$(\vec{a} \times \vec{b}) \times \vec{c}$ $\neq \vec{a} \times (\vec{b} \times \vec{c})$

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

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

よって

$(\vec{a} \times \vec{b}) \times \vec{c} \neq \vec{a} \times (\vec{b} \times \vec{c})$

となり，結合法則は成り立たない．

■説明

$\vec{a} = (a_{1}, a_{2}, a_{3})$ ， $\vec{b} = (b_{1}, b_{2}, b_{3})$ ， $\vec{c} = (c_{1}, c_{2}, c_{3})$

とおく．

● $(\vec{a} \times \vec{b}) \times \vec{c}$

$(\vec{a} \times \vec{b}) \times \vec{c} = \{(a_{1}, a_{2}, a_{3}) \times (b_{1}, b_{2}, b_{3})\} \times (c_{1}, c_{2}, c_{3})$

$= (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1}) \times (c_{1}, c_{2}, c_{3})$

$\begin{array}{l} = ((a_{3} b_{1} - a_{1} b_{3}) c_{3} - (a_{1} b_{2} - a_{2} b_{1}) c_{2} \\ , (a_{1} b_{2} - a_{2} b_{1}) c_{1} - (a_{2} b_{3} - a_{3} b_{2}) c_{3} \\ , (a_{2} b_{3} - a_{3} b_{2}) c_{2} - (a_{3} b_{1} - a_{1} b_{3}) c_{1}) \end{array}$

$\begin{array}{l} = (a_{3} b_{1} c_{3} - a_{1} b_{3} c_{3} - a_{1} b_{2} c_{2} + a_{2} b_{1} c_{2} \\ , a_{1} b_{2} c_{1} - a_{2} b_{1} c_{1} - a_{2} b_{3} c_{3} + a_{3} b_{2} c_{3} \\ , a_{2} b_{3} c_{2} - a_{3} b_{2} c_{2} - a_{3} b_{1} c_{1} + a_{1} b_{3} c_{1}) \end{array}$

$\begin{array}{l} = ((a_{3} b_{1} c_{3} + a_{2} b_{1} c_{2}) - (a_{1} b_{3} c_{3} + a_{1} b_{2} c_{2}) \\ , (a_{1} b_{2} c_{1} + a_{3} b_{2} c_{3}) - (a_{2} b_{1} c_{1} + a_{2} b_{3} c_{3}) \\ , (a_{2} b_{3} c_{2} + a_{1} b_{3} c_{1}) - (a_{3} b_{2} c_{2} + a_{3} b_{1} c_{1})) \end{array}$

$\begin{array}{l} = ((a_{3} c_{3} + a_{2} c_{2}) b_{1} - (b_{3} c_{3} + b_{2} c_{2}) a_{1} \\ , (a_{1} c_{1} + a_{3} c_{3}) b_{2} - (b_{1} c_{1} + b_{3} c_{3}) a_{2} \\ , (a_{2} c_{2} + a_{1} c_{1}) b_{3} - (b_{2} c_{2} + b_{1} c_{1}) a_{3}) \end{array}$

$\begin{array}{l} = ((a_{3} c_{3} + a_{2} c_{2}) b_{1} - (b_{3} c_{3} + b_{2} c_{2}) a_{1} + a_{1} b_{1} c_{1} - a_{1} b_{1} c_{1} \\ , (a_{1} c_{1} + a_{3} c_{3}) b_{2} - (b_{1} c_{1} + b_{3} c_{3}) a_{2} + a_{2} b_{2} c_{2} - a_{2} b_{2} c_{2} \\ , (a_{2} c_{2} + a_{1} c_{1}) b_{3} - (b_{2} c_{2} + b_{1} c_{1}) a_{3} + a_{3} b_{3} c_{3} - a_{3} b_{3} c_{3}) \end{array}$

$\begin{array}{l} = ((a_{3} c_{3} + a_{2} c_{2} + a_{1} c_{1}) b_{1} - (b_{3} c_{3} + b_{2} c_{2} + b_{1} c_{1}) a_{1} \\ , (a_{1} c_{1} + a_{3} c_{3} + a_{2} c_{2}) b_{2} - (b_{1} c_{1} + b_{3} c_{3} + b_{2} c_{2}) a_{2} \\ , (a_{2} c_{2} + a_{1} c_{1} + a_{3} c_{3}) b_{3} - (b_{2} c_{2} + b_{1} c_{1} + b_{3} c_{3}) a_{3}) \end{array}$

$\begin{array}{l} = ((a_{3} c_{3} + a_{2} c_{2} + a_{1} c_{1}) b_{1}, (a_{1} c_{1} + a_{3} c_{3} + a_{2} c_{2}) b_{2}, (a_{2} c_{2} + a_{1} c_{1} + a_{3} c_{3}) b_{3}) \\ - ((b_{3} c_{3} + b_{2} c_{2} + b_{1} c_{1}) a_{1}, (b_{1} c_{1} + b_{3} c_{3} + b_{2} c_{2}) a_{2}, (b_{2} c_{2} + b_{1} c_{1} + b_{3} c_{3}) a_{3}) \end{array}$

$= (a_{3} c_{3} + a_{2} c_{2} + a_{1} c_{1}) (b_{1}, b_{2}, b_{3})$ $- (b_{3} c_{3} + b_{2} c_{2} + b_{1} c_{1}) (a_{1}, a_{2}, a_{3})$

$= (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a}$

すなわち

$(\vec{a} \times \vec{b}) \times \vec{c}$ $= (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a}$ 　･･････(1)

● $\vec{a} \times (\vec{b} \times \vec{c})$

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

$\vec{a} \times (\vec{b} \times \vec{c}) = (a_{1}, a_{2}, a_{3}) \times \{(b_{1}, b_{2}, b_{3}) \times (c_{1}, c_{2}, c_{3})\}$

$= (a_{1}, a_{2}, a_{3}) \times (b_{2} c_{3} - b_{3} c_{2}, b_{3} c_{1} - b_{1} c_{3}, b_{1} c_{2} - b_{2} c_{1})$

$\begin{array}{l} = (a_{2} (b_{1} c_{2} - b_{2} c_{1}) - a_{3} (b_{3} c_{1} - b_{1} c_{3}) \\ , a_{3} (b_{2} c_{3} - b_{3} c_{2}) - a_{1} (b_{1} c_{2} - b_{2} c_{1}) \\ , a_{1} (b_{3} c_{1} - b_{1} c_{3}) - a_{2} (b_{2} c_{3} - b_{3} c_{2})) \end{array}$

$\begin{array}{l} = (a_{2} b_{1} c_{2} - a_{2} b_{2} c_{1} - a_{3} b_{3} c_{1} + a_{3} b_{1} c_{3} \\ , a_{3} b_{2} c_{3} - a_{3} b_{3} c_{2} - a_{1} b_{1} c_{2} + a_{1} b_{2} c_{1} \\ , a_{1} b_{3} c_{1} - a_{1} b_{1} c_{3} - a_{2} b_{2} c_{3} + a_{2} b_{3} c_{2}) \end{array}$

$\begin{array}{l} = ((a_{2} b_{1} c_{2} + a_{3} b_{1} c_{3}) - (a_{2} b_{2} c_{1} + a_{3} b_{3} c_{1}) \\ , (a_{3} b_{2} c_{3} + a_{1} b_{2} c_{1}) - (a_{3} b_{3} c_{2} + a_{1} b_{1} c_{2}) \\ , (a_{1} b_{3} c_{1} + a_{2} b_{3} c_{2}) - (a_{1} b_{1} c_{3} + a_{2} b_{2} c_{3})) \end{array}$

$\begin{array}{l} = ((a_{2} c_{2} + a_{3} c_{3}) b_{1} - (a_{2} b_{2} + a_{3} b_{3}) c_{1} \\ , (a_{3} c_{3} + a_{1} c_{1}) b_{2} - (a_{3} b_{3} + a_{1} b_{1}) c_{2} \\ , (a_{1} c_{1} + a_{2} c_{2}) b_{3} - (a_{1} b_{1} + a_{2} b_{2}) c_{3}) \end{array}$

$\begin{array}{l} = ((a_{2} c_{2} + a_{3} c_{3}) b_{1} - (a_{2} b_{2} + a_{3} b_{3}) c_{1} + a_{1} b_{1} c_{1} - a_{1} b_{1} c_{1} \\ , (a_{3} c_{3} + a_{1} c_{1}) b_{2} - (a_{3} b_{3} + a_{1} b_{1}) c_{2} + a_{2} b_{2} c_{2} - a_{2} b_{2} c_{2} \\ , (a_{1} c_{1} + a_{2} c_{2}) b_{3} - (a_{1} b_{1} + a_{2} b_{2}) c_{3} + a_{3} b_{3} c_{3} - a_{3} b_{3} c_{3}) \end{array}$

$\begin{array}{l} = ((a_{2} c_{2} + a_{3} c_{3} + a_{1} c_{1}) b_{1} - (a_{2} b_{2} + a_{3} b_{3} + a_{1} b_{1}) c_{1} \\ , (a_{3} c_{3} + a_{1} c_{1} + a_{2} c_{2}) b_{2} - (a_{3} b_{3} + a_{1} b_{1} + a_{2} b_{2}) c_{2} \\ , (a_{1} c_{1} + a_{2} c_{2} + a_{3} c_{3}) b_{3} - (a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}) c_{3}) \end{array}$

$\begin{array}{l} = ((a_{2} c_{2} + a_{3} c_{3} + a_{1} c_{1}) b_{1}, (a_{3} c_{3} + a_{1} c_{1} + a_{2} c_{2}) b_{2}, (a_{1} c_{1} + a_{2} c_{2} + a_{3} c_{3}) b_{3}) \\ - ((a_{2} b_{2} + a_{3} b_{3} + a_{1} b_{1}) c_{1}, (a_{3} b_{3} + a_{1} b_{1} + a_{2} b_{2}) c_{2}, (a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}) c_{3}) \end{array}$

$= (a_{2} c_{2} + a_{3} c_{3} + a_{1} c_{1}) (b_{1}, b_{2}, b_{3})$ $- (a_{2} b_{2} + a_{3} b_{3} + a_{1} b_{1}) (c_{1}, c_{2}, c_{3})$

$= (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}$

よって

$\vec{a} \times (\vec{b} \times \vec{c})$ $= (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}$ 　･･････(2)

(1)，(2)より

$(\vec{a} \times \vec{b}) \times \vec{c} \neq \vec{a} \times (\vec{b} \times \vec{c})$

ホーム>>カテゴリー分類>>ベクトル>>外積>> $(\vec{a} \times \vec{b}) \times \vec{c}$ $\neq \vec{a} \times (\vec{b} \times \vec{c})$

最終更新日 2023年2月20日

■説明

●(→a×→b)×→c<math><mstyle displaystyle="true"> <mfenced> <mrow> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mo>×</mo><mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> </mrow> </mfenced><mo>×</mo><mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> </mstyle></math>

●→a×(→b×→c)<math><mstyle displaystyle="true"> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mo>×</mo><mfenced> <mrow> <mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> <mo>×</mo><mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> </mrow> </mfenced> </mstyle></math>

(1)，(2)より

● $(\vec{a} \times \vec{b}) \times \vec{c}$

● $\vec{a} \times (\vec{b} \times \vec{c})$