If $[\mathbf{a} \mathbf{b} \mathbf{c}]=3$, then the volume (in cubic units) of the parallelopiped with $2 \mathbf{a}+\mathbf{b}, 2 \mathbf{b}+\mathbf{c}$ and $2 \mathbf{c}+\mathbf{a}$ as edges, is

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Aicharya · Accepted Answer

Given that, $ 	ext { [abc] }=3 $ Volume of the parallelopiped $ =[2 \mathbf{a}+\mathbf{b} \mathbf{2} \mathbf{b}+\mathbf{c}+2 \mathbf{c}+\mathbf{a}] $ $ \begin{aligned} & =[2 \mathrm{a} 2 \mathrm{~b}+\mathrm{c} 2 \mathrm{c}+\mathbf{a}]+[\mathrm{b} \mathbf{2 b}+\mathrm{c} 2 \mathrm{c}+\mathrm{a}] \ & =[2 \mathrm{a} 2 \mathrm{~b} 2 \mathrm{c} 2 \mathrm{a}]+[2 \mathrm{ac} 2 \mathrm{c}+\mathrm{a}] \ & +[b 2 b 2 c+a]+[b c 2 c+a] \ & =[2 \mathbf{a} 2 \mathbf{b ~} 2 \mathbf{c}]+[2 \mathbf{a} 2 \mathbf{b} \mathbf{a}]+0+0+[\mathbf{b} \mathbf{c} 2 \mathbf{c}]+[\mathrm{b} \mathbf{c} a] \ & =8[a b c]+0+0+[a b c] \ & =9[\mathbf{a} \mathbf{b} \mathbf{c}]=9 \cdot 3=27 \ & \end{aligned} $

If $[\mathbf{a} \mathbf{b} \mathbf{c}]=3$, then the volume (in cubic units) of the parallelopiped with $2…

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