If bar{a}, bar{b}, bar{c} are nonzero vectors | vedclass

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(C) The volume of a parallelepiped with coterminus edges $\bar{a}, \bar{b}, \bar{c}$ is given by the scalar triple product $[\bar{a} \bar{b} \bar{c}]

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$14$ cubic units

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(C) The volume of a parallelepiped with coterminus edges $\bar{a}, \bar{b}, \bar{c}$ is given by the scalar triple product $[\bar{a} \bar{b} \bar{c}] = |\bar{a} \cdot (\bar{b} 	imes \bar{c})| = 7$. We need to find the volume of a parallelepiped with edges $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$. This volume is given by the scalar triple product $[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}]$. Using the property of the scalar triple product,we have: $[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}] = (\bar{a}+\bar{b}) \cdot ((\bar{b}+\bar{c}) 	imes (\bar{c}+\bar{a}))$. Expanding the cross product: $(\bar{b}+\bar{c}) 	imes (\bar{c}+\bar{a}) = \bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{c} + \bar{c} 	imes \bar{a} = \bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{a}$ (since $\bar{c} 	imes \bar{c} = 0$). Now,taking the dot product with $(\bar{a}+\bar{b})$: $(\bar{a}+\bar{b}) \cdot (\bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{a}) = \bar{a} \cdot (\bar{b} 	imes \bar{c}) + \bar{a} \cdot (\bar{b} 	imes \bar{a}) + \bar{a} \cdot (\bar{c} 	imes \bar{a}) + \bar{b} \cdot (\bar{b} 	imes \bar{c}) + \bar{b} \cdot (\bar{b} 	imes \bar{a}) + \bar{b} \cdot (\bar{c} 	imes \bar{a})$. Since the scalar triple product is zero if any two vectors are the same,we have: $= [\bar{a} \bar{b} \bar{c}] + 0 + 0 + 0 + 0 + [\bar{b} \bar{c} \bar{a}] = [\bar{a} \bar{b} \bar{c}] + [\bar{a} \bar{b} \bar{c}] = 2[\bar{a} \bar{b} \bar{c}]$. Given $[\bar{a} \bar{b} \bar{c}] = 7$,the volume is $2 	imes 7 = 14$ cubic units.

If $\bar{a}, \bar{b}, \bar{c}$ are nonzero vectors along the coterminus edges of a parallelepiped with volume $7$ cubic units,then the volume of a parallelepiped with $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ as the coterminus edges is

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