Im trying to understand the steps to finding a point on the surface where the tangent plane is perpendicular to following tangent plane
$\displaystyle
x+y+z=11
$
This makes no sense. A "vector" does not have a surface! If your surface is given by f(x, y, z)= constant, then $\displaystyle \nabla f$ is perpendicular to the surface. That vector must be parallel (so a multiple of) <1, 1, 1>. Now, what surface are you talking about?