# 外積の成分表示

$\stackrel{\to }{a}=\left({a}_{x},\text{\hspace{0.17em}}{a}_{y},\text{\hspace{0.17em}}{a}_{z}\right)$$\stackrel{\to }{b}=\left({b}_{x},\text{\hspace{0.17em}}{b}_{y},\text{\hspace{0.17em}}{b}_{z}\right)$ のとき，

$\stackrel{\to }{a}×\stackrel{\to }{b}=\left({a}_{y}{b}_{z}-{a}_{z}{b}_{y},\text{\hspace{0.17em}}{a}_{z}{b}_{x}-{a}_{x}{b}_{z},\text{\hspace{0.17em}}{a}_{x}{b}_{y}-{a}_{y}{b}_{x}\right)$

となる．

## ■導出計算

 $\stackrel{\to }{a}×\stackrel{\to }{b}$ $=\left({a}_{x}\stackrel{\to }{{e}_{1}}+{a}_{y}\stackrel{\to }{{e}_{2}}+{a}_{z}\stackrel{\to }{{e}_{3}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}+{b}_{y}\stackrel{\to }{{e}_{2}}+{b}_{z}\stackrel{\to }{{e}_{3}}\right)$ $=\left({a}_{x}\stackrel{\to }{{e}_{1}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}+{b}_{y}\stackrel{\to }{{e}_{2}}+{b}_{z}\stackrel{\to }{{e}_{3}}\right)+\left({a}_{y}\stackrel{\to }{{e}_{2}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}+{b}_{y}\stackrel{\to }{{e}_{2}}+{b}_{z}\stackrel{\to }{{e}_{3}}\right)+\left({a}_{z}\stackrel{\to }{{e}_{3}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}+{b}_{y}\stackrel{\to }{{e}_{2}}+{b}_{z}\stackrel{\to }{{e}_{3}}\right)$ $=\left({a}_{x}\stackrel{\to }{{e}_{1}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}\right)+\left({a}_{x}\stackrel{\to }{{e}_{1}}\right)×\left({b}_{y}\stackrel{\to }{{e}_{2}}\right)+\left({a}_{x}\stackrel{\to }{{e}_{1}}\right)×\left({b}_{z}\stackrel{\to }{{e}_{3}}\right)$ $+\left({a}_{y}\stackrel{\to }{{e}_{2}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}\right)+\left({a}_{y}\stackrel{\to }{{e}_{2}}\right)×\left({b}_{y}\stackrel{\to }{{e}_{2}}\right)+\left({a}_{y}\stackrel{\to }{{e}_{2}}\right)×\left({b}_{z}\stackrel{\to }{{e}_{3}}\right)$ $+\left({a}_{z}\stackrel{\to }{{e}_{3}}\right)×\left({b}_{x}\stackrel{\to }{{e}_{1}}\right)+\left({a}_{z}\stackrel{\to }{{e}_{3}}\right)×\left({b}_{y}\stackrel{\to }{{e}_{2}}\right)+\left({a}_{z}\stackrel{\to }{{e}_{3}}\right)×\left({b}_{z}\stackrel{\to }{{e}_{3}}\right)$ $={a}_{x}{b}_{x}\left(\stackrel{\to }{{e}_{1}}×\stackrel{\to }{{e}_{1}}\right)+{a}_{x}{b}_{y}\left(\stackrel{\to }{{e}_{1}}×\stackrel{\to }{{e}_{2}}\right)+{a}_{x}{b}_{z}\left(\stackrel{\to }{{e}_{1}}×\stackrel{\to }{{e}_{3}}\right)$ $+{a}_{y}{b}_{x}\left(\stackrel{\to }{{e}_{2}}×\stackrel{\to }{{e}_{1}}\right)+{a}_{y}{b}_{y}\left(\stackrel{\to }{{e}_{2}}×\stackrel{\to }{{e}_{2}}\right)+{a}_{y}{b}_{z}\left(\stackrel{\to }{{e}_{2}}×\stackrel{\to }{{e}_{3}}\right)$ $+{a}_{z}{b}_{x}\left(\stackrel{\to }{{e}_{3}}×\stackrel{\to }{{e}_{1}}\right)+{a}_{z}{b}_{y}\left(\stackrel{\to }{{e}_{3}}×\stackrel{\to }{{e}_{2}}\right)+{a}_{z}{b}_{z}\left(\stackrel{\to }{{e}_{3}}×\stackrel{\to }{{e}_{3}}\right)$ $={a}_{x}{b}_{x}\stackrel{\to }{0}+{a}_{x}{b}_{y}\stackrel{\to }{{e}_{3}}+{a}_{x}{b}_{z}\left(-\stackrel{\to }{{e}_{2}}\right)+{a}_{y}{b}_{x}\left(-\stackrel{\to }{{e}_{3}}\right)+{a}_{y}{b}_{y}\stackrel{\to }{0}+{a}_{y}{b}_{z}\stackrel{\to }{{e}_{1}}+{a}_{z}{b}_{x}\stackrel{\to }{{e}_{2}}+{a}_{z}{b}_{y}\left(-\stackrel{\to }{{e}_{1}}\right)+{a}_{z}{b}_{z}\stackrel{\to }{0}$ $=\left({a}_{y}{b}_{z}-{a}_{z}{b}_{y}\right)\stackrel{\to }{{e}_{1}}+\left({a}_{z}{b}_{x}-{a}_{x}{b}_{z}\right)\stackrel{\to }{{e}_{2}}+\left({a}_{x}{b}_{y}-{a}_{y}{b}_{x}\right)\stackrel{\to }{{e}_{3}}$

となる．

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