Given a plane a x + b y + c z + d = 0 and a point x_0 = (x_0, y_0, z_0), the normal vector to the plane is given by v = [a b c], and a vector from the plane to the point is given by w = - [x - x_0 y - y_0 z - z_0].

coplanar | Hessian normal form | plane | point | projection theorem