How can I find a vector which is a unit normal vector to the surface $3x^2-yz+ z^2 = 0$ at the point $(1, -4, -3)$?
Hint: $(\nabla F)(a, b, c)$ is a normal vector to the surface $F(x, y, z) = 0$ at the point $(a, b, c)$.
For a surface given by the equation $\phi(x, y, z) = 0$, a normal at the point $(x, y, z)$ is $\nabla \phi (x, y, z)$. So find the gradient of the LHS and evaluate it at the given point. Then normalize it by dividing it by its magnitude.
I'll update this answer with the final answer once you get it yourself.