Given a normal vector to the plane , and , the position vectors of two points in the plane, then we have the vector form equation of a plane:
This can be expanded using commutativity:
in order to obtain:
Since each vector in the plane most be orthogonal to the normal vector , then it means that the vector is a vector in the plane.
With some simplification and substituting , and we obtain a parametric equation:
Since a plane is defined by three points in geometry, we can use the parametric equation form to determine the equation of the plane that contains all three points - there is only one such plane but can be described by several equations.
Suppose that the given points are:
We compute the normal vector by computing the cross-product of the vectors between any two points - for example, the vector from to and the vector from to :
and expanding the determinant by the cofactors of , and :
Next, in addition to the normal vector, we need a point to describe the plane by and we can pick any of the three points chosen. Any of the resulting equation will be valid. Let's pick the first one which will lead to the plane equation:
conversely, if we picked point , we would have obtained:
and, for point :
Any of these three equations describe a plane that contains all of the points , , .
We can transform the equations back into vector form by knowing the normal vector and making pairs of , and :