# 四维空间本文重定向自 四維

## 四维空间定义

${\displaystyle \mathbf {x} =(p,q,r,s)}$

${\displaystyle \|\mathbf {x} \|={\sqrt {p^{2}+q^{2}+r^{2}+s^{2}}}}$

### 向量

#### 向量运算

${\displaystyle \mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}.}$

${\displaystyle \mathbf {e} _{1}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\mathbf {e} _{2}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\mathbf {e} _{3}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\mathbf {e} _{4}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},}$

${\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}+a_{4}\mathbf {e} _{4}.}$

${\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.}$

${\displaystyle \left|\mathbf {a} \right|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}+{a_{4}}^{2}}},}$

${\displaystyle \theta =\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{\left|\mathbf {a} \right|\left|\mathbf {b} \right|}}.}$

${\displaystyle \mathbf {a} \wedge \mathbf {b} =(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{12}+(a_{1}b_{3}-a_{3}b_{1})\mathbf {e} _{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf {e} _{14}+(a_{2}b_{3}-a_{3}b_{2})\mathbf {e} _{23}+(a_{2}b_{4}-a_{4}b_{2})\mathbf {e} _{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.}$

## 几何

### 超球体

${\displaystyle V=2\pi ^{2}R^{3}}$