四重積 (ベクトル解析)
スカラー三重積の公式およびベクトル三重積の公式を使えば
( a × b ) ⋅ ( c × d ) = ( ( a × b ) × c ) ⋅ d = ( ( a ⋅ c ) b − ( b ⋅ c ) a ) ⋅ d = ( a ⋅ c ) ( b ⋅ d ) − ( a ⋅ d ) ( b ⋅ c ) {\displaystyle {\begin{aligned}({\boldsymbol {a}}\times {\boldsymbol {b}})\cdot ({\boldsymbol {c}}\times {\boldsymbol {d}})&=(({\boldsymbol {a}}\times {\boldsymbol {b}})\times {\boldsymbol {c}})\cdot {\boldsymbol {d}}\\&=(({\boldsymbol {a}}\cdot {\boldsymbol {c}}){\boldsymbol {b}}-({\boldsymbol {b}}\cdot {\boldsymbol {c}}){\boldsymbol {a}})\cdot {\boldsymbol {d}}\\&=({\boldsymbol {a}}\cdot {\boldsymbol {c}})({\boldsymbol {b}}\cdot {\boldsymbol {d}})-({\boldsymbol {a}}\cdot {\boldsymbol {d}})({\boldsymbol {b}}\cdot {\boldsymbol {c}})\end{aligned}}}