# 三重积

## 标量三重积

### 定義

${\displaystyle \mathbf {a} }$${\displaystyle \mathbf {b} }$${\displaystyle \mathbf {c} }$為三個向量，則标量三重积的定義為

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

### 特性

${\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 \mathbf {c} =c_{1}\mathbf {i} +c_{2}\mathbf {j} +c_{3}\mathbf {k} }$，則有

${\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{vmatrix}}}$

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

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

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

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

${\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )=\mathbf {b} \cdot (\mathbf {a} \times \mathbf {a} )=\mathbf {b} \cdot \mathbf {0} =0}$

### 其他記號

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

### 幾何意義

${\displaystyle V=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|=\left|{\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}\right|}$

${\displaystyle \mathbf {b} }$${\displaystyle \mathbf {c} }$ 来表示底面的边，则根据叉积的定义，底面的面积 ${\displaystyle A}$

${\displaystyle A=|\mathbf {b} ||\mathbf {c} |\sin \theta =|\mathbf {b} \times \mathbf {c} |}$

${\displaystyle h=|\mathbf {a} |\cos \alpha }$

${\displaystyle \cos \alpha =\pm \cos \phi =|\cos \phi |}$

${\displaystyle h=|\mathbf {a} ||\cos \phi |}$

${\displaystyle V=Ah=|\mathbf {a} ||\mathbf {b} \times \mathbf {c} ||\cos \phi |}$

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

## 向量三重积

### 定義

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

${\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\neq (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} }$

### 特性

${\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )}$
{\displaystyle {\begin{aligned}(\mathbf {a} \times \mathbf {b} )\times \mathbf {c} &=-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )\\&=-\mathbf {a} (\mathbf {c} \cdot \mathbf {b} )+\mathbf {b} (\mathbf {c} \cdot \mathbf {a} )\end{aligned}}}

• 兩個分項都帶有三個向量 （${\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }$
• 三重積一定是先做叉积的兩向量之線性組合
• 中間的向量所帶的係數一定為正（此處為向量${\displaystyle \mathbf {b} }$

### 證明

{\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\mathbf {w} _{y}-\mathbf {v} _{y}\mathbf {w} _{x})-\mathbf {u} _{z}(\mathbf {v} _{z}\mathbf {w} _{x}-\mathbf {v} _{x}\mathbf {w} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{x}\mathbf {w} _{x}+\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{x}\mathbf {v} _{x}+\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{x}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{x}\end{aligned}}}

{\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{y}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{y}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{y}\\(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{z}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{z}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{z}\end{aligned}}}

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

${\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\;+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )\;+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=0}$雅可比恆等式
${\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\;-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )}$

${\displaystyle \nabla \times (\nabla \times \mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} )-(\nabla \cdot \nabla )\mathbf {f} }$

## 參考文獻

1. ^ David K. Cheng. Field and Wave Electromagnetics. 2014: 第18頁. ISBN 9781292026565.