## The Vector Equation of a Plane

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 vectoris normal to this plane (the components of this vector are the coefficients of x, y and z). Any vectorsatisfyingis a vector in the plane. By inspection we can takeandas vectors satisfying this condition, andas a point in the plane, so that the vector equation of the plane can be written