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 must be linear independent of :
Cartesian form (general form) of a plane
where , and make up a vector that is perpendicular to the plane, called the normal. That’s why we also call it and write:
Normal form of a plane
The normal form is similar to the cartesian form and can be obtained by expanding it:
where is a point on the plane. If is normalized, this is the Hessian normal form.