The University of Sydney - School of Mathematics and Statistics

The Cartesian equation of a plane	Page 2 of 2

Suppose that P₀ has coordinates x₀,y₀,z₀ and n has components a,b,c.

Let P, with coordinates x,y,z, be an arbitrary point on the plane.

The position vectors of P₀ and P are

and

Substituting

into the vector equation, we obtain

which, when multiplied out, gives

This is called a Cartesian equation of the plane.

It simplifies to

where d is the constant ax₀ + by₀ + cz₀.

An equation of the form

where a,b,c and d are constants and not all a,b,c are zero, can be taken to be an equation of a plane in space.

The coefficients a, b and c are the components of a normal vector

for the plane described by the equation.