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## Points, lines, & planes

# Specifying planes in three dimensions

## Video transcript

We've already been exposed
to points and lines. Now let's think about planes. And you can view planes as
really a flat surface that exists in three dimensions, that
goes off in every direction. So for example, if I have
a flat surface like this, and it's not curved,
and it just keeps going on and on and
on in every direction. Now the question is, how
do you specify a plane? Well, you might say,
well, let's see. Let's think about
it a little bit. Could I specify a plane with
a one point, right over here? Let's call that point,
A. Would that, alone, be able to specify a plane? Well, there's an
infinite number of planes that could go
through that point. I could have a plane that
goes like this, where that point, A,
sits on that plane. I could have a plane like that. Or, I could have
a plane like this. I could have a plane
like this where point A sits on it, as well. So I could have a
plane like that. And I could just keep
rotating around A. So one point by
itself does not seem to be sufficient
to define a plane. Well, what about two points? Let's say I had a point,
B, right over here. Well, notice the way I
drew this, point A and B, they would define a line. For example, they would define
this line right over here. So they would define,
they could define, this line right over here. But both of these points and
in fact, this entire line, exists on both of these
planes that I just drew. And I could keep
rotating these planes. I could have a plane
that looks like this. I could have a plane
that looks like this, that both of these
points actually sit on. I'm essentially just
rotating around this line that is defined by
both of these points. So two points does not
seem to be sufficient. Let's try three. So there's no way
that I could put-- Well, let's be careful here. So I could put a third
point right over here, point C. And C
sits on that line, and C sits on all
of these planes. So it doesn't seem like
just a random third point is sufficient to define, to pick
out any one of these planes. But what if we
make the constraint that the three points are
not all on the same line. Obviously, two points
will always define a line. But what if the three
points are not collinear. So instead of
picking C as a point, what if we pick--
Is there any way to pick a point, D, that
is not on this line, that is on more than one
of these planes? We'll, no. If I say, well, let's see,
the point D-- Let's say point D is right over here. So it sits on this
plane right over here, one of the first
ones that I drew. So point D sits on that plane. Between point D, A, and
B, there's only one plane that all three of
those points sit on. So a plane is defined by
three non-colinear points. So D, A, and B, you see, do
not sit on the same line. A and B can sit
on the same line. D and A can sit
on the same line. D and B can sit
on the same line. But A, B, and D does not sit
on-- They are non-colinear. So for example, right
over here in this diagram, we have a plane. This plane is labeled, S. But another way that we
can specify plane S is we could say, plane--
And we just have to find three non-collinear
points on that plane. So we could call this plane AJB. We could call it plane JBW. We could call it plane-- and I
could keep going-- plane WJA. But I could not specify
this plane, uniquely, by saying plane ABW. And the reason why
I can't do this is because ABW are
all on the same line. And this line sits on an
infinite number of planes. I could keep rotating around the
line, just as we did over here. It does not specify
only one plane.