… is

\begin{equation*} \| \vec{u} - \vec{v} \| = \left\| \begin{pmatrix} u_1 \\ \vdots \\ u_m \end{pmatrix} - \begin{pmatrix} v_1 \\ \vdots \\ v_m \end{pmatrix} \right\| = \sqrt{{(u_1 - v_1)}^2 + \cdots + {(u_m - v_m)}^2}. \end{equation*}