The idea behind this is because the surface normal is by definition perpendicular to the surface, and the resulting vector of $v = p - p'$ where $p'$ is any point on the plane is a vector in the plane. All vectors in the plane must be perpendicular to the surface normal, meaning that their dot product is 0 (since cos(90) = 0).

The idea behind this is because the surface normal is by definition perpendicular to the surface, and the resulting vector of $v = p - p'$ where $p'$ is any point on the plane is a vector in the plane. All vectors in the plane must be perpendicular to the surface normal, meaning that their dot product is 0 (since cos(90) = 0).