Formally something that has both direction and length but in practice, computationally speaking, usually a set of Cartesian coordinates. For example
\[ \mathbf{v} = \begin{pmatrix} v_0\\ v_1\\ v_2 \end{pmatrix} \]
Given a number \(x\) and vectors \(\mathbf{v}\) and \(\mathbf{w}\) and denoting the \(i^{th}\) element of a vector \(\mathbf{v}\) with \(v_i\) the rules of vector arithmetic are given by
\[ \begin{align*} (\mathbf{v} \times x)_i &= v_i \times x\\ (x \times \mathbf{v})_i &= v_i \times x\\ \\ (\mathbf{v} \div x)_i &= v_i \div x\\ \\ (\mathbf{v} + \mathbf{w})_i &= v_i + w_i\\ (\mathbf{v} - \mathbf{w})_i &= v_i - w_i\\ \\ \mathbf{v} \times \mathbf{w} &= \sum_i v_i \times w_i \end{align*} \]
where \(\sum\) is the summation sign.