If vec{u}, vec{v}, and vec{w} are three non-c | vedclass

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(C) Let the given expression be $E = (\vec{u} + \vec{v} - \vec{w}) \cdot ((\vec{u} - \vec{v}) 	imes (\vec{v} - \vec{w}))$.First,expand the cross prod

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$\vec{u} \cdot (\vec{v} 	imes \vec{w})$

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(C) Let the given expression be $E = (\vec{u} + \vec{v} - \vec{w}) \cdot ((\vec{u} - \vec{v}) 	imes (\vec{v} - \vec{w}))$.First,expand the cross product term:$(\vec{u} - \vec{v}) 	imes (\vec{v} - \vec{w}) = \vec{u} 	imes \vec{v} - \vec{u} 	imes \vec{w} - \vec{v} 	imes \vec{v} + \vec{v} 	imes \vec{w}$.Since $\vec{v} 	imes \vec{v} = 0$,this simplifies to:$(\vec{u} 	imes \vec{v}) - (\vec{u} 	imes \vec{w}) + (\vec{v} 	imes \vec{w})$.Now,compute the dot product with $(\vec{u} + \vec{v} - \vec{w})$:$E = (\vec{u} + \vec{v} - \vec{w}) \cdot (\vec{u} 	imes \vec{v} - \vec{u} 	imes \vec{w} + \vec{v} 	imes \vec{w})$.Using the property of the scalar triple product $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} 	imes \vec{c})$:$E = [\vec{u} \vec{u} \vec{v}] - [\vec{u} \vec{u} \vec{w}] + [\vec{u} \vec{v} \vec{w}] + [\vec{v} \vec{u} \vec{v}] - [\vec{v} \vec{u} \vec{w}] + [\vec{v} \vec{v} \vec{w}] - [\vec{w} \vec{u} \vec{v}] + [\vec{w} \vec{u} \vec{w}] - [\vec{w} \vec{v} \vec{w}]$.Since any scalar triple product with two identical vectors is $0$:$E = 0 - 0 + [\vec{u} \vec{v} \vec{w}] + 0 - [\vec{v} \vec{u} \vec{w}] + 0 - [\vec{w} \vec{u} \vec{v}] + 0 - 0$.Using the cyclic property $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}]$ and anti-cyclic property $[\vec{a} \vec{b} \vec{c}] = -[\vec{b} \vec{a} \vec{c}]$:$[\vec{v} \vec{u} \vec{w}] = -[\vec{u} \vec{v} \vec{w}]$ and $[\vec{w} \vec{u} \vec{v}] = [\vec{u} \vec{v} \vec{w}]$.Substituting these back:$E = [\vec{u} \vec{v} \vec{w}] - (-[\vec{u} \vec{v} \vec{w}]) - [\vec{u} \vec{v} \vec{w}] = [\vec{u} \vec{v} \vec{w}] + [\vec{u} \vec{v} \vec{w}] - [\vec{u} \vec{v} \vec{w}] = [\vec{u} \vec{v} \vec{w}]$.Thus,the result is $\vec{u} \cdot (\vec{v} 	imes \vec{w})$.

If $\vec{u}, \vec{v},$ and $\vec{w}$ are three non-coplanar vectors,then $(\vec{u} + \vec{v} - \vec{w}) \cdot ((\vec{u} - \vec{v}) \times (\vec{v} - \vec{w})) = \dots$

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