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Vector addition

Define

… in terms of addition as the binary operation

\[\begin{equation*}
  \vec{u} + \vec{v}
  =
  \begin{pmatrix} u_1 \\ \vdots \\ u_n \end{pmatrix}
  + \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}
  =
  \begin{pmatrix} u_1 + v_1 \\ \vdots \\ u_n + v_n \end{pmatrix}
  \in V^n,
\end{equation*}
\]

with

\[\begin{equation*}
  \vec{u}, \vec{v} \in V^n, \quad
  \forall i : u_i, v_i \in F,
\end{equation*}
\]

where	\(\vec{u}, \vec{v}\)	are the terms,
	\(\vec{u} + \vec{v}\)	is the sum,
	\(V^n\)	is the vector space with \(\vec{v}, \vec{u}\), and
	\(F\)	is the field over which \(V^n\) is defined.

Imagine

… geometrically:

1. Draw \(\vec{u}\) and \(\vec{v}\) head to tail.
2. Draw \(\vec{u} + \vec{v}\) from the free tail to the free head.

(Weisstein, n.d.)

Constrain

• Additive identity

  … is the zero vector \(\vec{0}\), so

  \[\begin{gather*}
    \forall \vec{v} \in V, \\[1ex]
    \vec{v} + \vec{0} = \vec{v}.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall \vec{v} \in V, \> \exists! \vec{w} \in V : \\[1ex]
    \vec{v} + \vec{w} = \vec{v},
  \end{gather*}
  \]

  where the identity is

  \[\begin{equation*}
    \vec{w} = \vec{0}.
  \end{equation*}
  \]

• Additive inverse

  … is \(-\vec{v}\), so

  \[\begin{gather*}
    \forall \vec{v} \in V, \\[1ex]
    \vec{v} + (-\vec{v}) = \vec{0}.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall \vec{v} \in V, \> \exists! \vec{w} \in V : \\[1ex]
    \vec{v} + \vec{w} = \vec{0},
  \end{gather*}
  \]

  where the identity is

  \[\begin{equation*}
    w = -\vec{v}.
  \end{equation*}
  \]

• Commutative property

  … holds, so

  \[\begin{gather*}
    \forall \vec{u}, \vec{v} \in V, \\[1ex]
    \vec{u} + \vec{v} = \vec{v} + \vec{u}.
  \end{gather*}
  \]

• Associative property

  … holds, so

  \[\begin{gather*}
     \forall \vec{u}, \vec{v}, \vec{w} \in V, \\[1ex]
     (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}).
  \end{gather*}
  \]

Implement

… in Scheme as

• my-vector-add

  (define (my-vector-add vector . vectors)
    (if (null? vectors)
        vector
        (map + vector (apply my-vector-add vectors))))