Finding vector fields with given properties
Solution 1:
Use polar coordinates. Let $\hat{r}$ be the radial unit vector, then:
\begin{align*} \vec{F}(r) = -\hat{r} \end{align*}
is the required field. Now $\hat{r} = \cos \theta \hat{i} + \sin \theta \hat{j}$. Also, from basic trigonometry, you know: $$\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}$$ and $$\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}$$
So in Cartesian coordinate the field you want is: \begin{align*} \vec{F}(x,y) = -\left(\frac{x}{\sqrt{x^2 + y^2}} \hat{i} + \frac{y}{\sqrt{x^2 + y^2}} \hat{j}\right) \end{align*}