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

Question

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

Accepted Answer

Expanding $(\vec u-\vec v)\times(\vec v-\vec w) = \vec u\times\vec v-\vec u\times\vec w-\vec v\times\vec v+\vec v\times\vec w = \vec u\times\vec v-\vec u\times\vec w+\vec v\times\vec w$

Dotting with $(\vec u+\vec v-\vec w)$ and using $\vec a\cdot(\vec a\times\vec b)=0$ for repeated vectors, all terms cancel except:

$$\vec u\cdot(\vec v\times\vec w)-\vec v\cdot(\vec u\times\vec w)-\vec w\cdot(\vec u\times\vec v) = [\vec u\ \vec v\ \vec w]+[\vec v\ \vec w\ \vec u]-[\vec w\ \vec u\ \vec v] = [\vec u\ \vec v\ \vec w]$$

(using cyclic invariance $[\vec v\ \vec w\ \vec u]=[\vec w\ \vec u\ \vec v]=[\vec u\ \vec v\ \vec w]$)

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