How do I sketch a vector function such as $ \mathbf{v} = \frac{\mathbf{\hat{r}}}{r^2} $?

Question

I am studying Griffiths Introduction to electrodynamics and stumbled upon problem 1.16, it says,

Sketch the vector function $ \mathbf{v} = \frac{\mathbf{\hat{r}}}{r^2} $ and compute its divergence. The answer may surprise you. . . can you explain it?

I have calculated its divergence and it is $0$, however, I can't imagine how the vector field looks like. Can someone tell me how to?

Answer 1

I think you just have to remember that $$\hat{\mathbf{r}}=\frac{\mathbf{x}}{|\mathbf{x}|}=\frac{1}{\sqrt{x^2+y^2+z^2}}\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$ So that the vector field $\mathbf{v}$ has radial direction $($because it has the same direction as $\hat{\mathbf{r}})$ but it's multiplied by a factor $r^{-2}$ which goes to zero as $r$ approaches infinity. In the end, it's quiver plot looks something like this