Points, lines, & planes
Specifying planes in three dimensions
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.