… the closed form as

\begin{align*} \| \vec{u} - \vec{v} \| \tag{1} & = \left\| \begin{pmatrix} u_1 \\ \vdots \\ u_m \end{pmatrix} - \begin{pmatrix} v_1 \\ \vdots \\ v_m \end{pmatrix} \right\| \\[1ex] \tag{2} & = \sqrt{(\vec{u} - \vec{v}) \cdot (\vec{u} - \vec{v})} \\[1ex] \tag{3} & = \sqrt{\sum_{i = 1}^{m} (u_i - v_i) (u_i - v_i) } \\[1ex] \tag{4} & = \sqrt{\sum_{i = 1}^{m} {(u_i - v_i)}^2 } \\[1ex] \tag{5} & = \sqrt{{(u_1 - v_1)}^2 + \cdots + {(u_m - v_m)}^2} \end{align*}

with

1. by vector,
2. by vector length,
3. by dot product,
4. by exponentiation, and
5. by summation.