Finding the Vector Equation of a Plane
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 vectors
and
by some means.
If we have three points in the plane,
and
then we can take
to be any of
or
with
and
or similar. If
and
then
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 equation
This plane contains the points
and 
Then
then
so that withas 
