平行六面体
V = 1 8 | ∑ i = 1 8 u i ⊗ u i | = 1 8 | ∑ i = 1 8 u i u i ⊤ | = 1 8 | ∑ i = 1 8 ( u x u y u z ) i ( u x u y u z ) i | {\textstyle V={\sqrt {{\frac {1}{8}}{\begin{vmatrix}\sum _{i=1}^{8}\mathbf {u_{i}} \otimes \mathbf {u_{i}} \end{vmatrix}}}}={\sqrt {{\frac {1}{8}}{\begin{vmatrix}\sum _{i=1}^{8}\mathbf {u_{i}} \mathbf {u_{i}} ^{\top }\end{vmatrix}}}}={\sqrt {{\frac {1}{8}}{\begin{vmatrix}\sum _{i=1}^{8}{\begin{pmatrix}u_{x}\\u_{y}\\u_{z}\end{pmatrix}}_{i}{\begin{pmatrix}u_{x}&u_{y}&u_{z}\end{pmatrix}}_{i}\end{vmatrix}}}}} = 1 8 | ∑ i = 1 8 ( u x u x u x u y u x u z u y u x u y u y u y u z u z u x u z u y u z u z ) i | {\textstyle ={\sqrt {{\frac {1}{8}}{\begin{vmatrix}\sum _{i=1}^{8}{\begin{pmatrix}u_{x}u_{x}&u_{x}u_{y}&u_{x}u_{z}\\u_{y}u_{x}&u_{y}u_{y}&u_{y}u_{z}\\u_{z}u_{x}&u_{z}u_{y}&u_{z}u_{z}\end{pmatrix}}_{i}\end{vmatrix}}}}} = 1 8 | ∑ i = 1 8 u i x u i x ∑ i = 1 8 u i x u i y ∑ i = 1 8 u i x u i z ∑ i = 1 8 u i y u i x ∑ i = 1 8 u i y u i y ∑ i = 1 8 u i y u i z ∑ i = 1 8 u i z u i x ∑ i = 1 8 u i z u i y ∑ i = 1 8 u i z u i z | {\textstyle ={\sqrt {{\frac {1}{8}}{\begin{vmatrix}\sum _{i=1}^{8}u_{ix}u_{ix}&\sum _{i=1}^{8}u_{ix}u_{iy}&\sum _{i=1}^{8}u_{ix}u_{iz}\\\sum _{i=1}^{8}u_{iy}u_{ix}&\sum _{i=1}^{8}u_{iy}u_{iy}&\sum _{i=1}^{8}u_{iy}u_{iz}\\\sum _{i=1}^{8}u_{iz}u_{ix}&\sum _{i=1}^{8}u_{iz}u_{iy}&\sum _{i=1}^{8}u_{iz}u_{iz}\end{vmatrix}}}}}