What is proof of direction of Change in position vector $d \vec r$

Question

How can I prove that the direction of an infinitesimal change in position vector $\mathrm d\vec r$ is the same as that of the instantaneous velocity $ \vec v=\mathrm{d}\vec r/\mathrm{d}t$? What I found: $$ \mathrm{d} \hat r= \left(\frac{\| \vec v \|}{\|\mathrm{d} \vec r\|} \mathrm{d}t\right)\hat v $$ In this expression, the direction/unit vector $\mathrm{d}\hat r$ is related to some multiple of direction/unit vector instantaneous velocity $ \hat v$. How can they be the same and how can we define the value of $\mathrm{d}t$?