Chapter 13.7, Problem 37ES

Two surface $f\left(x,y,z\right)=0\text{and}g\left(x,y,z\right)=0$ are said to be orthogonal at a point P of intersection if $\nabla f\text{and}\nabla g$ are nonzero at P and the normal lines to the surfaces are perpendicular at P. Show that if $\nabla f\left(x_0,y_0,z_0\right)\ne 0\text{and}\nabla g\left(x_0,y_0,z_0\right)\ne 0,$ then the surfaces $f\left(x,y,z\right)=0\text{and}g\left(x,y,z\right)=0$ are orthogonal at the point $\left(x_0,y_0,z_0\right)$ if and only if

$f_x{g}_x+f_y{g}_y+f_z{g}_z=0$

at this point. [Note: This is a more general version of the result in Exercise 35.]