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Last edited by hbghlyj at 2024-12-26 13:29:00令$E^n$为$n$维欧氏空间,若存在运算$E^{n} \times E^{n} \stackrel{\times}{\longrightarrow} E^{n}$满足
(i)(双线性)$\forall \lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2} \in \mathbb{R}, u_{1}, u_{2}, v_{1}, v_{2} \in E^{n},$
$\left(\lambda_{1} u_{1}+\lambda_{2} u_{2}\right) \times\left(\mu_{1} v_{1}+\mu_{2} v_{2}\right)=\lambda_{1} \mu_{1}\left(u_{1} \times v_{1}\right)+\lambda_{1} \mu_{2}\left(u_{1} \times v_{2}\right)+\lambda_{2} \mu_{1}\left(u_{2} \times v_{1}\right)+\lambda_{2} \mu_{2}\left(u_{2} \times v_{2}\right)$
(ii)(垂直性)$(u \times v) \cdot u=(u \times v) \cdot v=0, \forall u, v \in E^{n}$
(iii)(平行四边形面积公式)$|u \times v|^{2}=|u|^{2} \cdot|v|^{2}-(u \cdot v)^{2}, \forall u, v \in E^{n}$
则$n=3,7$.
When Does a Cross Product on R^n Exist.pdf |
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