---

Derive

… 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.