If u, v and w are three non-coplanar vectors, then (u+v−w)⋅[(u−v)×(v−w)] equals:

(u+v−w)⋅[u−v×(v−w)]=(u+v−w)⋅[(u×v)−(u×w)−0+(v×w)]=[uuv]+[vuv]−[wuv]−[uw]−[vuw]+[wuw]+[uvw]+[vvw]−[wvw]=0+0−[uvw]−0+[uvw]+0+[uvw]+0−0=[uvw]=u⋅(v×w)

If u, v and w are three non-coplanar vectors, then (u+v−w)⋅[(u−v)×(v−w)] equals:

(u+v−w)⋅[u−v×(v−w)]=(u+v−w)⋅[(u×v)−(u×w)−0+(v×w)]=[uuv]+[vuv]−[wuv]−[uw]−[vuw]+[wuw]+[uvw]+[vvw]−[wvw]=0+0−[uvw]−0+[uvw]+0+[uvw]+0−0=[uvw]=u⋅(v×w)

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If $u, v$ and $w$ are three non-coplanar vectors, then $(u + v - w) \cdot [(u - v) \times (v - w)]$ equals:

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$0$

$u \cdot (v \times w)$

$u \cdot (w \times v)$

$3 u \cdot (v \times w)$

answer is B.

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Detailed Solution

$\begin{array}{l} (u + v - w) \cdot [u - v \times (v - w)] \\ = (u + v - w) \cdot [(u \times v) - (u \times w) - 0 + (v \times w)] \\ = [uuv] + [vuv] - [wuv] - [uw] - \\ [vuw] + [wuw] + [uvw] + [vvw] - [wvw] \\ = 0 + 0 - [uvw] - 0 + [uvw] + 0 + [uvw] + 0 - 0 = [uvw] \\ = u \cdot (v \times w) \end{array}$