The normal form of a plane uses a vector perpendicular ("normal") to the plane and one known point in the plane. If a point in the plane has position vector a and n is a normal vector, then any other point r lies in the plane if:

This equation expresses the idea that the vector from the known point to any other point in the plane is always perpendicular to n.

Since the normal vector is perpendicular to any line in the plane, we can find the normal vector by taking the vector product of the direction vectors in the form r=a+λb+μc:

Note that the normal is not unique - any vector parallel to this normal will also be normal to the plane.

Example

From the vector equation

the direction vectors are ⎝⎛​1−11​⎠⎞​ and ⎝⎛​20−2​⎠⎞​.

Their cross product is

Using a=⎝⎛​123​⎠⎞​ as a known point, the plane’s normal form follows from r⋅n=a⋅n. Since

the plane’s normal form is