A) \[u.\,\,w\times u\]
B) \[u.\,\,v\times w\]
C) \[0\]
D) \[3\,u.\,\,v\times w\]
Correct Answer: B
Solution :
\[(u+v-w).[(u-v)\times (v-w)]\] \[=(u+v-w).(u\times v-u\times w-v\times v+v\times w)\] \[(u+v-w).(u\times v-u\times w+v\times w)\] \[[\because \,\,\bar{v}\times \bar{v}=0]\] \[=u.(u\times v)-u.(u\times w)+u.(v\times w)+v.(u\times v)\] \[-v.(u\times w)+v.(v\times w)-w.(u\times v)+w.(u\times w)\] \[-w.(v\times w)\] \[=0-0+[u\,v\,w]+0-[v\,u\,w]+0-[w\,u\,v]\] \[+0-0\] \[=u.(v\times w)\]
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