$\left( {\vec u \,\, + \;\,\vec v \,\, - \,\,\vec w } \right).\,\,\left( {\vec u \, - \,\,\vec v } \right)\,\, \times \,\,\left( {\vec v \,\, - \,\,\vec w } \right)$
$ = \,\,\left( {\vec u \,\, + \;\,\vec v \,\, - \,\,\vec w } \right)\,\,.\,\,\left( {\vec u \,\, \times \,\vec v \,\, - \,\,\vec u \, \times \,\vec w \,\, - \,\,\vec v \,\, \times \,\,\vec v \, + \,\,\vec v \,\, \times \,\,\vec w } \right)$
$ = \,\,\left( {\vec u \,\, + \;\,\vec v \,\, - \,\,\vec w } \right)\,\,.\,\,\left( {\vec u \,\, \times \,\vec v \,\, - \,\,\vec u \, \times \,\vec w + \,\,\vec v \,\, \times \,\,\vec w } \right)$
$ = \,\,\vec u \,\,.\,\,\left( {\vec u \,\, \times \,\,\vec v } \right)\,\, - \,\,\vec u \,\,\left( {\vec u \,\, \times \,\,\,\vec w } \right)\,\, \times \,\,\vec u \,\, + \;\,\left( {\vec v \,\, \times \,\,\vec w } \right)\,\, + $ $\vec v \left( {\vec u \,\, \times \,\,\vec v } \right)\,\, - \,\,\vec v \,\,\left( {\vec u \,\, \times \,\,\vec w } \right)\,\, + \;\,\vec v \,\,\left( {\vec v \,\, \times \,\,\vec w } \right)\,\, - \,$ $\vec w \,\,\left( {\vec u \,\, \times \,\,\vec v } \right)\,\, + \;\,\vec w \,\,\left( {\vec u \,\, \times \,\,\vec w } \right)\,\, - \,\,\vec w \,\,\left( {\vec v \,\, \times \,\,\vec w } \right)$
$ = \,\,\vec u \,\,\left( {\vec v \,\, \times \,\,\vec w } \right)\,\, - \,\,\vec v \,\,\left( {\vec u \,\, \times \,\,\vec w } \right)\,\, - \,\,\vec w \,.\,\,\left( {\vec u \,\, \times \,\,\vec v } \right)\,$
$ = \,\,\left[ {\vec u \,\,\vec v \,\,\vec w } \right]\,\, - \,\,\left[ {\vec v \,\,\vec u \,\,\vec w } \right]\,\, - \,\,\left[ {\vec w \,\,\vec u \,\,\vec v } \right]$
$ = \,\,\left[ {\vec u \,\,\vec v \,\,\vec w } \right]\, + \,\,\left[ {\vec u \,\,\vec v \,\,\vec w } \right]\,\, - \,\,\left[ {\vec u \,\,\vec v \,\,\vec w } \right]$
$ = \,\,\left[ {\vec u \,\,\vec v \,\,\vec w } \right]\,\, $
$= \,\,\vec u \,\,\left( {\,\vec v \, \times \,\,\,\vec w } \right)$
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