Answer
See proof.
Work Step by Step
Both slope field and vector field keep track of direction; however, in a vector field, we also keep track of magnitude. That's why a vector field maps $\mathbb{R}^2 \to \mathbb{R}^2$, while a slope field maps $\mathbb{R}^2 \to \mathbb{R}$. We can say that in a vector field, we need two variables to keep track of, namely magnitude and direction, while in a slope field, we only care about the direction
