# 叉积

（重定向自叉積

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

## 定义

${\displaystyle \mathbf {a} \times \mathbf {b} =\|\mathbf {a} \|\|\mathbf {b} \|\sin(\theta )\ \mathbf {n} }$

## 计算

### 坐标表示

{\displaystyle {\begin{aligned}\mathbf {i} \times \mathbf {j} &=\mathbf {k} \\\mathbf {j} \times \mathbf {k} &=\mathbf {i} \\\mathbf {k} \times \mathbf {i} &=\mathbf {j} \end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {j\times i} &=-\mathbf {k} \\\mathbf {k\times j} &=-\mathbf {i} \\\mathbf {i\times k} &=-\mathbf {j} \end{aligned}}}

${\displaystyle \mathbf {i} \times \mathbf {i} =\mathbf {j} \times \mathbf {j} =\mathbf {k} \times \mathbf {k} =\mathbf {0} }$ 零向量）。

{\displaystyle {\begin{aligned}\mathbf {u} &=u_{1}\mathbf {i} +u_{2}\mathbf {j} +u_{3}\mathbf {k} \\\mathbf {v} &=v_{1}\mathbf {i} +v_{2}\mathbf {j} +v_{3}\mathbf {k} \end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {u} \times \mathbf {v} ={}&(u_{1}\mathbf {i} +u_{2}\mathbf {j} +u_{3}\mathbf {k} )\times (v_{1}\mathbf {i} +v_{2}\mathbf {j} +v_{3}\mathbf {k} )\\={}&u_{1}v_{1}(\mathbf {i} \times \mathbf {i} )+u_{1}v_{2}(\mathbf {i} \times \mathbf {j} )+u_{1}v_{3}(\mathbf {i} \times \mathbf {k} )+{}\\&u_{2}v_{1}(\mathbf {j} \times \mathbf {i} )+u_{2}v_{2}(\mathbf {j} \times \mathbf {j} )+u_{2}v_{3}(\mathbf {j} \times \mathbf {k} )+{}\\&u_{3}v_{1}(\mathbf {k} \times \mathbf {i} )+u_{3}v_{2}(\mathbf {k} \times \mathbf {j} )+u_{3}v_{3}(\mathbf {k} \times \mathbf {k} )\\\end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {u} \times \mathbf {v} ={}&-u_{1}v_{1}\mathbf {0} +u_{1}v_{2}\mathbf {k} -u_{1}v_{3}\mathbf {j} \\&-u_{2}v_{1}\mathbf {k} -u_{2}v_{2}\mathbf {0} +u_{2}v_{3}\mathbf {i} \\&+u_{3}v_{1}\mathbf {j} -u_{3}v_{2}\mathbf {i} -u_{3}v_{3}\mathbf {0} \\={}&(u_{2}v_{3}-u_{3}v_{2})\mathbf {i} +(u_{3}v_{1}-u_{1}v_{3})\mathbf {j} +(u_{1}v_{2}-u_{2}v_{1})\mathbf {k} \\\end{aligned}}}

{\displaystyle {\begin{aligned}s_{1}&=u_{2}v_{3}-u_{3}v_{2}\\s_{2}&=u_{3}v_{1}-u_{1}v_{3}\\s_{3}&=u_{1}v_{2}-u_{2}v_{1}\end{aligned}}}

${\displaystyle {\begin{pmatrix}s_{1}\\s_{2}\\s_{3}\end{pmatrix}}={\begin{pmatrix}u_{2}v_{3}-u_{3}v_{2}\\u_{3}v_{1}-u_{1}v_{3}\\u_{1}v_{2}-u_{2}v_{1}\end{pmatrix}}}$

### 矩阵表示

${\displaystyle \mathbf {u\times v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\\end{vmatrix}}}$ [1]

{\displaystyle {\begin{aligned}\mathbf {u\times v} &=(u_{2}v_{3}\mathbf {i} +u_{3}v_{1}\mathbf {j} +u_{1}v_{2}\mathbf {k} )-(u_{3}v_{2}\mathbf {i} +u_{1}v_{3}\mathbf {j} +u_{2}v_{1}\mathbf {k} )\\&=(u_{2}v_{3}-u_{3}v_{2})\mathbf {i} +(u_{3}v_{1}-u_{1}v_{3})\mathbf {j} +(u_{1}v_{2}-u_{2}v_{1})\mathbf {k} \end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {u\times v} &={\begin{vmatrix}u_{2}&u_{3}\\v_{2}&v_{3}\end{vmatrix}}\mathbf {i} -{\begin{vmatrix}u_{1}&u_{3}\\v_{1}&v_{3}\end{vmatrix}}\mathbf {j} +{\begin{vmatrix}u_{1}&u_{2}\\v_{1}&v_{2}\end{vmatrix}}\mathbf {k} \\&=(u_{2}v_{3}-u_{3}v_{2})\mathbf {i} -(u_{1}v_{3}-u_{3}v_{1})\mathbf {j} +(u_{1}v_{2}-u_{2}v_{1})\mathbf {k} \end{aligned}}}

## 性质

### 代数性质

• ${\displaystyle \mathbf {a} \times \mathbf {a} =\mathbf {0} }$
• ${\displaystyle \mathbf {a} \times \mathbf {0} =\mathbf {0} }$
• ${\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }$ 反交换律
• ${\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} }$ （加法的左分配律
• ${\displaystyle (\mathbf {a} +\mathbf {b} )\times \mathbf {c} =\mathbf {a} \times \mathbf {c} +\mathbf {b} \times \mathbf {c} }$ （加法的右分配律
• ${\displaystyle (\lambda \mathbf {a} )\times \mathbf {b} =\lambda (\mathbf {a} \times \mathbf {b} )=\mathbf {a} \times (\lambda \mathbf {b} )}$
• ${\displaystyle \mathbf {a} \times \mathbf {b} +\mathbf {c} \times \mathbf {d} =(\mathbf {a} -\mathbf {c} )\times (\mathbf {b} -\mathbf {d} )+\mathbf {a} \times \mathbf {d} +\mathbf {c} \times \mathbf {b} }$
• ${\displaystyle |\mathbf {a} \times \mathbf {b} |=|\mathbf {b} \times \mathbf {a} |}$
• ${\displaystyle |\mathbf {a} \times \mathbf {b} |^{2}=|\mathbf {a} |^{2}|\mathbf {b} |^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}={\begin{vmatrix}\mathbf {a} \cdot \mathbf {a} &\mathbf {a} \cdot \mathbf {b} \\\mathbf {a} \cdot \mathbf {b} &\mathbf {b} \cdot \mathbf {b} \\\end{vmatrix}}}$ 拉格朗日恆等式

• ${\displaystyle \mathbf {a} \times \mathbf {b} =\mathbf {0} }$  當且僅當 ${\displaystyle \mathbf {a} }$  平行於 ${\displaystyle \mathbf {b} }$

### 几何意义

${\displaystyle \left\|\mathbf {a} \times \mathbf {b} \right\|=\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin \theta .}$

${\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} ).}$

${\displaystyle V=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}$

### 向量微分

• ${\displaystyle {\frac {d}{dt}}(\mathbf {a} \times \mathbf {b} )={\frac {d\mathbf {a} }{dt}}\times \mathbf {b} +\mathbf {a} \times {\frac {d\mathbf {b} }{dt}}}$

## 三維坐標

${\displaystyle \mathbf {i} \times \mathbf {j} =\mathbf {k} }$ ${\displaystyle \mathbf {j} \times \mathbf {k} =\mathbf {i} }$ ${\displaystyle \mathbf {k} \times \mathbf {i} =\mathbf {j} }$

${\displaystyle \mathbf {a} =a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k} }$
${\displaystyle \mathbf {b} =b_{1}\mathbf {i} +b_{2}\mathbf {j} +b_{3}\mathbf {k} }$

{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} &=(a_{2}b_{3}-a_{3}b_{2})\mathbf {i} +(a_{3}b_{1}-a_{1}b_{3})\mathbf {j} +(a_{1}b_{2}-a_{2}b_{1})\mathbf {k} \\&={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\\end{vmatrix}}\end{aligned}}}

## 高维情形

${\displaystyle \mathbf {x} \times (a\mathbf {y} +b\mathbf {z} )=a\mathbf {x} \times \mathbf {y} +b\mathbf {x} \times \mathbf {z} }$
${\displaystyle (a\mathbf {y} +b\mathbf {z} )\times \mathbf {x} =a\mathbf {y} \times \mathbf {x} +b\mathbf {z} \times \mathbf {x} }$
${\displaystyle \mathbf {x} \times \mathbf {y} +\mathbf {y} \times \mathbf {x} =\mathbf {0} }$
• ${\displaystyle \mathbf {x} \times \mathbf {y} }$  同时与 ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {y} }$  垂直：
${\displaystyle \mathbf {x} \cdot (\mathbf {x} \times \mathbf {y} )=\mathbf {y} \cdot (\mathbf {x} \times \mathbf {y} )=\mathbf {0} }$
${\displaystyle |\mathbf {x} \times \mathbf {y} |^{2}=|\mathbf {x} |^{2}|\mathbf {y} |^{2}-(\mathbf {x} \cdot \mathbf {y} )^{2}}$
${\displaystyle \mathbf {x} \times (\mathbf {y} \times \mathbf {z} )\;+\mathbf {y} \times (\mathbf {z} \times \mathbf {x} )\;+\mathbf {z} \times (\mathbf {x} \times \mathbf {y} )\neq \mathbf {0} }$

## 历史

1773年，约瑟夫·拉格朗日引入了点积和叉积的概念来研究三维空间中的四面体。1843年，威廉·哈密顿引入了四元数乘法，同时区分了“向（矢）量”和“标量”的概念。给定两个四元数[0,u]和[0,v]，其中u和v是${\displaystyle R^{3}}$ 空间中的向量，使得其乘积可以写成为${\displaystyle [-\mathbf {u} \cdot \mathbf {v} ,\mathbf {u} \times \mathbf {v} ]}$ 的形式。詹姆斯·克拉克·麦克斯韦在四元数的基础建立了著名的麦克斯韦方程组。四元数因此（同时也因为其他方面的）应用，在很长一段时间内都是物理学教育的必备内容。

• 两个向量的直接乘标量乘或者点乘
• 两个向量的斜乘向量乘叉乘

## 参考文献

1. ^ David K. Cheng. Field and Wave Electromagnetics. 2014: 第21頁. ISBN 9781292026565.
2. ^ Dennis G. Zill; Michael R. Cullen. Equation 7: a × b as sum of determinants. cited work. Jones & Bartlett Learning. 2006: 321. ISBN 0-7637-4591-X.