外積の計算則 結合法則は不成立

a×b×ca×b×c

a × b × c = a · c b b · c a

a × b × c = a · c b a · b c

よって

a × b × c a × b × c

となり,結合法則は成り立たない

■説明

a = a 1 , a 2 , a 3 b = b 1 , b 2 , b 3 c = c 1 , c 2 , c 3

とおく.

a × b × c

a × b × c = a 1 , a 2 , a 3 × b 1 , b 2 , b 3 × 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 × c 1 , c 2 , c 3

= 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

= 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

= 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

= 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

= 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

= 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

= 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

= 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

= a · c b b · c a

すなわち

a × b × c = a · c b b · c a  ・・・・・・(1)

a × b × c

a × b × c = a · c b a · b c

a × b × c = a 1 , a 2 , a 3 × b 1 , b 2 , b 3 × c 1 , c 2 , c 3

= a 1 , a 2 , a 3 × b 2 c 3 b 3 c 2 , b 3 c 1 b 1 c 3 , b 1 c 2 b 2 c 1

= 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

= 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

= 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

= 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

= 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

= 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

= 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

= 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

= a · c b a · b c

よって

a × b × c = a · c b a · b c  ・・・・・・(2)

(1),(2)より

a × b × c a × b × c

 

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最終更新日 2023年2月20日