A plane in three dimensions can be defined by three points in the plane, in Cartesian form or in terms of vectors drawn in the plane and a point in the plane. From each of the definition, each of the others can be given.
An example of a plane in vector notation is
Suppose A(2,4,0), B(1,0,4) and C(1,1,0) are three points in a plane. We can find two vectors in the plane by find vectors between the points.
and 
We can choose any of A, B or C as a point in the plane. If we choose A then the equation of the plane can be written
Given the cartesian equation of a plane,
we can find the vector equation of the plane.
The vector
is normal to this plane (the components of this vector are the coefficients of x, y and z). Any vector
satisfying
is a vector in the plane. By inspection we can take
and
as vectors satisfying this condition, and
as a point in the plane, so that the vector equation of the plane can be written