(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F(x, y, z) = 0 and G(x, y, z) = 0 are orthogonal at a point P where ∇F ≠ 0 and ∇G ≠ 0 if and only if

F_xG_x + F_yG_y + F_zG_z = 0 at P

(b) Use part (a) to show that the surfaces z² = x² + y² and x² + y² + z² = r² are orthogonal at every point of intersection. Can you see why this is true without using calculus?