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If the volume of the parallelopiped with \(\vec{a} \times \vec{b}, \vec{b} \times \vec{c}\) and \(\vec{c} \times \vec{a}\) as coterminous edges is 9 cu. units., then the volume of the parallelopiped with \((\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})\) and \((\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})\) as coterminous edges is
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Verified Answer
The correct answer is:
81 cu. units
Hint : Volume of Parallelopiped whose coterminous edges are \(\vec{a}, \vec{b}\) and \(\vec{c}=[\vec{a} \vec{b} \vec{c}]\)
So, Given that
\(\therefore[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=3
\quad \ldots \ldots (II)\)
Now volume of parallelopiped whose coterminous edges are
\(\begin{aligned} & (\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a}),(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b}) \\ & {[(\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c})(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})]} \\ & {[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]^2}\end{aligned}\)
\(\begin{aligned}
& =(9)^2 \quad \text{(from (1))}\\
& =81
\end{aligned}\)
So, Given that
\(\therefore[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=3
\quad \ldots \ldots (II)\)
Now volume of parallelopiped whose coterminous edges are
\(\begin{aligned} & (\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a}),(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b}) \\ & {[(\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c})(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})]} \\ & {[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]^2}\end{aligned}\)
\(\begin{aligned}
& =(9)^2 \quad \text{(from (1))}\\
& =81
\end{aligned}\)
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