No, we can not divide vector by vector. Here is possibly the simplest explanation. To explain this consider the dot product of two vectors \vec{A} and \vec{B} which gives the scalar m.

\vec{A}\cdot \vec{B}=mHere \vec{A} doesn’t equal to 0, and \vec{B} is an unknown vector.

Considering division as the inverse of multiplication, we have to find \vec{B} such that it gives scalar m when the dot product is applied with \vec{A}. But this has infinitely many solutions. Since the projection of \vec{B} onto the direction of \vec{A} is not unique (shown in the figure above). Hence the operation of division is avoided in vector algebra.