Question

If $u,v$ and $w$ are the three non-coplanar vectors, then $\left(u+v–w\right)\bullet \left(u–v\right)\times \left(v–w\right)$ is equal to

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

Answer 1

The correct option is B

$u\bullet v\times w$

Explanation for the correct option:

Finding the value of the expression:

$\begin{array}{rcl}\left(u+v–w\right)\bullet \left[(u–v)\times (v–w)\right]& =& (u+v–w)\bullet (u\times v–u\times w–v\times v+v\times w)\\ & =& u.(u\times v)–u.(u\times w)+u.(v\times w)+v.(u\times v)–v.(u\times w)+v.(v\times w)–w.(u\times v)+w.(u\times v)–w.(v\times w)\\ & =& u.v\times w–v.u\times w–w.u\times v\\ & =& u.v\times w+w.u\times v–w.u\times v\\ & =& u.v\times w\end{array}$

Hence, the correct option is B.