Examples of vector fields directed towards origin I'm asked to state a vector field equation directed radially in towards origin, would $-\langle x,y\rangle $ suffice, or do I need to divide it by its magnitude r i.e. $-\left\langle \frac{x}{\sqrt {x^2+y^2}},\frac{y}{\sqrt {x^2+y^2}}\right\rangle $ or should I divide it by $r^2$?

Any vector field of the form
$$
F(\mathbf r) = -f(|\mathbf r|)\, \mathbf r = -f(r)\, \mathbf r \ ; \qquad f(r) > 0
$$
is directed towards the origin, where $f$ is function of one argument, and its argument here is the distance $r$ from the origin, which is the length of $|\mathbf r|$. For example, in $\mathbb R^2$ we have
$$
r = \sqrt{x^2 +y^2}\ ;\quad  \mathbf r = (x,y)^T
$$
one can take
\begin{align}
f(r) &= \cos^2(r^2) = \cos^2(x^2 +y^2)\\
f(r) &= \ln^2(r^2-3r) = \ln^2(x^2 +y^2-3\sqrt{x^2 +y^2}) \\
f(r) &= \frac{1}{r^3} = (x^2 +y^2)^{-\frac{3}{2}} \\
\end{align}
and $F(\mathbf r)$ above will be as required
