There are different ways to define a plane using vectors.

Parametric form of a plane

This form is similar to the parametric form of a line. We add a second dimension to the line in order to make it a plane by adding another vector multiplied by a scalar. This new vector equation must be linear independent of equation:

    equation

plane

Cartesian form (general form) of a plane

    equation

where equation, equation and equation make up a vector that is perpendicular to the plane, called the normal. That’s why we also call it equation and write:

(1)   equation

Normal form of a plane

The normal form is similar to the cartesian form and can be obtained by expanding it:

    equation

where equation is a point on the plane. If equation is normalized, this is the Hessian normal form.