Find the directional derivative of $ f(x,y,z) = (xy + z^2) $ at the point $(2,3,2)$ in the direction of a vector making an angle of $\theta = 3π/4 $ with gradient vector $(2,3,2)$.
The directional derivative is defined as $\nabla f(x,y,z) ⋅ u$, so I tried to find the vector $u$ first:
$$\cos(3π/4) = \frac{u⋅v}{||v||||u||}$$ $$-\frac{1}{2} = \frac{u⋅(2,3,2)}{||\sqrt17||}$$ $$-\frac{\sqrt(17)}{2} = {u⋅(2,3,2)}$$
which doesn't seem to really give me a definite vector for $u$, and I'm stuck here. Is there a way to find $u$ or am I approaching this entirely wrong?