外積の成分表示

$3$\displaystyle{{a}\limits^{\rightarrow }= $ \hspace{1}{a}_{x},\text{\hspace{1}}\hspace{1}{a}_{y},\text{\hspace{1}}\hspace{1}{a}_{z} $ }$ ， $3$\displaystyle{{b}\limits^{\rightarrow }= $ \hspace{1}{b}_{x},\text{\hspace{1}}\hspace{1}{b}_{y},\text{\hspace{1}}\hspace{1}{b}_{z} $ }$ のとき

$3$\displaystyle{{a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow }}$ $3$\displaystyle{= $ \hspace{1}{a}_{y}\hspace{1}{b}_{z}- \hspace{1}{a}_{z}\hspace{1}{b}_{y},\text{\hspace{1}}\hspace{1}{a}_{z}\hspace{1}{b}_{x}- \hspace{1}{a}_{x}\hspace{1}{b}_{z},\text{\hspace{1}}\hspace{1}{a}_{x}\hspace{1}{b}_{y}- \hspace{1}{a}_{y}\hspace{1}{b}_{x} $ }$

となる．

各成分の幾何学的意味はここ参照

■導出計算

成分表示を基本ベクトル表示に直し，外積の定義に従って計算すると

$3$\displaystyle{{a}\limits^{\rightarrow }\times {b}\limits^{\rightarrow }}$ $3$\displaystyle{= $ \hspace{1}{a}_{x}{ \hspace{1}{e}_{1} }\limits^{\rightarrow }+\hspace{1}{a}_{y}{ \hspace{1}{e}_{2} }\limits^{\rightarrow }+\hspace{1}{a}_{z}{ \hspace{1}{e}_{3} }\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{ \hspace{1}{e}_{1} }\limits^{\rightarrow }+\hspace{1}{b}_{y}{ \hspace{1}{e}_{2} }\limits^{\rightarrow }+\hspace{1}{b}_{z}{ \hspace{1}{e}_{3} }\limits^{\rightarrow } $ }$

分配法則が成り立つことより

$3$\displaystyle{ = $ \hspace{1}{a}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow }+ \hspace{1}{b}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow }+ \hspace{1}{b}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + $ \hspace{1}{a}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow }+ \hspace{1}{b}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow }+ \hspace{1}{b}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + $ \hspace{1}{a}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow }+ \hspace{1}{b}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow }+ \hspace{1}{b}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$

$3$\displaystyle{ = $ \hspace{1}{a}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ + $ \hspace{1}{a}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + $ \hspace{1}{a}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ + $ \hspace{1}{a}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + $ \hspace{1}{a}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ + $ \hspace{1}{a}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + $ \hspace{1}{a}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{x}{\hspace{1}{e}_{1}}\limits^{\rightarrow } $ + $ \hspace{1}{a}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{y}{\hspace{1}{e}_{2}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + $ \hspace{1}{a}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ \times $ \hspace{1}{b}_{z}{\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$

外積のスカラー倍の性質より

$3$\displaystyle{ =\hspace{1}{a}_{x}\hspace{1}{b}_{x} $ {\hspace{1}{e}_{1}}\limits^{\rightarrow }\times {\hspace{1}{e}_{1}}\limits^{\rightarrow } $ + \hspace{1}{a}_{x}\hspace{1}{b}_{y} $ {\hspace{1}{e}_{1}}\limits^{\rightarrow }\times {\hspace{1}{e}_{2}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + \hspace{1}{a}_{x}\hspace{1}{b}_{z} $ {\hspace{1}{e}_{1}}\limits^{\rightarrow }\times {\hspace{1}{e}_{3}}\limits^{\rightarrow } $ v+ \hspace{1}{a}_{y}\hspace{1}{b}_{x} $ {\hspace{1}{e}_{2}}\limits^{\rightarrow }\times {\hspace{1}{e}_{1}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + \hspace{1}{a}_{y}\hspace{1}{b}_{y} $ {\hspace{1}{e}_{2}}\limits^{\rightarrow }\times {\hspace{1}{e}_{2}}\limits^{\rightarrow } $ + \hspace{1}{a}_{y}\hspace{1}{b}_{z} $ {\hspace{1}{e}_{2}}\limits^{\rightarrow }\times {\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + \hspace{1}{a}_{z}\hspace{1}{b}_{x} $ {\hspace{1}{e}_{3}}\limits^{\rightarrow }\times {\hspace{1}{e}_{1}}\limits^{\rightarrow } $ + \hspace{1}{a}_{z}\hspace{1}{b}_{y} $ {\hspace{1}{e}_{3}}\limits^{\rightarrow }\times {\hspace{1}{e}_{2}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + \hspace{1}{a}_{z}\hspace{1}{b}_{z} $ {\hspace{1}{e}_{3}}\limits^{\rightarrow }\times {\hspace{1}{e}_{3}}\limits^{\rightarrow } $ }$

基本ベクトルの外積の結果より

$3$\displaystyle{ =\hspace{1}{a}_{x}\hspace{1}{b}_{x}{0}\limits^{\rightarrow }+ \hspace{1}{a}_{x}\hspace{1}{b}_{y}{\hspace{1}{e}_{3}}\limits^{\rightarrow }+ \hspace{1}{a}_{x}\hspace{1}{b}_{z} $ - {\hspace{1}{e}_{2}}\limits^{\rightarrow } $ }$ $3$\displaystyle{ + \hspace{1}{a}_{y}\hspace{1}{b}_{x} $ - {\hspace{1}{e}_{3}}\limits^{\rightarrow } $ + \hspace{1}{a}_{y}\hspace{1}{b}_{y}{0}\limits^{\rightarrow }+ \hspace{1}{a}_{y}\hspace{1}{b}_{z}{\hspace{1}{e}_{1}}\limits^{\rightarrow } }$ $3$\displaystyle{ + \hspace{1}{a}_{z}\hspace{1}{b}_{x}{\hspace{1}{e}_{2}}\limits^{\rightarrow }+ \hspace{1}{a}_{z}\hspace{1}{b}_{y} $ - {\hspace{1}{e}_{1}}\limits^{\rightarrow } $ + \hspace{1}{a}_{z}\hspace{1}{b}_{z}{0}\limits^{\rightarrow } }$

$3$\displaystyle{ = $ \hspace{1}{a}_{y}\hspace{1}{b}_{z}- \hspace{1}{a}_{z}\hspace{1}{b}_{y} $ {\hspace{1}{e}_{1}}\limits^{\rightarrow }+ $ \hspace{1}{a}_{z}\hspace{1}{b}_{x}- \hspace{1}{a}_{x}\hspace{1}{b}_{z} $ {\hspace{1}{e}_{2}}\limits^{\rightarrow } }$ $3$\displaystyle{ + $ \hspace{1}{a}_{x}\hspace{1}{b}_{y}- \hspace{1}{a}_{y}\hspace{1}{b}_{x} $ {\hspace{1}{e}_{3}}\limits^{\rightarrow } }$

となる．

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最終更新日 2026年3月26日