Let $f(x, y) = x^3 - xy + \cos(\pi(x + y))$.
(a) Find a vector normal to the level curve $f(x, y) = 1$ at the point $(1, 1)$.
(b) Find the equation of the line tangent to the level curve $f(x, y) = 1$ at the point $(1, 1)$.
(c) Find a vector normal to the graph $z = f(x, y)$ at the point $(1, 1)$.
(d) Find the equation of the plane tangent to the graph $z = f(x, y)$ at the point $(1, 1)$.