# 七维叉积

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

## 乘法表

× e1 e2 e3 e4 e5 e6 e7
e1 0 e3 e2 e5 e4 e7 e6
e2 e3 0 e1 e6 e7 e4 e5
e3 e2 e1 0 e7 e6 e5 e4
e4 e5 e6 e7 0 e1 e2 e3
e5 e4 e7 e6 e1 0 e3 e2
e6 e7 e4 e5 e2 e3 0 e1
e7 e6 e5 e4 e3 e2 e1 0

${\displaystyle \mathbf {e} _{1}\times \mathbf {e} _{2}=\mathbf {e} _{3}=-\mathbf {e} _{2}\times \mathbf {e} _{1}}$

${\displaystyle \left(\mathbf {x\times y} \right)_{1}=x_{2}y_{3}-x_{3}y_{2}+x_{4}y_{5}-x_{5}y_{4}+x_{7}y_{6}-x_{6}y_{7}}$

${\displaystyle \mathbf {e} _{i}\mathbf {\times } \mathbf {e} _{j}=\varepsilon _{ijk}\mathbf {e} _{k}}$

## 定义

${\displaystyle \mathbf {x} \cdot (\mathbf {x} \times \mathbf {y} )=(\mathbf {x} \times \mathbf {y} )\cdot \mathbf {y} =0}$
${\displaystyle |\mathbf {x} \times \mathbf {y} |^{2}=|\mathbf {x} |^{2}|\mathbf {y} |^{2}-(\mathbf {x} \cdot \mathbf {y} )^{2}}$

${\displaystyle |\mathbf {x} \times \mathbf {y} |=|\mathbf {x} ||\mathbf {y} |\sin \theta }$

${\displaystyle \left(\mathbf {x} \cdot \mathbf {y} \right)=0}$ ，则${\displaystyle |\mathbf {x} \times \mathbf {y} |=|\mathbf {x} ||\mathbf {y} |}$

（如果假设x × x = 0是另一个公理。[10]

## 定义性质的内涵

1. 反交换律
${\displaystyle \mathbf {x} \times \mathbf {y} =-\mathbf {y} \times \mathbf {x} }$
2. 标量三重积
${\displaystyle \mathbf {x} \cdot (\mathbf {y} \times \mathbf {z} )=\mathbf {y} \cdot (\mathbf {z} \times \mathbf {x} )=\mathbf {z} \cdot (\mathbf {x} \times \mathbf {y} )}$
3. [8]
${\displaystyle (\mathbf {x} \times \mathbf {y} )\times (\mathbf {x} \times \mathbf {z} )=((\mathbf {x} \times \mathbf {y} )\times \mathbf {z} )\times \mathbf {x} +((\mathbf {y} \times \mathbf {z} )\times \mathbf {x} )\times \mathbf {x} +((\mathbf {z} \times \mathbf {x} )\times \mathbf {x} )\times \mathbf {y} }$
${\displaystyle \mathbf {x} \times (\mathbf {x} \times \mathbf {y} )=-|\mathbf {x} |^{2}\mathbf {y} +(\mathbf {x} \cdot \mathbf {y} )\mathbf {x} }$

1. 向量三重积
${\displaystyle \mathbf {x} \times (\mathbf {y} \times \mathbf {z} )=(\mathbf {x} \cdot \mathbf {z} )\mathbf {y} -(\mathbf {x} \cdot \mathbf {y} )\mathbf {z} }$
2. 雅可比恒等式[8]
${\displaystyle \mathbf {x} \times (\mathbf {y} \times \mathbf {z} )+\mathbf {y} \times (\mathbf {z} \times \mathbf {x} )+\mathbf {z} \times (\mathbf {x} \times \mathbf {y} )\neq 0}$

## 坐标表示式

× e1 e2 e3 e4 e5 e6 e7
e1 0 e4 e7 e2 e6 e5 e3
e2 e4 0 e5 e1 e3 e7 e6
e3 e7 e5 0 e6 e2 e4 e1
e4 e2 e1 e6 0 e7 e3 e5
e5 e6 e3 e2 e7 0 e1 e4
e6 e5 e7 e4 e3 e1 0 e2
e7 e3 e6 e1 e5 e4 e2 0

${\displaystyle \mathbf {e} _{1}\times \mathbf {e} _{2}=\mathbf {e} _{4},\quad \mathbf {e} _{2}\times \mathbf {e} _{4}=\mathbf {e} _{1},\quad \mathbf {e} _{4}\times \mathbf {e} _{1}=\mathbf {e} _{2},}$
${\displaystyle \mathbf {e} _{2}\times \mathbf {e} _{3}=\mathbf {e} _{5},\quad \mathbf {e} _{3}\times \mathbf {e} _{5}=\mathbf {e} _{2},\quad \mathbf {e} _{5}\times \mathbf {e} _{2}=\mathbf {e} _{3},}$
${\displaystyle \mathbf {e} _{3}\times \mathbf {e} _{4}=\mathbf {e} _{6},\quad \mathbf {e} _{4}\times \mathbf {e} _{6}=\mathbf {e} _{3},\quad \mathbf {e} _{6}\times \mathbf {e} _{3}=\mathbf {e} _{4},}$
${\displaystyle \mathbf {e} _{4}\times \mathbf {e} _{5}=\mathbf {e} _{7},\quad \mathbf {e} _{5}\times \mathbf {e} _{7}=\mathbf {e} _{4},\quad \mathbf {e} _{7}\times \mathbf {e} _{4}=\mathbf {e} _{5},}$
${\displaystyle \mathbf {e} _{5}\times \mathbf {e} _{6}=\mathbf {e} _{1},\quad \mathbf {e} _{6}\times \mathbf {e} _{1}=\mathbf {e} _{5},\quad \mathbf {e} _{1}\times \mathbf {e} _{5}=\mathbf {e} _{6},}$
${\displaystyle \mathbf {e} _{6}\times \mathbf {e} _{7}=\mathbf {e} _{2},\quad \mathbf {e} _{7}\times \mathbf {e} _{2}=\mathbf {e} _{6},\quad \mathbf {e} _{2}\times \mathbf {e} _{6}=\mathbf {e} _{7},}$
${\displaystyle \mathbf {e} _{7}\times \mathbf {e} _{1}=\mathbf {e} _{3},\quad \mathbf {e} _{1}\times \mathbf {e} _{3}=\mathbf {e} _{7},\quad \mathbf {e} _{3}\times \mathbf {e} _{7}=\mathbf {e} _{1}}$

${\displaystyle \mathbf {e} _{i}\times \mathbf {e} _{i+1}=\mathbf {e} _{i+3}}$

${\displaystyle \mathbf {e} _{i}\times \left(\mathbf {e} _{i}\times \mathbf {e} _{i+1}\right)=-\mathbf {e} _{i+1}=\mathbf {e} _{i}\times \mathbf {e} _{i+3}}$

{\displaystyle {\begin{aligned}\mathbf {x} \times \mathbf {y} =(x_{2}y_{4}-x_{4}y_{2}+x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})\,&\mathbf {e} _{1}\\{}+(x_{3}y_{5}-x_{5}y_{3}+x_{4}y_{1}-x_{1}y_{4}+x_{6}y_{7}-x_{7}y_{6})\,&\mathbf {e} _{2}\\{}+(x_{4}y_{6}-x_{6}y_{4}+x_{5}y_{2}-x_{2}y_{5}+x_{7}y_{1}-x_{1}y_{7})\,&\mathbf {e} _{3}\\{}+(x_{5}y_{7}-x_{7}y_{5}+x_{6}y_{3}-x_{3}y_{6}+x_{1}y_{2}-x_{2}y_{1})\,&\mathbf {e} _{4}\\{}+(x_{6}y_{1}-x_{1}y_{6}+x_{7}y_{4}-x_{4}y_{7}+x_{2}y_{3}-x_{3}y_{2})\,&\mathbf {e} _{5}\\{}+(x_{7}y_{2}-x_{2}y_{7}+x_{1}y_{5}-x_{5}y_{1}+x_{3}y_{4}-x_{4}y_{3})\,&\mathbf {e} _{6}\\{}+(x_{1}y_{3}-x_{3}y_{1}+x_{2}y_{6}-x_{6}y_{2}+x_{4}y_{5}-x_{5}y_{4})\,&\mathbf {e} _{7}\end{aligned}}}

### 其他乘法表

${\displaystyle \mathbf {e} _{6}\times \left(\mathbf {e} _{6}\times \mathbf {e} _{1}\right)=-\mathbf {e} _{1}=\mathbf {e} _{6}\times \mathbf {e} _{5}}$

${\displaystyle \mathbf {e} _{5}\times \mathbf {e} _{6}=\mathbf {e} _{1}}$

### 利用几何代数

${\displaystyle \mathbf {B} =\mathbf {x} \wedge \mathbf {y} ={\frac {1}{2}}(\mathbf {xy} -\mathbf {yx} )}$

${\displaystyle \mathbf {v} =\mathbf {e} _{124}+\mathbf {e} _{235}+\mathbf {e} _{346}+\mathbf {e} _{457}+\mathbf {e} _{561}+\mathbf {e} _{672}+\mathbf {e} _{713}}$

${\displaystyle \mathbf {x} \times \mathbf {y} =-(\mathbf {x} \wedge \mathbf {y} )~\lrcorner ~\mathbf {v} }$

## 与八元数的关系

${\displaystyle \mathbf {x} \times \mathbf {y} =\mathrm {Im} (\mathbf {xy} )={\frac {1}{2}}(\mathbf {xy} -\mathbf {yx} )}$

${\displaystyle (a,\mathbf {x} )(b,\mathbf {y} )=(ab-\mathbf {x} \cdot \mathbf {y} ,a\mathbf {y} +b\mathbf {x} +\mathbf {x} \times \mathbf {y} )}$

${\displaystyle \mathbf {x} \times (\mathbf {y} \times \mathbf {z} )+\mathbf {y} \times (\mathbf {z} \times \mathbf {x} )+\mathbf {z} \times (\mathbf {x} \times \mathbf {y} )=-{\frac {3}{2}}[\mathbf {x} ,\mathbf {y} ,\mathbf {z} ]}$

## 广义化

• 正交：
${\displaystyle \left(\mathbf {a} _{1}\times \ \cdots \ \times \mathbf {a} _{k}\right)\cdot \mathbf {a} _{i}=0}$

对于${\displaystyle i=1,\ \dots \ ,k}$
• 格拉姆行列式：
${\displaystyle |\mathbf {a} _{1}\times \cdots \times \mathbf {a} _{k}|^{2}=\det(\mathbf {a} _{i}\cdot \mathbf {a} _{j})={\begin{vmatrix}\mathbf {a} _{1}\cdot \mathbf {a} _{1}&\mathbf {a} _{1}\cdot \mathbf {a} _{2}&\cdots &\mathbf {a} _{1}\cdot \mathbf {a} _{k}\\\mathbf {a} _{2}\cdot \mathbf {a} _{1}&\mathbf {a} _{2}\cdot \mathbf {a} _{2}&\cdots &\mathbf {a} _{2}\cdot \mathbf {a} _{k}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {a} _{k}\cdot \mathbf {a} _{1}&\mathbf {a} _{k}\cdot \mathbf {a} _{2}&\cdots &\mathbf {a} _{k}\cdot \mathbf {a} _{k}\\\end{vmatrix}}}$

• 在三维和七维中是二元积；
• n ≥ 3维中是n − 1个向量的积，而它是这些向量的外积的霍奇对偶
• 在八维中是三个向量的积。

${\displaystyle \mathbf {a} \times \mathbf {b} \times \mathbf {c} =(\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} )~\lrcorner ~(\mathbf {w} -\mathbf {ve} _{8})}$

1. 叉积必定垂直于所有输入函数。
2. 如果输入函数线性无关，则其叉积必定非零。

## 参考文献

1. WS Massey. Cross products of vectors in higher dimensional Euclidean spaces. The American Mathematical Monthly (Mathematical Association of America). 1983, 90 (10): 697–701. JSTOR 2323537. doi:10.2307/2323537.
2. WS Massey. Cross products of vectors in higher dimensional Euclidean spaces. The American Mathematical Monthly. 1983, 90 (10): 697–701. JSTOR 2323537. doi:10.2307/2323537. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.
3. ^ G Gentili, C Stoppato, DC Struppa and F Vlacci. Recent developments for regular functions of a hypercomplex variable. Irene Sabadini; M Shapiro; F Sommen (编). Hypercomplex analysis Conference on quaternionic and Clifford analysis; proceedings. Birkhäuser. 2009: 168. ISBN 978-3-7643-9892-7.
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6. ^ Mappings are restricted to be bilinear by （Massey 1993） and Robert B Brown & Alfred Gray. Vector cross products. Commentarii Mathematici Helvetici (Birkhäuser Basel). 1967, 42 (1/December): 222–236. S2CID 121135913. doi:10.1007/BF02564418..
7. ^ Francis Begnaud Hildebrand. Methods of applied mathematics Reprint of Prentice-Hall 1965 2nd. Courier Dover Publications. 1992: 24. ISBN 0-486-67002-3.
8. Lounesto, pp. 96–97
9. ^ Kendall, M. G. A Course in the Geometry of N Dimensions. Courier Dover Publications. 2004: 19. ISBN 0-486-43927-5.
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12. ^ Rafał Abłamowicz; Bertfried Fauser. Clifford Algebras and Their Applications in Mathematical Physics: Algebra and physics. Springer. 2000: 26. ISBN 0-8176-4182-3.
13. Jörg Schray; Corinne A. Manogue. Octonionic representations of Clifford algebras and triality. Foundations of Physics. 1996, 26 (1/January): 17–70. Bibcode:1996FoPh...26...17S. S2CID 119604596. . doi:10.1007/BF02058887. Available as ArXive preprint Figure 1 is located here.
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15. John C. Baez. The Octonions (PDF). Bull. Amer. Math. Soc. 2002, 39 (2): 145–205. S2CID 586512. . doi:10.1090/s0273-0979-01-00934-x. （原始内容 (PDF)存档于2010-07-07）.
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18. ^ Lounesto, §7.5: Cross products of k vectors in ${\displaystyle \mathbb {R} ^{n}}$ , p. 98
19. ^