User:Just granpa/sandbox



\phantom{-}\frac{8}{2}=\phantom{-}4 $$


 * and


 * $$\frac{-8}{-2}=\phantom{-}4$$

If dividend and divisor have different signs, the result is always negative.



\frac{\phantom{-}8}{-2}=-4 $$


 * and



\frac{-8}{\phantom{-}2}=-4 $$


 * and



\begin{array}[l] ({\color{blue} \mathbf{u} \, \lrcorner \, (\alpha \wedge \beta)} &= {\color{red} (\mathbf{u} \, \lrcorner \, \alpha)} \wedge \beta &- \alpha \wedge (\mathbf{u} \, \lrcorner \, \beta) \\ &= ( \mathbf{u} \cdot \alpha ) \cdot \beta &- \alpha \cdot ( \mathbf{u} \cdot \beta ) \\ &= ( \mathbf{u} \cdot \alpha ) \cdot \beta &- ( \mathbf{u} \cdot \beta ) \cdot \alpha \end{array} $$


 * and



\begin{array}[l] \mathbf{u} \, \lrcorner \, (\alpha \wedge \beta \wedge \gamma)

&= {\color{blue} (\mathbf{u} \, \lrcorner \, (\alpha \wedge \beta))} \wedge \gamma && +(\alpha \wedge \beta) \wedge (\mathbf{u} \, \lrcorner \, \gamma) \\

&= {\color{blue} ( \, ( \mathbf{u} \, \lrcorner \, \alpha ) \wedge \beta} & {\color{blue} - \alpha \wedge (\mathbf{u} \, \lrcorner \, \beta) \, ) } \wedge \gamma &+ (\alpha \wedge \beta) \wedge (\mathbf{u} \, \lrcorner \, \gamma) \\

&= ( \mathbf{u} \cdot \alpha ) \cdot \beta \wedge \gamma &- \alpha \cdot ( \mathbf{u} \cdot \beta ) \wedge \gamma &+ (\alpha \wedge \beta) \cdot (\mathbf{u} \cdot \gamma) \\

&= ( \mathbf{u} \cdot \alpha ) \cdot \beta \wedge \gamma &+ ( \mathbf{u} \cdot \beta ) \cdot \gamma \wedge \alpha &+ (\mathbf{u} \cdot \gamma) \cdot (\alpha \wedge \beta) \end{array} $$