User:Just granpa/sandbox

::$$

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

:and

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

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}