Finding the Vector Equation of a Plane

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

1. Three points in the plane

2. Two vectors and a point in the plane

3. Two points and a vector in the plane

4. 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 oforwithandor 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