(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 FxGx + FyGy + FzGz = 0 atP(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?

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