Consider the surface $x^3 + y^3 z = 3$. Find tangent vector at the point $(1,1,2)$ that has $\mathbf{i}$ component 1 and $\mathbf{j}$ component 1. To find it, first find a normal vector.
Solution
Let $\bfv = \bfi + \bfj + c \ \bfk$ be the vector we seek.