If the volume of the parallelopiped with vec{ | vedclass

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(C) The volume of a parallelopiped with coterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $|[\vec{u} \vec{v} \vec{w}

Vedclass · Accepted Answer

$81 	ext{ cu. units}$

Vedclass · Answer

(C) The volume of a parallelopiped with coterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $|[\vec{u} \vec{v} \vec{w}]|$.Given that the volume of the parallelopiped with edges $\vec{a} 	imes \vec{b}, \vec{b} 	imes \vec{c}, \vec{c} 	imes \vec{a}$ is $9$, we have:$|[(\vec{a} 	imes \vec{b}) \quad (\vec{b} 	imes \vec{c}) \quad (\vec{c} 	imes \vec{a})]| = 9$We know that $[(\vec{a} 	imes \vec{b}) \quad (\vec{b} 	imes \vec{c}) \quad (\vec{c} 	imes \vec{a})] = [\vec{a} \vec{b} \vec{c}]^2$.Thus, $[\vec{a} \vec{b} \vec{c}]^2 = 9$.Now, we need to find the volume of the parallelopiped with edges $\vec{u}' = (\vec{a} 	imes \vec{b}) 	imes(\vec{b} 	imes \vec{c})$, $\vec{v}' = (\vec{b} 	imes \vec{c}) 	imes(\vec{c} 	imes \vec{a})$, and $\vec{w}' = (\vec{c} 	imes \vec{a}) 	imes(\vec{a} 	imes \vec{b})$.Using the property $(\vec{a} 	imes \vec{b}) 	imes(\vec{b} 	imes \vec{c}) = [\vec{a} \vec{b} \vec{c}] \vec{b}$, we have:$\vec{u}' = [\vec{a} \vec{b} \vec{c}] \vec{b}$, $\vec{v}' = [\vec{a} \vec{b} \vec{c}] \vec{c}$, $\vec{w}' = [\vec{a} \vec{b} \vec{c}] \vec{a}$.The volume is $|[\vec{u}' \vec{v}' \vec{w}']| = |[([\vec{a} \vec{b} \vec{c}] \vec{b}) \quad ([\vec{a} \vec{b} \vec{c}] \vec{c}) \quad ([\vec{a} \vec{b} \vec{c}] \vec{a})]|$.$= |[\vec{a} \vec{b} \vec{c}]^3 [\vec{b} \vec{c} \vec{a}]| = |[\vec{a} \vec{b} \vec{c}]^4|$.Since $[\vec{a} \vec{b} \vec{c}]^2 = 9$, then $[\vec{a} \vec{b} \vec{c}]^4 = (9)^2 = 81$.Therefore, the volume is $81 	ext{ cu. units}$.

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