---

Derive again

… the closed form as

\[\begin{align*}
  \| \vec{u} - \vec{v} \|
  \tag{1}
  & = \left\|
  \begin{pmatrix} u_1 \\ \vdots \\ u_m \end{pmatrix}
  -
  \begin{pmatrix} v_1 \\ \vdots \\ v_m \end{pmatrix}
  \right\|
  \\[1ex]
  \tag{2}
  & = \sqrt{(\vec{u} - \vec{v}) \cdot (\vec{u} - \vec{v})}
  \\[1ex]
  \tag{3}
  & = \sqrt{{(\vec{u} - \vec{v})}^\mathsf{T} (\vec{u} - \vec{v})}
  \\[1ex]
  \tag{4}
  & = \sqrt{
    \begin{pmatrix}
      \vphantom{\bigg(}
      u_1 - v_1 & \cdots & u_m - v_m
    \end{pmatrix}
    \begin{pmatrix}
      u_1 - v_1 \\ \vdots \\ u_m - v_m
    \end{pmatrix}
  }
  \\[1ex]
  \tag{5}
  & = \sqrt{(u_1 - v_1) (u_1 - v_1) + \cdots + (u_m - v_m) (u_m - v_m)}
  \\[1ex]
  \tag{6}
  & = \sqrt{{(u_1 - v_1)}^2 + \cdots + {(u_m - v_m)}^2}
\end{align*}
\]

with

1. by vector,
2. by vector length,
3. by the dot product in terms of the matrix transpose,
4. by the matrix transpose,
5. by matrix multiplication, and
6. by exponentiation.