The University of Sydney - School of Mathematics and Statistics

The vector equation of a plane	Page 1 of 2

A plane can be described in many ways. The plane, for example, can be specified by three non-collinear points of the plane: there is a unique plane containing a given set of three non-collinear points in space.

An alternative way to specify a plane is given as follows.

Select a point P₀ in the plane. There is a unique line through P₀ perpendicular to the plane. This line is called the normal to to the plane at P₀. A vector n0 parallel to this normal is called a normal vector for the plane.

There is a unique plane which passes through P₀ and has n as a normal vector.

Now P lies in the plane through P₀ perpendicular to n if and only if and n are perpendicular.

As = r - r₀, this condition is equivalent to

This is a vector equation of the plane.