Examples of vector fields directed towards origin

Question

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 $r^2$?

Answer 1

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

Answer 2

Both would be directed toward the origin, one of magnitude the distance from the origin, the other of magnitude one. Take your choice. They are scalar multiples of one another.

So, $F(x,y)=(-x,-y)$ should suffice, say.