The value of (vec{a} + 2vec{b} - vec{c}) cdot | vedclass

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(C) Let the expression be $E = (\vec{a} + 2\vec{b} - \vec{c}) \cdot \{(\vec{a} - \vec{b}) 	imes (\vec{a} - \vec{b} - \vec{c})\}$.First,simplify the c

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$3[\vec{a} \, \vec{b} \, \vec{c}]$

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(C) Let the expression be $E = (\vec{a} + 2\vec{b} - \vec{c}) \cdot \{(\vec{a} - \vec{b}) 	imes (\vec{a} - \vec{b} - \vec{c})\}$.First,simplify the cross product term: $(\vec{a} - \vec{b}) 	imes (\vec{a} - \vec{b} - \vec{c}) = (\vec{a} - \vec{b}) 	imes \vec{a} - (\vec{a} - \vec{b}) 	imes \vec{b} - (\vec{a} - \vec{b}) 	imes \vec{c}$.$= (\vec{a} 	imes \vec{a} - \vec{b} 	imes \vec{a}) - (\vec{a} 	imes \vec{b} - \vec{b} 	imes \vec{b}) - (\vec{a} 	imes \vec{c} - \vec{b} 	imes \vec{c})$.Since $\vec{a} 	imes \vec{a} = 0$ and $\vec{b} 	imes \vec{b} = 0$,this becomes: $0 - (-\vec{a} 	imes \vec{b}) - (\vec{a} 	imes \vec{b} - 0) - (\vec{a} 	imes \vec{c} - \vec{b} 	imes \vec{c}) = \vec{a} 	imes \vec{b} - \vec{a} 	imes \vec{b} - \vec{a} 	imes \vec{c} + \vec{b} 	imes \vec{c} = \vec{b} 	imes \vec{c} - \vec{a} 	imes \vec{c}$.Now,calculate the dot product: $E = (\vec{a} + 2\vec{b} - \vec{c}) \cdot (\vec{b} 	imes \vec{c} - \vec{a} 	imes \vec{c})$.$= \vec{a} \cdot (\vec{b} 	imes \vec{c}) - \vec{a} \cdot (\vec{a} 	imes \vec{c}) + 2\vec{b} \cdot (\vec{b} 	imes \vec{c}) - 2\vec{b} \cdot (\vec{a} 	imes \vec{c}) - \vec{c} \cdot (\vec{b} 	imes \vec{c}) + \vec{c} \cdot (\vec{a} 	imes \vec{c})$.Using the properties of scalar triple products,$\vec{a} \cdot (\vec{a} 	imes \vec{c}) = 0$,$\vec{b} \cdot (\vec{b} 	imes \vec{c}) = 0$,$\vec{c} \cdot (\vec{b} 	imes \vec{c}) = 0$,and $\vec{c} \cdot (\vec{a} 	imes \vec{c}) = 0$.So,$E = [\vec{a} \, \vec{b} \, \vec{c}] - 0 + 0 - 2[\vec{b} \, \vec{a} \, \vec{c}] - 0 + 0$.$= [\vec{a} \, \vec{b} \, \vec{c}] - 2(-[\vec{a} \, \vec{b} \, \vec{c}]) = [\vec{a} \, \vec{b} \, \vec{c}] + 2[\vec{a} \, \vec{b} \, \vec{c}] = 3[\vec{a} \, \vec{b} \, \vec{c}]$.

The value of $(\vec{a} + 2\vec{b} - \vec{c}) \cdot \{(\vec{a} - \vec{b}) \times (\vec{a} - \vec{b} - \vec{c})\}$ is equal to

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