Abstellraum: Strukturwissenschaften: Vektoren

Euklidischer Raum

$𝐯\in {ℝ}^n⟺𝐯=\left(\begin{array}{c}{𝑣}_0\\ {𝑣}_1\\ ⋮\\ {𝑣}_{n-1}\end{array}\right)\mathrm{m}\mathrm{i}\mathrm{t}{𝑣}_{𝑖}\mathrm{\in }ℝ,𝑖\mathrm{=}\mathrm{0},\mathrm{\cdots },\mathrm{n}\mathrm{-}\mathrm{1}$

Addition

$𝐮+𝐯=\left(\begin{array}{c}{𝑢}_0\\ {𝑢}_1\\ ⋮\\ {𝑢}_{n-1}\end{array}\right)+\left(\begin{array}{c}{𝑣}_0\\ {𝑣}_1\\ ⋮\\ {𝑣}_{n-1}\end{array}\right)=\left(\begin{array}{c}{𝑢}_0+{𝑣}_0\\ {𝑢}_1+{𝑣}_1\\ ⋮\\ {𝑢}_{n-1}+{𝑣}_{n-1}\end{array}\right)\in {ℝ}^n$

Subtraktion

$𝐮-𝐯=\left(\begin{array}{c}{𝑢}_0\\ {𝑢}_1\\ ⋮\\ {𝑢}_{n-1}\end{array}\right)-\left(\begin{array}{c}{𝑣}_0\\ {𝑣}_1\\ ⋮\\ {𝑣}_{n-1}\end{array}\right)=\left(\begin{array}{c}{𝑢}_0-{𝑣}_0\\ {𝑢}_1-{𝑣}_1\\ ⋮\\ {𝑢}_{n-1}-{𝑣}_{n-1}\end{array}\right)\in {ℝ}^n$

Multiplikation mit einem Skalar

$𝑎𝐮=\left(\begin{array}{c}{𝑎𝑢}_0\\ {𝑎𝑢}_1\\ ⋮\\ {𝑎𝑢}_{n-1}\end{array}\right)\in {ℝ}^n$

Kreuzprodukt

$𝐰=\left(\begin{array}{c}{𝑤}_x\\ {𝑤}_y\\ {𝑤}_z\end{array}\right)=𝐮\times 𝐯=\left(\begin{array}{c}{𝑢}_{𝑦}{𝑣}_{𝑧}-{𝑢}_{𝑧}{𝑣}_{𝑦}\\ {𝑢}_{𝑧}{𝑣}_{𝑥}-{𝑢}_{𝑥}{𝑣}_{𝑧}\\ {𝑢}_{𝑥}{𝑣}_{𝑦}-{𝑢}_{𝑦}{𝑣}_{𝑥}\end{array}\right)$