$3$\displaystyle{\left({ {a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow } }\right)\times {c}\limits^{\rightarrow }}$ $3$\displaystyle{\neq {a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)}$

$3$\displaystyle{\left({ {a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow } }\right)\times {c}\limits^{\rightarrow }}$ $3$\displaystyle{=\left({ {a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){b}\limits^{\rightarrow }- \left({ {b}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){a}\limits^{\rightarrow }}$

$3$\displaystyle{{a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)}$ $3$\displaystyle{=\left({ {a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){b}\limits^{\rightarrow }- \left({ {a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow } }\right){c}\limits^{\rightarrow }}$

よって

$3$\displaystyle{\left({ {a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow } }\right)\times {c}\limits^{\rightarrow }\neq {a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)}$

となり，結合法則は成り立たない．

■説明

$3$\displaystyle{{a}\limits^{\rightarrow }=\left({ \hspace{1}{a}_{1},\hspace{1}{a}_{2},\hspace{1}{a}_{3} }\right)}$ ， $3$\displaystyle{{b}\limits^{\rightarrow }=\left({ \hspace{1}{b}_{1},\hspace{1}{b}_{2},\hspace{1}{b}_{3} }\right)}$ ， $3$\displaystyle{{c}\limits^{\rightarrow }=\left({ \hspace{1}{c}_{1},\hspace{1}{c}_{2},\hspace{1}{c}_{3} }\right)}$

とおく．

● $3$\displaystyle{\left({ {a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow } }\right)\times {c}\limits^{\rightarrow }}$

$3$\displaystyle{\left({ {a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow } }\right)\times {c}\limits^{\rightarrow }=\left{{ \left({ \hspace{1}{a}_{1},\hspace{1}{a}_{2},\hspace{1}{a}_{3} }\right)\times \left({ \hspace{1}{b}_{1},\hspace{1}{b}_{2},\hspace{1}{b}_{3} }\right) }\right}\times \left({ \hspace{1}{c}_{1},\hspace{1}{c}_{2},\hspace{1}{c}_{3} }\right)}$

$3$\displaystyle{=\left({ \hspace{1}{a}_{2}\hspace{1}{b}_{3}- \hspace{1}{a}_{3}\hspace{1}{b}_{2},\hspace{1}{a}_{3}\hspace{1}{b}_{1}- \hspace{1}{a}_{1}\hspace{1}{b}_{3},\hspace{1}{a}_{1}\hspace{1}{b}_{2}- \hspace{1}{a}_{2}\hspace{1}{b}_{1} }\right)\times \left({ \hspace{1}{c}_{1},\hspace{1}{c}_{2},\hspace{1}{c}_{3} }\right)}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{1}- \hspace{1}{a}_{1}\hspace{1}{b}_{3} }\right)\hspace{1}{c}_{3}- \left({ \hspace{1}{a}_{1}\hspace{1}{b}_{2}- \hspace{1}{a}_{2}\hspace{1}{b}_{1} }\right)\hspace{1}{c}_{2} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{b}_{2}- \hspace{1}{a}_{2}\hspace{1}{b}_{1} }\right)\hspace{1}{c}_{1}- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{3}- \hspace{1}{a}_{3}\hspace{1}{b}_{2} }\right)\hspace{1}{c}_{3}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\left({ \hspace{1}{a}_{2}\hspace{1}{b}_{3}- \hspace{1}{a}_{3}\hspace{1}{b}_{2} }\right)\hspace{1}{c}_{2}- \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{1}- \hspace{1}{a}_{1}\hspace{1}{b}_{3} }\right)\hspace{1}{c}_{1} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \hspace{1}{a}_{3}\hspace{1}{b}_{1}\hspace{1}{c}_{3}- \hspace{1}{a}_{1}\hspace{1}{b}_{3}\hspace{1}{c}_{3}- \hspace{1}{a}_{1}\hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{2}\hspace{1}{b}_{1}\hspace{1}{c}_{2} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\hspace{1}{a}_{1}\hspace{1}{b}_{2}\hspace{1}{c}_{1}- \hspace{1}{a}_{2}\hspace{1}{b}_{1}\hspace{1}{c}_{1}- \hspace{1}{a}_{2}\hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{3}\hspace{1}{b}_{2}\hspace{1}{c}_{3}& \vspace{6}\\ \left.{ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}},\hspace{1}{a}_{2}\hspace{1}{b}_{3}\hspace{1}{c}_{2}- \hspace{1}{a}_{3}\hspace{1}{b}_{2}\hspace{1}{c}_{2}- \hspace{1}{a}_{3}\hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{1}\hspace{1}{b}_{3}\hspace{1}{c}_{1} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{1}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{b}_{1}\hspace{1}{c}_{2} }\right)- \left({ \hspace{1}{a}_{1}\hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{2}\hspace{1}{c}_{2} }\right) & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{b}_{2}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{b}_{2}\hspace{1}{c}_{3} }\right)- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right)& \vspace{6}\\ \left.{ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{2}\hspace{1}{b}_{3}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{b}_{3}\hspace{1}{c}_{1} }\right)- \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{1}\hspace{1}{c}_{1} }\right) }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{1}- \left({ \hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{a}_{1} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{2}- \left({ \hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{a}_{2}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{3}- \left({ \hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{a}_{3} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{1}- \left({ \hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{a}_{1}+\hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{1}- \hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{1} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{2}- \left({ \hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{a}_{2}+\hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{2}- \hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{2}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{3}- \left({ \hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{a}_{3}+\hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{3}- \hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{1}- \left({ \hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{a}_{1} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{2}- \left({ \hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{a}_{2}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{3}- \left({ \hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{a}_{3} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{1},\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{2},\left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{3} }\right)& \vspace{6}\\ \text{\hspace{5}}- \left({ \left({ \hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{a}_{1},\left({ \hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{a}_{2},\left({ \hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1}+\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{a}_{3} }\right)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{=\left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\left({ \hspace{1}{b}_{1},\hspace{1}{b}_{2},\hspace{1}{b}_{3} }\right)}$ $3$\displaystyle{- \left({ \hspace{1}{b}_{3}\hspace{1}{c}_{3}+\hspace{1}{b}_{2}\hspace{1}{c}_{2}+\hspace{1}{b}_{1}\hspace{1}{c}_{1} }\right)\left({ \hspace{1}{a}_{1},\hspace{1}{a}_{2},\hspace{1}{a}_{3} }\right)}$

$3$\displaystyle{=\left({ {a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){b}\limits^{\rightarrow }- \left({ {b}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){a}\limits^{\rightarrow }}$

すなわち

● $3$\displaystyle{{a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)}$

$3$\displaystyle{{a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)=\left({ {a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){b}\limits^{\rightarrow }- \left({ {a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow } }\right){c}\limits^{\rightarrow }}$

$3$\displaystyle{{a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)=\left({ \hspace{1}{a}_{1},\hspace{1}{a}_{2},\hspace{1}{a}_{3} }\right)\times \left{{ \left({ \hspace{1}{b}_{1},\hspace{1}{b}_{2},\hspace{1}{b}_{3} }\right)\times \left({ \hspace{1}{c}_{1},\hspace{1}{c}_{2},\hspace{1}{c}_{3} }\right) }\right}}$

$3$\displaystyle{=\left({ \hspace{1}{a}_{1},\hspace{1}{a}_{2},\hspace{1}{a}_{3} }\right)\times \left({ \hspace{1}{b}_{2}\hspace{1}{c}_{3}- \hspace{1}{b}_{3}\hspace{1}{c}_{2},\hspace{1}{b}_{3}\hspace{1}{c}_{1}- \hspace{1}{b}_{1}\hspace{1}{c}_{3},\hspace{1}{b}_{1}\hspace{1}{c}_{2}- \hspace{1}{b}_{2}\hspace{1}{c}_{1} }\right)}$

$3$\displaystyle{\begin{array}{lll}=\left({ \hspace{1}{a}_{2}\left({ \hspace{1}{b}_{1}\hspace{1}{c}_{2}- \hspace{1}{b}_{2}\hspace{1}{c}_{1} }\right)- \hspace{1}{a}_{3}\left({ \hspace{1}{b}_{3}\hspace{1}{c}_{1}- \hspace{1}{b}_{1}\hspace{1}{c}_{3} }\right) & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\hspace{1}{a}_{3}\left({ \hspace{1}{b}_{2}\hspace{1}{c}_{3}- \hspace{1}{b}_{3}\hspace{1}{c}_{2} }\right)- \hspace{1}{a}_{1}\left({ \hspace{1}{b}_{1}\hspace{1}{c}_{2}- \hspace{1}{b}_{2}\hspace{1}{c}_{1} }\right)& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\hspace{1}{a}_{1}\left({ \hspace{1}{b}_{3}\hspace{1}{c}_{1}- \hspace{1}{b}_{1}\hspace{1}{c}_{3} }\right)- \hspace{1}{a}_{2}\left({ \hspace{1}{b}_{2}\hspace{1}{c}_{3}- \hspace{1}{b}_{3}\hspace{1}{c}_{2} }\right) }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \hspace{1}{a}_{2}\hspace{1}{b}_{1}\hspace{1}{c}_{2}- \hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{1}- \hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{b}_{1}\hspace{1}{c}_{3} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\hspace{1}{a}_{3}\hspace{1}{b}_{2}\hspace{1}{c}_{3}- \hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{2}- \hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{b}_{2}\hspace{1}{c}_{1}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\hspace{1}{a}_{1}\hspace{1}{b}_{3}\hspace{1}{c}_{1}- \hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{3}- \hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{b}_{3}\hspace{1}{c}_{2} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{1}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{1}\hspace{1}{c}_{3} }\right)- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{1}+\hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{1} }\right) & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{3}\hspace{1}{b}_{2}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{2}\hspace{1}{c}_{1} }\right)- \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{2}+\hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{2} }\right)& \vspace{6}\\ \left.{ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{b}_{3}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{3}\hspace{1}{c}_{2} }\right)- \left({ \hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{3}+\hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{3} }\right) }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{1}- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3} }\right)\hspace{1}{c}_{1} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{2}- \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1} }\right)\hspace{1}{c}_{2}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{3}- \left({ \hspace{1}{a}_{1}\hspace{1}{b}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{2} }\right)\hspace{1}{c}_{3} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{1}- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3} }\right)\hspace{1}{c}_{1}+\hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{1}- \hspace{1}{a}_{1}\hspace{1}{b}_{1}\hspace{1}{c}_{1} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{2}- \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1} }\right)\hspace{1}{c}_{2}+\hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{2}- \hspace{1}{a}_{2}\hspace{1}{b}_{2}\hspace{1}{c}_{2}& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}}\left.{ ,\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{3}- \left({ \hspace{1}{a}_{1}\hspace{1}{b}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{2} }\right)\hspace{1}{c}_{3}+\hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{3}- \hspace{1}{a}_{3}\hspace{1}{b}_{3}\hspace{1}{c}_{3} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{1}- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1} }\right)\hspace{1}{c}_{1} & \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{2}- \left({ \hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{2} }\right)\hspace{1}{c}_{2}& \vspace{6}\\ \left.{ \text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}},\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{3}- \left({ \hspace{1}{a}_{1}\hspace{1}{b}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3} }\right)\hspace{1}{c}_{3} }\right\)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}=\left({ \left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\hspace{1}{b}_{1},\left({ \hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{c}_{2} }\right)\hspace{1}{b}_{2},\left({ \hspace{1}{a}_{1}\hspace{1}{c}_{1}+\hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3} }\right)\hspace{1}{b}_{3} }\right)& \vspace{6}\\ \text{\hspace{5}}- \left({ \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1} }\right)\hspace{1}{c}_{1},\left({ \hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{2} }\right)\hspace{1}{c}_{2},\left({ \hspace{1}{a}_{1}\hspace{1}{b}_{1}+\hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3} }\right)\hspace{1}{c}_{3} }\right)& \vspace{6}\\ \end{array}}$

$3$\displaystyle{=\left({ \hspace{1}{a}_{2}\hspace{1}{c}_{2}+\hspace{1}{a}_{3}\hspace{1}{c}_{3}+\hspace{1}{a}_{1}\hspace{1}{c}_{1} }\right)\left({ \hspace{1}{b}_{1},\hspace{1}{b}_{2},\hspace{1}{b}_{3} }\right)}$ $3$\displaystyle{- \left({ \hspace{1}{a}_{2}\hspace{1}{b}_{2}+\hspace{1}{a}_{3}\hspace{1}{b}_{3}+\hspace{1}{a}_{1}\hspace{1}{b}_{1} }\right)\left({ \hspace{1}{c}_{1},\hspace{1}{c}_{2},\hspace{1}{c}_{3} }\right)}$

$3$\displaystyle{=\left({ {a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }\right){b}\limits^{\rightarrow }- \left({ {a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow } }\right){c}\limits^{\rightarrow }}$

よって

(1)，(2)より

ホーム>>カテゴリー分類>>ベクトル>>外積>> $3$\displaystyle{\left({ {a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow } }\right)\times {c}\limits^{\rightarrow }}$ $3$\displaystyle{\neq {a}\limits^{\rightarrow }\times \left({ {b}\limits^{\rightarrow }\times {c}\limits^{\rightarrow } }\right)}$

最終更新日 2023年2月20日