A vector field $\mathbf{F}(x,y)$ is a function that returns a vector for every point $(x,y)$.
In order to specify a vector field $\mathbf{F}(x,y)$, we need to find the $\mathbf{i}$ and $\mathbf{j}$ components of $\mathbf{F}$ at all positions $(x,y)$.