コーシー・ビネの公式
1 = 1 ( m = 0 ) a ⋅ x = a 1 x 1 + a 2 x 2 + a 3 x 3 ( m = 1 ) | a ⋅ x a ⋅ y b ⋅ x b ⋅ y | = | a 2 a 3 b 2 b 3 | | x 2 y 2 x 3 y 3 | + | a 3 a 1 b 3 b 1 | | x 3 y 3 x 1 y 1 | + | a 1 a 2 b 1 b 2 | | x 1 y 1 x 2 y 2 | = ( a × b ) ⋅ ( x × y ) ( m = 2 ) | a ⋅ x a ⋅ y a ⋅ z b ⋅ x b ⋅ y b ⋅ z c ⋅ x c ⋅ y c ⋅ z | = | a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 | | x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 | = { a ⋅ ( b × c ) } { x ⋅ ( y × z ) } ( m = 3 ) | a ⋅ x a ⋅ y a ⋅ z a ⋅ w b ⋅ x b ⋅ y b ⋅ z b ⋅ w c ⋅ x c ⋅ y c ⋅ z c ⋅ w d ⋅ x d ⋅ y d ⋅ z d ⋅ w | = 0 ( m = 4 ) {\displaystyle {\begin{aligned}1&=1&(m=0)\\{\boldsymbol {a}}\cdot {\boldsymbol {x}}&=a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}&(m=1)\\{\begin{vmatrix}{\boldsymbol {a}}\cdot {\boldsymbol {x}}&{\boldsymbol {a}}\cdot {\boldsymbol {y}}\\{\boldsymbol {b}}\cdot {\boldsymbol {x}}&{\boldsymbol {b}}\cdot {\boldsymbol {y}}\end{vmatrix}}&={\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}{\begin{vmatrix}x_{2}&y_{2}\\x_{3}&y_{3}\end{vmatrix}}+{\begin{vmatrix}a_{3}&a_{1}\\b_{3}&b_{1}\end{vmatrix}}{\begin{vmatrix}x_{3}&y_{3}\\x_{1}&y_{1}\end{vmatrix}}+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}{\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}\\&=({\boldsymbol {a}}\times {\boldsymbol {b}})\cdot ({\boldsymbol {x}}\times {\boldsymbol {y}})&(m=2)\\{\begin{vmatrix}{\boldsymbol {a}}\cdot {\boldsymbol {x}}&{\boldsymbol {a}}\cdot {\boldsymbol {y}}&{\boldsymbol {a}}\cdot {\boldsymbol {z}}\\{\boldsymbol {b}}\cdot {\boldsymbol {x}}&{\boldsymbol {b}}\cdot {\boldsymbol {y}}&{\boldsymbol {b}}\cdot {\boldsymbol {z}}\\{\boldsymbol {c}}\cdot {\boldsymbol {x}}&{\boldsymbol {c}}\cdot {\boldsymbol {y}}&{\boldsymbol {c}}\cdot {\boldsymbol {z}}\end{vmatrix}}&={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}{\begin{vmatrix}x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\end{vmatrix}}\\&=\{{\boldsymbol {a}}\cdot ({\boldsymbol {b}}\times {\boldsymbol {c}})\}\{{\boldsymbol {x}}\cdot ({\boldsymbol {y}}\times {\boldsymbol {z}})\}&(m=3)\\{\begin{vmatrix}{\boldsymbol {a}}\cdot {\boldsymbol {x}}&{\boldsymbol {a}}\cdot {\boldsymbol {y}}&{\boldsymbol {a}}\cdot {\boldsymbol {z}}&{\boldsymbol {a}}\cdot {\boldsymbol {w}}\\{\boldsymbol {b}}\cdot {\boldsymbol {x}}&{\boldsymbol {b}}\cdot {\boldsymbol {y}}&{\boldsymbol {b}}\cdot {\boldsymbol {z}}&{\boldsymbol {b}}\cdot {\boldsymbol {w}}\\{\boldsymbol {c}}\cdot {\boldsymbol {x}}&{\boldsymbol {c}}\cdot {\boldsymbol {y}}&{\boldsymbol {c}}\cdot {\boldsymbol {z}}&{\boldsymbol {c}}\cdot {\boldsymbol {w}}\\{\boldsymbol {d}}\cdot {\boldsymbol {x}}&{\boldsymbol {d}}\cdot {\boldsymbol {y}}&{\boldsymbol {d}}\cdot {\boldsymbol {z}}&{\boldsymbol {d}}\cdot {\boldsymbol {w}}\end{vmatrix}}&=0&(m=4)\end{aligned}}}