Can we divide vector by vector?

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}=m

Here \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.