Showing that multiple points are coplanar

[ZeroSum]

I recently solved a problem involving multiple points that were intended to be proven to be coplanar. Someone else suggested to me that I should be using the much messier scalar triple product.

However, I worked the problem in a different way. I treated it like a conjecture (I assumed that the points are coplanar). I crossed two vectors formed by the points to get a normal vector and then used that to create a formula for a plane. I then plugged each of the points into that equation to show that the points were all solutions for the formula for the plane (each side of the equation zeroed out, showing that the points were solutions for the plane equation).

Can anyone tell me if there is something wrong with me doing this as a general method of solution for this sort of problem?

 

[Hyperbolful]

I'm not an expert, but first let me see if I get the gist of your problem
You are given a set of points, and asked to show that the lie in the same plane
So you took the points, and constructed a plane that went through all of them
and Tah Dah they must be coplanar.

This sounds like it works, but only if the question was for specific points. If the problem was to find necessary and sufficent conditions that the points be coplanar the proof would involve more about the points themselves, and not a construction of a plane through them.

 

[ZeroSum]

Thank you for your reply, Hyperbolful. You are correct, the question was involving given specific points. It wasn't intended to be a formal proof, just an exercise.

 

[AlephZero]

You would have hit a problem if the points were collinear (or coincident) as well as coplanar, because the "equation of the plane" would not be defined.

The scalar triple product is equivalent to finding the volume of the tetrahedron defined by 4 points, and the volume is 0 if the points are coplanar. The scalar triple product can be calculated for ANY 4 points, so in that is a more "general" method.

But if your method worked for the points you were given, it is a perfectly good proof.

 

[Hyperbolful]

No problem Zero
AlephZero,

If the equation of the plane is not well defined, would that imply that the points must all be colinear? Assuming they were all finite and none were infinite or anything strange like that.

 

[ZeroSum]

Thank you for your reply, AlephZero. It's interesting to consider the edge case of points that happen to also be collinear. I'll keep that in mind in case anyone throws that one at me as a curveball. :)