I've just started looking at the axioms of 3D Geometry. The first one that I encountered is this one:
"Three non collinear points define a plane" or " Given three non collinear points, only one plane goes through them"
I know that it is an axiom and it is taken to be true but I don't understand the intuition behind it. I understand that if I take one point or any number of collinear points, then I can draw infinite planes just by rotating around the line that connects these points, but why do we need 3 non collinear points to define a plane, why not more? And why, given three non collinear points, does only one plane go through them? Why not two or three?
3 Answers
Two points determine a line (shown in the center). There are infinitely many infinite planes that contain that line. Only one plane passes through a point not collinear with the original two points:
Two points determine a line $l$. Thus, as you say, you can draw infinitely many planes containing these points just by rotating the line containing the two points. So you find a set of infinitely many planes containing a common line. For any third point not on $l$ then there is only one of these planes containing it.
An analogy is the same problem is lower dimension. Take a point in a plane. There are infinitely many lines through it. Now take a second point different from the first. Then there is a unique line among the infinitely many given that contains the two points.
A plane is a vectorial space whose dimension is $ 2$. its base contains exactly two independent vectors. If your three points $ A,B,C $ do not lie in the same line, you can take as a base, the couple $ (\vec{AB},\vec{AC}) $.
