17:33.000 --> 00:17:46.000
same time.
So...
00:17:46.000 --> 00:17:52.000
Let's say that we have a 3x3
linear system.
00:17:52.000 --> 00:18:04.000
Just to take an example -- it
doesn't really matter what I
00:18:04.000 --> 00:18:14.000
give you, but let's say I give
you x z = 1, x y = 2,
00:18:14.000 --> 00:18:21.000
x 2y 3z = 3.
What does it mean to solve this?
00:18:21.000 --> 00:18:27.000
It means we want to find x,
y, z which satisfy all of these
00:18:27.000 --> 00:18:30.000
conditions.
Let's just look at the first
00:18:30.000 --> 00:18:33.000
equation, first.
Well, the first equation says
00:18:33.000 --> 00:18:37.000
our point should be on the plane
which has this equation.
00:18:37.000 --> 00:18:42.000
Then, the second equation says
that our point should also be on
00:18:42.000 --> 00:18:46.000
that plane.
So, if you just look at the
00:18:46.000 --> 00:18:50.000
first two equations,
you have two planes.
00:18:50.000 --> 00:19:08.000
And the solutions -- these two
equations determine for you two
00:19:08.000 --> 00:19:22.000
planes, and two planes intersect
in a line.
00:19:22.000 --> 00:19:27.000
Now, what happens with the
third equation?
00:19:27.000 --> 00:19:30.000
That's actually going to be a
third plane.
00:19:30.000 --> 00:19:33.000
So, if we want to solve the
first two equations,
00:19:33.000 --> 00:19:37.000
we have to be on this line.
And if we want to solve the
00:19:37.000 --> 00:19:41.000
third one, we also need to be on
another plane.
00:19:41.000 --> 00:19:52.000
And, in general,
the three planes intersect in a
00:19:52.000 --> 00:20:02.000
point because this line of
intersection...
00:20:02.000 --> 00:20:04.000
Three planes intersect in a
point,
00:20:04.000 --> 00:20:09.000
and one way to think about it
is that the line where the first
00:20:09.000 --> 00:20:14.000
two planes intersect meets the
third plane in a point.
00:20:14.000 --> 00:20:21.000
And, that point is the solution
to the linear system.
00:20:21.000 --> 00:20:28.000
The line -- this is
mathematical notation for the
00:20:28.000 --> 00:20:36.000
intersection between the first
two planes -- intersects the
00:20:36.000 --> 00:20:46.000
third plane in a point,
which is going to be the
00:20:46.000 --> 00:20:53.000
solution.
So, how do we find the solution?
00:20:53.000 --> 00:20:58.000
One way is to draw pictures and
try to figure out where the
00:20:58.000 --> 00:21:03.000
solution is, but that's not how
we do it in practice if we are
00:21:03.000 --> 00:21:07.000
given the equations.
Let me use matrix notation.
00:21:07.000 --> 00:21:17.000
Remember, we saw on Tuesday
that the solution to AX = B is
00:21:17.000 --> 00:21:23.000
given by X = A inverse B.
We got from here to there by
00:21:23.000 --> 00:21:26.000
multiplying on the left by A
inverse.
00:21:26.000 --> 00:21:32.000
A inverse AX simplifies to X
equals A inverse B.
00:21:32.000 --> 00:21:35.000
And, once again,
it's A inverse B and not BA
00:21:35.000 --> 00:21:37.000
inverse.
If you try to set up the
00:21:37.000 --> 00:21:39.000
multiplication,
BA inverse doesn't work.
00:21:39.000 --> 00:21:47.000
The sizes are not compatible,
you can't multiply the other
00:21:47.000 --> 00:21:54.000
way around.
OK, that's pretty good --
00:21:54.000 --> 00:22:03.000
unless it doesn't work that way.
What could go wrong?
00:22:03.000 --> 00:22:07.000
Well, let's say that our first
two planes do intersect nicely
00:22:07.000 --> 00:22:10.000
in a line, but let's think about
the third plane.
00:22:10.000 --> 00:22:13.000
Maybe the third plane does not
intersect that line nicely in a
00:22:13.000 --> 00:22:19.000
point.
Maybe it's actually parallel to
00:22:19.000 --> 00:22:26.000
that line.
Let's try to think about this
00:22:26.000 --> 00:22:33.000
question for a second.
Let's say that the set of
00:22:33.000 --> 00:22:39.000
solutions to a 3x3 linear system
is not just one point.
00:22:39.000 --> 00:22:43.000
So, we don't have a unique
solution that we can get this
00:22:43.000 --> 00:22:53.000
way.
What do you think could happen?
00:22:53.000 --> 00:22:58.000
OK, I see that answers number
three and five seem to be
00:22:58.000 --> 00:23:03.000
dominating.
There's also a bit of answer
00:23:03.000 --> 00:23:06.000
number one.
In fact, these are pretty good
00:23:06.000 --> 00:23:08.000
answers.
I see that some of you figured
00:23:08.000 --> 00:23:12.000
out that you can answer one and
three at the same time,
00:23:12.000 --> 00:23:15.000
or three and five at the same
time.
00:23:15.000 --> 00:23:18.000
I yet have to see somebody with
three hands answer all three
00:23:18.000 --> 00:23:20.000
numbers at the same time.
OK.
00:23:20.000 --> 00:23:26.000
Indeed, we'll see very soon
that we could have either no
00:23:26.000 --> 00:23:29.000
solution, a line,
or a plane.
00:23:29.000 --> 00:23:33.000
The other answers:
"two points"
00:23:33.000 --> 00:23:35.000
(two solutions),
we will see,
00:23:35.000 --> 00:23:37.000
is actually not a possibility
because if you have two
00:23:37.000 --> 00:23:40.000
different solutions,
then the entire line through
00:23:40.000 --> 00:23:44.000
these two points is also going
to be made of solutions.
00:23:44.000 --> 00:23:47.000
"A tetrahedron"
is just there to amuse you,
00:23:47.000 --> 00:23:51.000
it's actually not a good answer
to the question.
00:23:51.000 --> 00:23:54.000
It's not very likely that you
will get a tetrahedron out of
00:23:54.000 --> 00:23:56.000
intersecting planes.
"A plane"
00:23:56.000 --> 00:23:58.000
is indeed possible,
and "I don't know"
00:23:58.000 --> 00:24:00.000
is still OK for a few more
minutes,
00:24:00.000 --> 00:24:04.000
but we're going to get to the
bottom of this,
00:24:04.000 --> 00:24:09.000
and then we will know.
OK, let's try to figure out
00:24:09.000 --> 00:24:16.000
what can happen.
Let me go back to my picture.
00:24:16.000 --> 00:24:20.000
I had my first two planes;
they determine a line.
00:24:20.000 --> 00:24:23.000
And now I have my third plane.
Maybe my third plane is
00:24:23.000 --> 00:24:29.000
actually parallel to the line
but doesn't pass through it.
00:24:29.000 --> 00:24:32.000
Well, then, there's no
solutions because,
00:24:32.000 --> 00:24:37.000
to solve the system of
equations, I need to be in the
00:24:37.000 --> 00:24:40.000
first two planes.
So, that means I need to be in
00:24:40.000 --> 00:24:43.000
that vertical line.
(That line was supposed to be
00:24:43.000 --> 00:24:47.000
red, but I guess it doesn't
really show up as red).
00:24:47.000 --> 00:24:49.000
And it also needs to be in the
third plane.
00:24:49.000 --> 00:24:52.000
But the line and the plane are
parallel to each other.
00:24:52.000 --> 00:24:55.000
There's just no place where
they intersect.
00:24:55.000 --> 00:24:59.000
So there's no way to solve all
the equations.
00:24:59.000 --> 00:25:03.000
On the other hand,
the other thing that could
00:25:03.000 --> 00:25:07.000
happen is that actually the line
is contained in the plane.
00:25:07.000 --> 00:25:13.000
And then, any point on that
line will automatically solve
00:25:13.000 --> 00:25:19.000
the third equation.
So if you try solving a system
00:25:19.000 --> 00:25:23.000
that looks like this by hand,
if you do substitutions,
00:25:23.000 --> 00:25:25.000
eliminations,
and so on,
00:25:25.000 --> 00:25:28.000
what you will notice is that,
after you have dealt with two
00:25:28.000 --> 00:25:31.000
of the equations,
the third one would actually
00:25:31.000 --> 00:25:35.000
turn out to be the same as what
you got out of the first two.
00:25:35.000 --> 00:25:36.000
It doesn't give you any
additional information.
00:25:36.000 --> 00:25:41.000
It's as if you had only two
equations.
00:25:41.000 --> 00:25:45.000
The previous case would be when
actually the third equation
00:25:45.000 --> 00:25:49.000
contradicts something that you
can get out of the first two.
00:25:49.000 --> 00:25:51.000
For example,
maybe out of the first two,
00:25:51.000 --> 00:25:54.000
you got that x plus z equals
one, and the third equation is x
00:25:54.000 --> 00:25:57.000
plus z equals two.
Well, it can't be one and two
00:25:57.000 --> 00:26:00.000
at the same time.
Another way to say it is that
00:26:00.000 --> 00:26:04.000
this picture is one where you
can get out of the equations
00:26:04.000 --> 00:26:07.000
that a number equals a different
number.
00:26:07.000 --> 00:26:10.000
That's impossible.
And, that picture is one where
00:26:10.000 --> 00:26:12.000
out of the equations you get
zero equals zero,
00:26:12.000 --> 00:26:15.000
which is certainly true,
but isn't a very useful
00:26:15.000 --> 00:26:19.000
equation.
So, you can't actually finish
00:26:19.000 --> 00:26:27.000
solving.
OK, let me write that down.
00:26:27.000 --> 00:26:48.000
unless the third plane is
parallel to the line where P1
00:26:48.000 --> 00:26:58.000
and P2 intersect.
Then there's two subcases.
00:26:58.000 --> 00:27:11.000
If the line of intersections of
P1 and P2 is actually contained
00:27:11.000 --> 00:27:22.000
in P3 (the third plane),
then we have infinitely many
00:27:22.000 --> 00:27:26.000
solutions.
Namely, any point on the line
00:27:26.000 --> 00:27:29.000
will automatically solve the
third equation.
00:27:49.000 --> 00:28:05.000
The other subcase is if the
line of the intersection of P1
00:28:05.000 --> 00:28:19.000
and P2 is parallel to P3 and not
contained in it.
00:28:19.000 --> 00:28:35.000
Then we get no solutions.
Just to show you the pictures
00:28:35.000 --> 00:28:38.000
once again: when we have the
first two planes,
00:28:38.000 --> 00:28:42.000
they give us a line.
And now, depending on what
00:28:42.000 --> 00:28:45.000
happens to that line in relation
to the third plane,
00:28:45.000 --> 00:28:50.000
various situations can happen.
If the line hits the third
00:28:50.000 --> 00:28:55.000
plane in a point,
then that's going to be our
00:28:55.000 --> 00:28:58.000
solution.
If that line,
00:28:58.000 --> 00:29:01.000
instead, is parallel to the
third plane, well,
00:29:01.000 --> 00:29:05.000
if it's parallel and outside of
it, then we have no solution.
00:29:05.000 --> 00:29:16.000
If it's parallel and contained
in it, then we have infinitely
00:29:16.000 --> 00:29:23.000
many solutions.
So, going back to our list of
00:29:23.000 --> 00:29:29.000
possibilities,
let's see what can happen.
00:29:29.000 --> 00:29:32.000
No solution:
we've seen that it happens when
00:29:32.000 --> 00:29:37.000
the line where the first two
planes intersect is parallel to
00:29:37.000 --> 00:29:40.000
the third one.
Two points: well,
00:29:40.000 --> 00:29:45.000
that didn't come up.
As I said, the problem is that,
00:29:45.000 --> 00:29:49.000
if the line of intersections of
the first two planes has two
00:29:49.000 --> 00:29:52.000
points that are in the third
plane,
00:29:52.000 --> 00:29:55.000
then that means the entire line
must actually be in the third
00:29:55.000 --> 00:29:58.000
plane.
So, if you have two solutions,
00:29:58.000 --> 00:30:03.000
then you have more than two.
In fact, you have infinitely
00:30:03.000 --> 00:30:05.000
many, and we've seen that can
happen.
00:30:05.000 --> 00:30:10.000
A tetrahedron:
still doesn't look very
00:30:10.000 --> 00:30:13.000
promising.
What about a plane?
00:30:13.000 --> 00:30:17.000
Well, that's a case that I
didn't explain because I've been
00:30:17.000 --> 00:30:20.000
assuming that P1 and P2 are
different planes and they
00:30:20.000 --> 00:30:23.000
intersect in a line.
But, in fact,
00:30:23.000 --> 00:30:26.000
they could be parallel,
in which case we already have
00:30:26.000 --> 00:30:28.000
no solution to the first two
equations;
00:30:28.000 --> 00:30:32.000
or they could be the same plane.
And now, if the third plane is
00:30:32.000 --> 00:30:36.000
also the same plane -- if all
three planes are the same plane,
00:30:36.000 --> 00:30:38.000
then you have a plane of
solutions.
00:30:38.000 --> 00:30:40.000
If I give you three times the
same equation,
00:30:40.000 --> 00:30:44.000
that is a linear system.
It's not a very interesting
00:30:44.000 --> 00:30:50.000
one, but it's a linear system.
And "I don't know"
00:30:50.000 --> 00:30:58.000
is no longer a solution either.
OK, any questions?
00:30:58.000 --> 00:31:01.000
[STUDENT QUESTION:]
What's the geometric
00:31:01.000 --> 00:31:04.000
significance of the plane x y z
equals 1, as opposed to 2,
00:31:04.000 --> 00:31:07.000
or 3?
That's a very good question.
00:31:07.000 --> 00:31:10.000
The question is,
what is the geometric
00:31:10.000 --> 00:31:14.000
significance of an equation like
x y z equals to 1,2,
00:31:14.000 --> 00:31:19.000
3, or something else?
Well, if the equation is x y z
00:31:19.000 --> 00:31:23.000
equals zero, it means that our
plane is passing through the
00:31:23.000 --> 00:31:25.000
origin.
And then, if we change the
00:31:25.000 --> 00:31:28.000
constant, it means we move to a
parallel plane.
00:31:28.000 --> 00:31:31.000
So, the first guess that you
might have is that this number
00:31:31.000 --> 00:31:35.000
on the right-hand side is the
distance between the origin and
00:31:35.000 --> 00:31:37.000
the plane.
It tells us how far from the
00:31:37.000 --> 00:31:42.000
origin we are.
That is not quite true.
00:31:42.000 --> 00:31:47.000
In fact, that would be true if
the coefficients here formed a
00:31:47.000 --> 00:31:50.000
unit vector.
Then this would just be the
00:31:50.000 --> 00:31:55.000
distance to the origin.
Otherwise, you have to actually
00:31:55.000 --> 00:31:57.000
scale by the length of this
normal vector.
00:31:57.000 --> 00:32:01.000
And, I think there's a problem
in the Notes that will show you
00:32:01.000 --> 00:32:05.000
exactly how this works.
You should think of it roughly
00:32:05.000 --> 00:32:09.000
as how much we have moved the
plane away from the origin.
00:32:09.000 --> 00:32:13.000
That's the meaning of the last
term, D, in the right-hand side
00:32:13.000 --> 00:32:14.000
of the equation.
00:32:29.000 --> 00:32:34.000
So, let's try to think about
what exactly these cases are --
00:32:34.000 --> 00:32:38.000
how do we detect in which
situation we are?
00:32:38.000 --> 00:32:43.000
It's all very nice in the
picture, but it's difficult to
00:32:43.000 --> 00:32:46.000
draw planes.
In fact, when I draw these
00:32:46.000 --> 00:32:48.000
pictures, I'm always very
careful not to actually pretend
00:32:48.000 --> 00:32:51.000
to draw an actual plane given by
an equation.
00:32:51.000 --> 00:32:56.000
When I do, then it's blatantly
false -- it's difficult to draw
00:32:56.000 --> 00:32:58.000
a plane correctly.
So, instead,
00:32:58.000 --> 00:33:02.000
let's try to think about it in
terms of matrices.
00:33:02.000 --> 00:33:04.000
In particular,
what's wrong with this?
00:33:04.000 --> 00:33:09.000
Why can't we always say the
solution is X = A inverse B?
00:33:09.000 --> 00:33:19.000
Well, the point is that,
actually, you cannot always
00:33:19.000 --> 00:33:26.000
invert a matrix.
Recall we've seen this formula:
00:33:26.000 --> 00:33:32.000
A inverse is one over
determinant of A times the
00:33:32.000 --> 00:33:36.000
adjoint matrix.
And we've learned how to
00:33:36.000 --> 00:33:39.000
compute this thing:
remember, we had to take
00:33:39.000 --> 00:33:43.000
minors, then flip some signs,
and then transpose.
00:33:43.000 --> 00:33:46.000
That step we can always do.
We can always do these
00:33:46.000 --> 00:33:48.000
calculations.
But then, at the end,
00:33:48.000 --> 00:33:51.000
we have to divide by the
determinant.
00:33:51.000 --> 00:33:53.000
That's fine if the determinant
is not zero.
00:33:53.000 --> 00:34:00.000
But, if the determinant is
zero, then certainly we cannot
00:34:00.000 --> 00:34:05.000
do that.
What I didn't mention last time
00:34:05.000 --> 00:34:11.000
is that the matrix is invertible
-- that means it has an inverse
00:34:11.000 --> 00:34:16.000
-- exactly when its determinant
is not zero.
00:34:16.000 --> 00:34:20.000
That's something we should
remember.
00:34:20.000 --> 00:34:24.000
So, if the determinant is not
zero, then we can use our method
00:34:24.000 --> 00:34:28.000
to find the inverse.
And then we can solve using
00:34:28.000 --> 00:34:31.000
this method.
If not, then not.
00:34:31.000 --> 00:34:33.000
Yes?
[STUDENT QUESTION:]
00:34:33.000 --> 00:34:36.000
Sorry, can you reexplain that?
You can invert A if the
00:34:36.000 --> 00:34:38.000
determinant of A is not equal to
zero?
00:34:38.000 --> 00:34:41.000
That's correct.
We can invert the matrix A if
00:34:41.000 --> 00:34:46.000
the determinant is not zero.
If you look again at the method
00:34:46.000 --> 00:34:49.000
that we saw last time:
first we had to compute the
00:34:49.000 --> 00:34:52.000
adjoint matrix.
And, these are operations we
00:34:52.000 --> 00:34:54.000
can always do.
If we are given a 3x3 matrix,
00:34:54.000 --> 00:34:56.000
we can always compute the
adjoint.
00:34:56.000 --> 00:34:59.000
And then, the last step to find
the inverse was to divide by the
00:34:59.000 --> 00:35:02.000
determinant.
And that we can only do if the
00:35:02.000 --> 00:35:06.000
determinant is not zero.
So, if we have a matrix whose
00:35:06.000 --> 00:35:09.000
determinant is not zero,
then we know how to find the
00:35:09.000 --> 00:35:11.000
inverse.
If the determinant is zero,
00:35:11.000 --> 00:35:14.000
then of course this method
doesn't work.
00:35:14.000 --> 00:35:17.000
I'm actually saying even more:
there isn't an inverse at all.
00:35:17.000 --> 00:35:19.000
It's not just that our method
fails.
00:35:19.000 --> 00:35:27.000
I cannot take the inverse of a
matrix with determinant zero.
00:35:27.000 --> 00:35:30.000
Geometrically,
the situation where the
00:35:30.000 --> 00:35:34.000
determinant is not zero is
exactly this nice usual
00:35:34.000 --> 00:35:39.000
situation where the three planes
intersect in a point,
00:35:39.000 --> 00:35:45.000
while the situation where the
determinant is zero is this
00:35:45.000 --> 00:35:52.000
situation here where the line
determined by the first two
00:35:52.000 --> 00:35:56.000
planes is parallel to the third
plane.
00:35:56.000 --> 00:36:06.000
Let me emphasize this again,
and let's see again what
00:36:06.000 --> 00:36:19.000
happens.
Let's start with an easier case.
00:36:19.000 --> 00:36:21.000
It's called the case of a
homogeneous system.
00:36:21.000 --> 00:36:27.000
It's called homogeneous because
it's the situation where the
00:36:27.000 --> 00:36:31.000
equations are invariant under
scaling.
00:36:31.000 --> 00:36:35.000
So, a homogeneous system is one
where the right hand side is
00:36:35.000 --> 00:36:38.000
zero -- there's no B.
If you want,
00:36:38.000 --> 00:36:42.000
the constant terms here are all
zero: 0,0, 0.
00:36:42.000 --> 00:36:46.000
OK, so this one is not
homogenous.
00:36:46.000 --> 00:36:57.000
So, let's see what happens
there.
00:36:57.000 --> 00:37:02.000
Let's take an example.
Instead of this system,
00:37:02.000 --> 00:37:10.000
we could take x z = 0,
x y = 0, and x 2y 3z also
00:37:10.000 --> 00:37:16.000
equals zero.
Can we solve these equations?
00:37:16.000 --> 00:37:20.000
I think actually you already
know a very simple solution to
00:37:20.000 --> 00:37:23.000
these equations.
Yeah, you can just take x,
00:37:23.000 --> 00:37:34.000
y, and z all to be zero.
So, there's always an obvious
00:37:34.000 --> 00:37:44.000
solution -- -- namely,
(0,0, 0).
00:37:44.000 --> 00:37:53.000
And, in mathematical jargon,
this is called the trivial
00:37:53.000 --> 00:37:57.000
solution.
There's always this trivial
00:37:57.000 --> 00:37:59.000
solution.
What's the geometric
00:37:59.000 --> 00:38:01.000
interpretation?
Well, having zeros here means
00:38:01.000 --> 00:38:04.000
that all three planes pass
through the origin.
00:38:04.000 --> 00:38:07.000
So, certainly the origin is
always a solution.
00:38:21.000 --> 00:38:35.000
The origin is always a solution
because the three planes -- --
00:38:35.000 --> 00:38:45.000
pass through the origin.
Now there's two subcases.
00:38:45.000 --> 00:38:52.000
One case is if the determinant
of the matrix A is nonzero.
00:38:52.000 --> 00:39:01.000
That means that we can invert A.
So, if we can invert A,
00:39:01.000 --> 00:39:07.000
then we can solve the system by
multiplying by A inverse.
00:39:07.000 --> 00:39:13.000
If we multiply by A inverse,
we'll get X equals A inverse
00:39:13.000 --> 00:39:21.000
times zero, which is zero.
That's the only solution
00:39:21.000 --> 00:39:24.000
because,
if AX is zero,
00:39:24.000 --> 00:39:27.000
then let's multiply by A
inverse: we get that A inverse
00:39:27.000 --> 00:39:29.000
AX, which is X,
equals A inverse zero,
00:39:29.000 --> 00:39:32.000
which is zero.
We get that X equals zero.
00:39:32.000 --> 00:39:42.000
We've solved it,
there's no other solution.
00:39:42.000 --> 00:39:55.000
To go back to these pictures
that we all enjoy,
00:39:55.000 --> 00:40:03.000
it's this case.
Now the other case,
00:40:03.000 --> 00:40:13.000
if the determinant of A equals
zero, then this method doesn't
00:40:13.000 --> 00:40:18.000
quite work.
What does it mean that the
00:40:18.000 --> 00:40:22.000
determinant of A is zero?
Remember, the entries in A are
00:40:22.000 --> 00:40:25.000
the coefficients in the
equations.
00:40:25.000 --> 00:40:29.000
But now, the coefficients in
the equations are exactly the
00:40:29.000 --> 00:40:36.000
normal vectors to the planes.
So, that's the same thing as
00:40:36.000 --> 00:40:47.000
saying that the determinant of
the three normal vectors to our
00:40:47.000 --> 00:40:54.000
three planes is 0.
That means that N1,
00:40:54.000 --> 00:41:02.000
N2, and N3 are actually in a
same plane -- they're coplanar.
00:41:02.000 --> 00:41:06.000
These three vectors are
coplanar.
00:41:06.000 --> 00:41:14.000
So, let's see what happens.
I claim it will correspond to
00:41:14.000 --> 00:41:20.000
this situation here.
Let's draw the normal vectors
00:41:20.000 --> 00:41:27.000
to these three planes.
(Well, it's not very easy to
00:41:27.000 --> 00:41:33.000
see, but I've tried to draw the
normal vectors to my planes.)
00:41:33.000 --> 00:41:37.000
They are all in the direction
that's perpendicular to the line
00:41:37.000 --> 00:41:40.000
of intersection.
They are all in the same plane.
00:41:40.000 --> 00:41:44.000
So, if I try to form a
parallelepiped with these three
00:41:44.000 --> 00:41:47.000
normal vectors,
well, I will get something
00:41:47.000 --> 00:41:50.000
that's completely flat,
and has no volume,
00:41:50.000 --> 00:42:04.000
has volume zero.
So the parallelepiped -- -- has
00:42:04.000 --> 00:42:11.000
volume 0.
And the fact that the normal
00:42:11.000 --> 00:42:19.000
vectors are coplanar tells us
that, in fact -- (well,
00:42:19.000 --> 00:42:25.000
let me start a new blackboard).
Let's say that our normal
00:42:25.000 --> 00:42:28.000
vectors, N1, N2,
N3, are all in the same plane.
00:42:28.000 --> 00:42:32.000
And let's think about the
direction that's perpendicular
00:42:32.000 --> 00:42:35.000
to N1, N2, and N3 at the same
time.
00:42:35.000 --> 00:42:37.000
I claim that it will be the
line of intersection.
00:43:08.000 --> 00:43:12.000
So, let me try to draw that
picture again.
00:43:12.000 --> 00:43:26.000
We have three planes -- (now
you see why I prepared a picture
00:43:26.000 --> 00:43:31.000
in advance.
It's easier to draw it
00:43:31.000 --> 00:43:37.000
beforehand).
And I said their normal vectors
00:43:37.000 --> 00:43:41.000
are all in the same plane.
What else do I know?
00:43:41.000 --> 00:43:45.000
I know that all these planes
pass through the origin.
00:43:45.000 --> 00:43:50.000
So the origin is somewhere in
the intersection of the three
00:43:50.000 --> 00:43:59.000
planes.
Now, I said that the normal
00:43:59.000 --> 00:44:13.000
vectors to my three planes are
all actually coplanar.
00:44:13.000 --> 00:44:23.000
So N1, N2, N3 determine a plane.
Now, if I look at the line
00:44:23.000 --> 00:44:27.000
through the origin that's
perpendicular to N1,
00:44:27.000 --> 00:44:33.000
N2, and N3,
so, perpendicular to this red
00:44:33.000 --> 00:44:39.000
plane here,
it's supposed to be in all the
00:44:39.000 --> 00:44:44.000
planes.
(You can see that better on the
00:44:44.000 --> 00:44:47.000
side screens).
And why is that?
00:44:47.000 --> 00:44:51.000
Well, that's because my line is
perpendicular to the normal
00:44:51.000 --> 00:44:54.000
vectors, so it's parallel to the
planes.
00:44:54.000 --> 00:44:58.000
It's parallel to all the planes.
Now, why is it in the planes
00:44:58.000 --> 00:45:01.000
instead of parallel to them?
Well, that's because my line
00:45:01.000 --> 00:45:03.000
goes through the origin,
and the origin is on the
00:45:03.000 --> 00:45:07.000
planes.
So, certainly my line has to be
00:45:07.000 --> 00:45:11.000
contained in the planes,
not parallel to them.
00:45:11.000 --> 00:45:26.000
So the line through the origin
and perpendicular to the plane
00:45:26.000 --> 00:45:39.000
of N1, N2, N3 -- -- is parallel
to all three planes.
00:45:39.000 --> 00:45:47.000
And, because the planes go
through the origin,
00:45:47.000 --> 00:45:58.000
it's contained in them.
So what happens here is I have,
00:45:58.000 --> 00:46:06.000
in fact, infinitely many
solutions.
00:46:06.000 --> 00:46:09.000
How do I find these solutions?
Well, if I want to find
00:46:09.000 --> 00:46:13.000
something that's perpendicular
to N1, N2, and N3 -- if I just
00:46:13.000 --> 00:46:16.000
want to be perpendicular to N1
and N2,
00:46:16.000 --> 00:46:29.000
I can take their cross product.
So, for example,
00:46:29.000 --> 00:46:38.000
N1 cross N2 is perpendicular to
N1 and to N2,
00:46:38.000 --> 00:46:43.000
and also to N3,
because N3 is in the same plane
00:46:43.000 --> 00:46:46.000
as N1 and N2,
so, if you're perpendicular to
00:46:46.000 --> 00:46:49.000
N1 and N2, you are also
perpendicular to N3.
00:46:49.000 --> 00:47:03.000
It's automatic.
So, it's a nontrivial solution.
00:47:03.000 --> 00:47:09.000
This vector goes along the line
of intersections.
00:47:09.000 --> 00:47:13.000
OK, that's the case of
homogeneous systems.
00:47:13.000 --> 00:47:24.000
And then, let's finish with the
other case, the general case.
00:47:24.000 --> 00:47:32.000
If we look at a system,
AX = B, with B now anything,
00:47:32.000 --> 00:47:41.000
there's two cases.
If the determinant of A is not
00:47:41.000 --> 00:47:51.000
zero, then there is a unique
solution -- -- namely,
00:47:51.000 --> 00:47:59.000
X equals A inverse B.
If the determinant of A is
00:47:59.000 --> 00:48:02.000
zero,
then it means we have the
00:48:02.000 --> 00:48:06.000
situation with planes that are
all parallel to a same line,
00:48:06.000 --> 00:48:18.000
and then we have either no
solution or infinitely many
00:48:18.000 --> 00:48:23.000
solutions.
It cannot be a single solution.
00:48:23.000 --> 00:48:26.000
Now, whether you have no
solutions or infinitely many
00:48:26.000 --> 00:48:30.000
solutions, we haven't actually
developed the tools to answer
00:48:30.000 --> 00:48:32.000
that.
But, if you try solving the
00:48:32.000 --> 00:48:34.000
system by hand,
by elimination,
00:48:34.000 --> 00:48:37.000
you will see that you end up
maybe with something that says
00:48:37.000 --> 00:48:40.000
zero equals zero,
and you have infinitely many
00:48:40.000 --> 00:48:42.000
solutions.
Actually, if you can find one
00:48:42.000 --> 00:48:45.000
solution, then you know that
there's infinitely many.
00:48:45.000 --> 00:48:48.000
On the other hand,
if you end up with something
00:48:48.000 --> 00:48:51.000
that's a contradiction,
like one equals two,
00:48:51.000 --> 00:48:54.000
then you know there's no
solutions.
00:48:54.000 --> 00:48:58.000
That's the end for today.
Tomorrow, we will learn about
00:48:58.000 --> 00:49:01.000
parametric equations for lines
and curves.