外積の成分表示

$\vec{a} = (a_{x}, a_{y}, a_{z})$ ， $\vec{b} = (b_{x}, b_{y}, b_{z})$ のとき

$\vec{a} \times \vec{b}$ $= (a_{y} b_{z} - a_{z} b_{y}, a_{z} b_{x} - a_{x} b_{z}, a_{x} b_{y} - a_{y} b_{x})$

となる．

各成分の幾何学的意味はここ参照

■導出計算

$\vec{a} \times \vec{b}$ $= (a_{x} \vec{e_{1}} + a_{y} \vec{e_{2}} + a_{z} \vec{e_{3}}) \times (b_{x} \vec{e_{1}} + b_{y} \vec{e_{2}} + b_{z} \vec{e_{3}})$

分配法則が成り立つことより

$= (a_{x} \vec{e_{1}}) \times (b_{x} \vec{e_{1}} + b_{y} \vec{e_{2}} + b_{z} \vec{e_{3}})$ $+ (a_{y} \vec{e_{2}}) \times (b_{x} \vec{e_{1}} + b_{y} \vec{e_{2}} + b_{z} \vec{e_{3}})$ $+ (a_{z} \vec{e_{3}}) \times (b_{x} \vec{e_{1}} + b_{y} \vec{e_{2}} + b_{z} \vec{e_{3}})$

$= (a_{x} \vec{e_{1}}) \times (b_{x} \vec{e_{1}}) + (a_{x} \vec{e_{1}}) \times (b_{y} \vec{e_{2}})$ $+ (a_{x} \vec{e_{1}}) \times (b_{z} \vec{e_{3}}) + (a_{y} \vec{e_{2}}) \times (b_{x} \vec{e_{1}})$ $+ (a_{y} \vec{e_{2}}) \times (b_{y} \vec{e_{2}}) + (a_{y} \vec{e_{2}}) \times (b_{z} \vec{e_{3}})$ $+ (a_{z} \vec{e_{3}}) \times (b_{x} \vec{e_{1}}) + (a_{z} \vec{e_{3}}) \times (b_{y} \vec{e_{2}})$ $+ (a_{z} \vec{e_{3}}) \times (b_{z} \vec{e_{3}})$

外積のスカラー倍の性質より

$= a_{x} b_{x} (\vec{e_{1}} \times \vec{e_{1}}) + a_{x} b_{y} (\vec{e_{1}} \times \vec{e_{2}})$ $+ a_{x} b_{z} (\vec{e_{1}} \times \vec{e_{3}}) v + a_{y} b_{x} (\vec{e_{2}} \times \vec{e_{1}})$ $+ a_{y} b_{y} (\vec{e_{2}} \times \vec{e_{2}}) + a_{y} b_{z} (\vec{e_{2}} \times \vec{e_{3}})$ $+ a_{z} b_{x} (\vec{e_{3}} \times \vec{e_{1}}) + a_{z} b_{y} (\vec{e_{3}} \times \vec{e_{2}})$ $+ a_{z} b_{z} (\vec{e_{3}} \times \vec{e_{3}})$

基本ベクトルの外積の結果より

$= a_{x} b_{x} \vec{0} + a_{x} b_{y} \vec{e_{3}} + a_{x} b_{z} (- \vec{e_{2}})$ $+ a_{y} b_{x} (- \vec{e_{3}}) + a_{y} b_{y} \vec{0} + a_{y} b_{z} \vec{e_{1}}$ $+ a_{z} b_{x} \vec{e_{2}} + a_{z} b_{y} (- \vec{e_{1}}) + a_{z} b_{z} \vec{0}$

$= (a_{y} b_{z} - a_{z} b_{y}) \vec{e_{1}} + (a_{z} b_{x} - a_{x} b_{z}) \vec{e_{2}}$ $+ (a_{x} b_{y} - a_{y} b_{x}) \vec{e_{3}}$

となる．

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最終更新日 2026年3月26日