We can find the vector equation of a three dimensional plane given either

Three points in the plane

Two vectors and a point in the plane

Two points and a vector in the plane

The equation of the plane in some formulation
The vector equation of a plane is of the form(1)
We must obtain the vectorsandby some means.
If we have three points in the plane,andthen we can taketo be any of orwithandor similar. Ifandthen so that withas
The second case above is obvious, and immediate from (1).
The third case follows since we can construct a second vector by subtracting the two points to give a second vector in the plane so that we have two vectors and a point in the plane.
All equations of a plane define a plane, and from some formulation we must be able to find three points in the plane, then use the method above. Suppose for example that we have the plane with cartesian equationThis plane contains the pointsand Then
then so that withas