双対基底
e 1 = ( e 2 × e 3 V ) ⊤ , e 2 = ( e 3 × e 1 V ) ⊤ , e 3 = ( e 1 × e 2 V ) ⊤ {\displaystyle {\boldsymbol {e}}^{1}=\left({\frac {{\boldsymbol {e}}_{2}\times {\boldsymbol {e}}_{3}}{V}}\right)^{\top },\mathbf {e} ^{2}=\left({\frac {{\boldsymbol {e}}_{3}\times {\boldsymbol {e}}_{1}}{V}}\right)^{\top },{\boldsymbol {e}}^{3}=\left({\frac {{\boldsymbol {e}}_{1}\times {\boldsymbol {e}}_{2}}{V}}\right)^{\top }}
ここで は転置を表し、
V = ( e 1 ; e 2 ; e 3 ) = e 1 ⋅ ( e 2 × e 3 ) = e 2 ⋅ ( e 3 × e 1 ) = e 3 ⋅ ( e 1 × e 2 ) {\displaystyle V=({\boldsymbol {e}}_{1};{\boldsymbol {e}}_{2};{\boldsymbol {e}}_{3})={\boldsymbol {e}}_{1}\cdot ({\boldsymbol {e}}_{2}\times {\boldsymbol {e}}_{3})={\boldsymbol {e}}_{2}\cdot ({\boldsymbol {e}}_{3}\times {\boldsymbol {e}}_{1})={\boldsymbol {e}}_{3}\cdot ({\boldsymbol {e}}_{1}\times {\boldsymbol {e}}_{2})}
は基底ベクトル e1, e2, e3 によって構成される平行六面体の体積である。