If there is a plane that contains every point of a given set, those points are called coplanar.
We want to check if the points and are coplanar.
First consider the plane The cross product of the vectors and is normal to the plane they are on.
Therefore, if the vector is normal to all four points are on the same plane. That is, we want to prove:
Are and on the same plane?
First we want to find the cross product The vectors are and
The cross product is then
The vector is We then just need the dot product of this with
Since the four points are not coplanar.