Section 14.6: Directional Derivatives & Gradient Vector

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 \(\nabla F \neq \mathbf{0}\) and \(\nabla G \neq \mathbf{0}\) if and only if \(F_x G_x + F_y G_y + F_z G_z = 0\) at P

Use part (a) to show that the surfaces \(z^2 = x^2 + y^2\) and \(x^2 + y^2 + z^2 = r^2\) are orthogonal at every point of intersection. Can you see why this is true without using calculus?