Can you explain the meaning of $v\vec{v}$?

A vector can in general be written $\vec{a} = a \hat{a}$, where $\hat{a}$ is a unit vector (has unit length) in the direction of $\vec{a}$, and $a$ is the magnitude of the vector. So a vector $v\vec{v}$ can be written $v v \hat{v} = v^2 \hat{v}$: it therefore has magnitude $v^2$.