${\displaystyle \mathbf {v} \in \mathbb {R} ^{n}\Longleftrightarrow \mathbf {v} ={\begin{pmatrix}{\mathit {v}}_{0}\\{\mathit {v}}_{1}\\\vdots \\{\mathit {v}}_{n-1}\end{pmatrix}}{\rm {{mit}{\mathit {v_{i}}}\in \mathbb {R} ,{\mathit {i}}=0,\cdots ,n-1}}}$

${\displaystyle \mathbf {u} +\mathbf {v} ={\begin{pmatrix}{\mathit {u}}_{0}\\{\mathit {u}}_{1}\\\vdots \\{\mathit {u}}_{n-1}\end{pmatrix}}+{\begin{pmatrix}{\mathit {v}}_{0}\\{\mathit {v}}_{1}\\\vdots \\{\mathit {v}}_{n-1}\end{pmatrix}}={\begin{pmatrix}{\mathit {u}}_{0}+{\mathit {v}}_{0}\\{\mathit {u}}_{1}+{\mathit {v}}_{1}\\\vdots \\{\mathit {u}}_{n-1}+{\mathit {v}}_{n-1}\end{pmatrix}}\in \mathbb {R} ^{n}}$

${\displaystyle \mathbf {u} -\mathbf {v} ={\begin{pmatrix}{\mathit {u}}_{0}\\{\mathit {u}}_{1}\\\vdots \\{\mathit {u}}_{n-1}\end{pmatrix}}-{\begin{pmatrix}{\mathit {v}}_{0}\\{\mathit {v}}_{1}\\\vdots \\{\mathit {v}}_{n-1}\end{pmatrix}}={\begin{pmatrix}{\mathit {u}}_{0}-{\mathit {v}}_{0}\\{\mathit {u}}_{1}-{\mathit {v}}_{1}\\\vdots \\{\mathit {u}}_{n-1}-{\mathit {v}}_{n-1}\end{pmatrix}}\in \mathbb {R} ^{n}}$

${\displaystyle {\mathit {a}}\mathbf {u} ={\begin{pmatrix}{\mathit {au}}_{0}\\{\mathit {au}}_{1}\\\vdots \\{\mathit {au}}_{n-1}\end{pmatrix}}\in \mathbb {R} ^{n}}$

${\displaystyle \mathbf {w} ={\begin{pmatrix}{\mathit {w}}_{x}\\{\mathit {w}}_{y}\\{\mathit {w}}_{z}\end{pmatrix}}=\mathbf {u} \times \mathbf {v} ={\begin{pmatrix}{\mathit {u_{y}v_{z}}}-{\mathit {u_{z}v_{y}}}\\{\mathit {u_{z}v_{x}}}-{\mathit {u_{x}v_{z}}}\\{\mathit {u_{x}v_{y}}}-{\mathit {u_{y}v_{x}}}\end{pmatrix}}}$