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

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(C) Let the expression be $E = (\bar{u} + \bar{v} - \bar{w}) \cdot ((\bar{u} - \bar{v}) 	imes (\bar{v} - \bar{w}))$.First,compute the cross product:

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

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(C) Let the expression be $E = (\bar{u} + \bar{v} - \bar{w}) \cdot ((\bar{u} - \bar{v}) 	imes (\bar{v} - \bar{w}))$.First,compute the cross product: $(\bar{u} - \bar{v}) 	imes (\bar{v} - \bar{w}) = \bar{u} 	imes \bar{v} - \bar{u} 	imes \bar{w} - \bar{v} 	imes \bar{v} + \bar{v} 	imes \bar{w}$.Since $\bar{v} 	imes \bar{v} = 0$,this simplifies to $\bar{u} 	imes \bar{v} + \bar{w} 	imes \bar{u} + \bar{v} 	imes \bar{w}$.Now,take the dot product with $(\bar{u} + \bar{v} - \bar{w})$:$E = (\bar{u} + \bar{v} - \bar{w}) \cdot (\bar{u} 	imes \bar{v} + \bar{w} 	imes \bar{u} + \bar{v} 	imes \bar{w})$.Distributing the dot product:$= \bar{u} \cdot (\bar{u} 	imes \bar{v}) + \bar{u} \cdot (\bar{w} 	imes \bar{u}) + \bar{u} \cdot (\bar{v} 	imes \bar{w}) + \bar{v} \cdot (\bar{u} 	imes \bar{v}) + \bar{v} \cdot (\bar{w} 	imes \bar{u}) + \bar{v} \cdot (\bar{v} 	imes \bar{w}) - \bar{w} \cdot (\bar{u} 	imes \bar{v}) - \bar{w} \cdot (\bar{w} 	imes \bar{u}) - \bar{w} \cdot (\bar{v} 	imes \bar{w})$.Using the property that scalar triple product is zero if any two vectors are the same:$= 0 + 0 + [\bar{u} \bar{v} \bar{w}] + 0 + [\bar{v} \bar{w} \bar{u}] + 0 - [\bar{w} \bar{u} \bar{v}] - 0 - 0$.Since $[\bar{u} \bar{v} \bar{w}] = [\bar{v} \bar{w} \bar{u}] = [\bar{w} \bar{u} \bar{v}]$,we get:$E = [\bar{u} \bar{v} \bar{w}] + [\bar{u} \bar{v} \bar{w}] - [\bar{u} \bar{v} \bar{w}] = [\bar{u} \bar{v} \bar{w}] = \bar{u} \cdot (\bar{v} 	imes \bar{w})$.

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

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