ビネ・コーシーの恒等式
{ ( a × b ) × c } ⋅ d = { ( a ⋅ c ) b − ( b ⋅ c ) a } ⋅ d ∴ ( a × b ) × c = ( a ⋅ c ) b − ( b ⋅ c ) a {\displaystyle {\begin{aligned}\{({\boldsymbol {a}}\times {\boldsymbol {b}})\times {\boldsymbol {c}}\}\cdot {\boldsymbol {d}}&=\left\{({\boldsymbol {a}}\cdot {\boldsymbol {c}}){\boldsymbol {b}}-({\boldsymbol {b}}\cdot {\boldsymbol {c}}){\boldsymbol {a}}\right\}\cdot {\boldsymbol {d}}\\\therefore ~({\boldsymbol {a}}\times {\boldsymbol {b}})\times {\boldsymbol {c}}&=({\boldsymbol {a}}\cdot {\boldsymbol {c}}){\boldsymbol {b}}-({\boldsymbol {b}}\cdot {\boldsymbol {c}}){\boldsymbol {a}}\end{aligned}}}
とベクトル三重積の公式が得られる。
また、c = a, d = b とおくと、
‖ a × b ‖ 2 = ‖ a ‖ 2 ‖ b ‖ 2 − ( a ⋅ b ) 2 ( a , b ∈ R 3 ) {\displaystyle \|{\boldsymbol {a}}\times {\boldsymbol {b}}\|^{2}=\|{\boldsymbol {a}}\|^{2}\|{\boldsymbol {b}}\|^{2}-({\boldsymbol {a}}\cdot {\boldsymbol {b}})^{2}\quad ({\boldsymbol {a}},{\boldsymbol {b}}\in \mathbb {R} ^{3})}
と、ベクトル解析におけるラグランジュの恒等式が得られる。