二項積
二項積を基本ベクトルで展開してベクトルの併置を点乗積に置き換えれば、蹟 (spur) あるいは展開因子 (expansion factor) が得られる。
| A | = A 11 i ⋅ i + A 12 i ⋅ j + A 13 i ⋅ k + A 21 j ⋅ i + A 22 j ⋅ j + A 23 j ⋅ k + A 31 k ⋅ i + A 32 k ⋅ j + A 33 k ⋅ k = A 11 + A 22 + A 33 . {\displaystyle {\begin{aligned}|\mathbf {A} |&=\quad A_{11}\,\mathbf {i\cdot i} +A_{12}\,\mathbf {i\cdot j} +A_{13}\,\mathbf {i\cdot k} \\&\quad +A_{21}\,\mathbf {j\cdot i} +A_{22}\,\mathbf {j\cdot j} +A_{23}\,\mathbf {j\cdot k} \\&\quad +A_{31}\mathbf {k\cdot i} +A_{32}\mathbf {k\cdot j} +A_{33}\mathbf {k\cdot k} \\&=A_{11}+A_{22}+A_{33}.\end{aligned}}}