Let $f(x, y)$ be a smooth function. If the directional derivative of $f$ at the point $P(1, 2)$ is $2$ in the direction of $\vec{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle$ and $-1$ in the direction of $\vec{v} = \langle \frac{4}{5}, -\frac{3}{5} \rangle$, find the gradient $\nabla f(1, 2)$.