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(A) Given,the volume of the parallelepiped with coterminous edges $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ is given by the scalar

Vedclass · Accepted Answer

$80$

Vedclass · Answer

(A) Given,the volume of the parallelepiped with coterminous edges $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ is given by the scalar triple product $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] = 40 	ext{ cubic units}$.The volume of the parallelepiped with coterminous edges $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}$ is given by the scalar triple product $[\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}]$.Using the property of the scalar triple product:$[\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}] = (\overrightarrow{b}+\overrightarrow{c}) \cdot ((\overrightarrow{c}+\overrightarrow{a}) 	imes (\overrightarrow{a}+\overrightarrow{b}))$$= (\overrightarrow{b}+\overrightarrow{c}) \cdot (\overrightarrow{c} 	imes \overrightarrow{a} + \overrightarrow{c} 	imes \overrightarrow{b} + \overrightarrow{a} 	imes \overrightarrow{a} + \overrightarrow{a} 	imes \overrightarrow{b})$Since $\overrightarrow{a} 	imes \overrightarrow{a} = 0$,this simplifies to:$= (\overrightarrow{b}+\overrightarrow{c}) \cdot (\overrightarrow{c} 	imes \overrightarrow{a} + \overrightarrow{c} 	imes \overrightarrow{b} + \overrightarrow{a} 	imes \overrightarrow{b})$$= \overrightarrow{b} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) + \overrightarrow{b} \cdot (\overrightarrow{c} 	imes \overrightarrow{b}) + \overrightarrow{b} \cdot (\overrightarrow{a} 	imes \overrightarrow{b}) + \overrightarrow{c} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) + \overrightarrow{c} \cdot (\overrightarrow{c} 	imes \overrightarrow{b}) + \overrightarrow{c} \cdot (\overrightarrow{a} 	imes \overrightarrow{b})$$= [\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}] + 0 + 0 + 0 + 0 + [\overrightarrow{c} \overrightarrow{a} \overrightarrow{b}]$$= [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] + [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] = 2[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$$= 2 	imes 40 = 80 	ext{ cubic units}$.

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