We are tasked with determining whether or not $\exists V:\mathbb{R}^3\to\mathbb{R}$ such that $\vec{F}(t)=\vec{r}(t)/|\vec{r}(t)|^3$ satisfies $$ \vec{F}(t)=-\nabla V $$ Intuitively I wish to therefore integrate the given expression for $\vec{F}(t)$ with respect to $t$, in hopes of obtaining another expression in terms of $t$, which I may then claim is equal to $-V$. I know that the expression I obtain from integrating $\vec{F}(t)$ must be a scalar, if $\vec{F}(t)$ is to satisfy the given relationship, since $V$ is a scalar function, but since the expression given for $\vec{F}(t)$ is a vector quantity, I'm a little stomped on how to proceed.
Any hints and suggestions are appreciated. Thank you.
Given the symmetry use spherical coordinates: $$ F=\frac{1}{|\mathbf{r}|^2} \hat{\mathbf r} $$ so you can integrate and find $$ F=-\nabla \frac{1}{|\mathbf r|} $$