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Euclidean distance

a.k.a.

• Vector distance
• Distance between vectors

Define

… as the length of the vector difference

\[\begin{equation*}
  \| \vec{u} - \vec{v} \|.
\end{equation*}
\]

Imagine

… in the real plane \(\mathbb{R}^2\):

1. Draw \(\vec{u}, \vec{v}\) as points \(P\) and \(Q\).
2. Measure the length of the line segment \(\overline{PQ}\).

Closed form

… is

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

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.

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.

Denote

… as shown above, or

\[\begin{equation*}
  \| \vec{u} - \vec{v} \|
  = d(\vec{u}, \vec{v})
  = \operatorname{dist}(\vec{u}, \vec{v}).
\end{equation*}
\]

Implement

… in Scheme as

• my-vector-euclidean-distance

  (define (my-vector-euclidean-distance vector other-vector)
    (my-vector-length (my-vector-subtract vector other-vector)))
  

  in terms of my-vector-length and my-vector-subtract.