Tip:
Highlight text to annotate it
X
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 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
Also, I can 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 2 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, 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- well, let's be careful here
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.
Well, 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?
Well, 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 noncollinear 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 noncollinear.
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 the three noncollinear 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-
so I could not say plane "ABW"
And the reason why I can't do this is 'cause
A, B, and W 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.