If $\vec{u} \in V$, then there is a unique way to express $\vec{u}$ as a linear combination of the vectors $\vec{v_1},\vec{v_2},\vec{v_3}$. So, if $\vec{u} = a\vec{v_1} + b\vec{v_2} + c\vec{v_3}$, then $[\vec{u}]_\beta = \begin{pmatrix}a \\ b \\ c\end{pmatrix}$