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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?

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  • $\begingroup$ Because any one point off the axis of rotation defined by the first two points fixes the position of the rotating plane. Another way to think about it is to connect those points by segments to get a triangle, and a triangle uniquely specifies the plane it is in. Yet another way to see it is to take one of the points as the origin, and form two vectors connecting it to the other two. Two non-collinear vectors span a plane. $\endgroup$
    – Conifold
    Jul 2, 2020 at 21:50
  • $\begingroup$ Intuitively it's because the dimension of a plane is 2 so you need exactly two linearly independent vectors to generate a plane. The three points are the origin and the tips of the two vectors, you wouldn't have two linearly independent vectors if the three points weren't non collinear. Is this useful at all? Maybe you should clarify the reason why it doesn't convince you... $\endgroup$
    – pink frog
    Jul 2, 2020 at 21:53

3 Answers 3

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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:

enter image description here

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  • $\begingroup$ Thanks your drawing made me intuitively understand this idea $\endgroup$
    – Tom Avery
    Jul 2, 2020 at 22:05
  • $\begingroup$ @TomAvery: You're welcome. (Perhaps an upvote or $\checkmark$ is in order...) $\endgroup$
    – David G. Stork
    Jul 2, 2020 at 22:06
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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.

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  • $\begingroup$ In general, $(n+1)$ points in "general position" are needed to determine a unique $n$-dimensional "hyperplane" (where a $1$-dimensional hyperplane is a line and a $2$-dimensional hyperplane is just a normal plane). "General position" here means that they aren't redundant in any way: e.g. no two are the same, no three are collinear, no four lie in the same plane (= coplanar), etc. $\endgroup$
    – Noah Schweber
    Jul 2, 2020 at 21:52
  • $\begingroup$ I am trying to explain this to someone who seems to not understand the intuition behind the statement. I prefer to give an analogy in low dimension than introducing rigorous terminology and concepts in dimensions we cannot visualize. $\endgroup$
    – Gibbs
    Jul 2, 2020 at 21:54
  • $\begingroup$ Oh indeed - I didn't mean my comment as criticism, just a comment the OP might find helpful after they understand your answer. $\endgroup$
    – Noah Schweber
    Jul 2, 2020 at 21:55
  • $\begingroup$ Fair enough, I misunderstood your intention. $\endgroup$
    – Gibbs
    Jul 2, 2020 at 21:56
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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}) $.

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