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(D) The volume of a parallelepiped with conterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $[\vec{u} \vec{v} \vec{w}

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(D) The volume of a parallelepiped with conterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $[\vec{u} \vec{v} \vec{w}] = \vec{u} \cdot (\vec{v} 	imes \vec{w})$.Here,the edges are $\vec{u} = \vec{a} - \vec{b}$,$\vec{v} = \vec{b} - \vec{c}$,and $\vec{w} = \vec{c} - \vec{a}$.Volume $= (\vec{a} - \vec{b}) \cdot ((\vec{b} - \vec{c}) 	imes (\vec{c} - \vec{a}))$.First,calculate the cross product: $(\vec{b} - \vec{c}) 	imes (\vec{c} - \vec{a}) = \vec{b} 	imes \vec{c} - \vec{b} 	imes \vec{a} - \vec{c} 	imes \vec{c} + \vec{c} 	imes \vec{a} = \vec{b} 	imes \vec{c} + \vec{a} 	imes \vec{b} + \vec{c} 	imes \vec{a}$ (since $\vec{c} 	imes \vec{c} = 0$ and $-\vec{b} 	imes \vec{a} = \vec{a} 	imes \vec{b}$).Now,take the dot product with $(\vec{a} - \vec{b})$:$(\vec{a} - \vec{b}) \cdot (\vec{b} 	imes \vec{c} + \vec{a} 	imes \vec{b} + \vec{c} 	imes \vec{a})$$= \vec{a} \cdot (\vec{b} 	imes \vec{c}) + \vec{a} \cdot (\vec{a} 	imes \vec{b}) + \vec{a} \cdot (\vec{c} 	imes \vec{a}) - \vec{b} \cdot (\vec{b} 	imes \vec{c}) - \vec{b} \cdot (\vec{a} 	imes \vec{b}) - \vec{b} \cdot (\vec{c} 	imes \vec{a})$.Using the property that the scalar triple product is zero if any two vectors are identical,we have:$\vec{a} \cdot (\vec{a} 	imes \vec{b}) = 0$,$\vec{a} \cdot (\vec{c} 	imes \vec{a}) = 0$,$\vec{b} \cdot (\vec{b} 	imes \vec{c}) = 0$,$\vec{b} \cdot (\vec{a} 	imes \vec{b}) = 0$.Thus,the expression simplifies to $\vec{a} \cdot (\vec{b} 	imes \vec{c}) - \vec{b} \cdot (\vec{c} 	imes \vec{a}) = [\vec{a} \vec{b} \vec{c}] - [\vec{b} \vec{c} \vec{a}] = [\vec{a} \vec{b} \vec{c}] - [\vec{a} \vec{b} \vec{c}] = 0$.

If three conterminous edges of a parallelepiped are represented by $\vec{a} - \vec{b}$,$\vec{b} - \vec{c}$,and $\vec{c} - \vec{a}$,then its volume is

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