Question

If $\vec {u} , \vec {v} \,and\, \vec{w}$ are three non-coplanar vectors, then $ (\vec {u} +\vec {v}- \vec {w}).\,[(\vec {u} -\vec {v})\times (\vec {v} - \vec {w})] $ equals to

• A. $ u.\,\,w\times u $
• B. $ u.\,\,v\times w $
• C. $ 0 $
• D. $ 3\,u.\,\,v\times w $

Answer 1

Answer: (B)

$ (u+v-w).[(u-v)\times (v-w)] $
$=(u+v-w).(u\times v-u\times w-v\times v+v\times w) $ $ (u+v-w).(u\times v-u\times w+v\times w) $ $ [\because \,\,\bar{v}\times \bar{v}=0] $
$=u.(u\times v)-u.(u\times w)+u.(v\times w)+v.(u\times v) $ $ -v.(u\times w)+v.(v\times w)-w.(u\times v)+w.(u\times w) $ $ -w.(v\times w) $
$=0-0+[u\,v\,w]+0-[v\,u\,w]+0-[w\,u\,v] $ $ +0-0 $
$=u.(v\times w) $