18. The line normal to the surface 3x + y² – z² = 0 at the point (3,0,3) also intersects the surface at what other point?

First find the parametric equations for the line that is normal to the given surface and intersects the given point. Then substitute the parametric equations into the equation of the given surface to find the values of the parameter that satisfies that equation. One parameter will correspond to the given point and the other will yield the other point we are trying to find.

The normal vector of the given surface at the point (3,0,3) is <F_x,F_y,F_z>

The parametric equations of the line that is normal to the given surface and intersects the given point are given by:

Where t is the parameter and the normal vector gives the coefficients of t and the constant terms are from the given point.

Substitute the above equations into the equation of the surface and solve for the parameter t.

Hence t = 0 corresponds to the given point (3,0,3) and t = 5/4 yields the other point in question, (27/4,0,-9/2).