How do I find the normal vector to this surface?

Question

I would like to find the normal vector to the surface $$xz + z^2 - xy^2 = 5$$

at the point $(1,1,2)$, and further deduce the equation of the tangent plane at this point.

By letting the surface be some $f$, I have found $\nabla f$ to be the vector $i -2 j + 5k$, but I have no idea whether this vector is normal or the tangential one, since ordinarily when differentiating we get a tangential line. How should I further proceed?

Answer 1

$\nabla f$ is a vector normal to the surface.

You use $\nabla f$ as your normal to the tangent plain at $(1,1,2)$.

The equation of your tangent plain is

$$ 1(x-1)-2(y-1)+5(z-2)=0 $$ You may solve for $z$ if you wish.