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Question

# Find the volume of the parallelopiped whose coterminous edges are represented by the vectors: (i) $\stackrel{\to }{a}=2\stackrel{^}{i}+3\stackrel{^}{j}+4\stackrel{^}{k},\stackrel{\to }{b}=\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k},\stackrel{\to }{c}=3\stackrel{^}{i}-\stackrel{^}{j}+2\stackrel{^}{k}$ (ii) $\stackrel{\to }{a}=2\stackrel{^}{i}-3\stackrel{^}{j}+4\stackrel{^}{k},\stackrel{\to }{b}=\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k},\stackrel{\to }{c}=3\stackrel{^}{i}-\stackrel{^}{j}-2\stackrel{^}{k}$ (iii) $\stackrel{\to }{a}=11\stackrel{^}{i},\stackrel{\to }{b}=2\stackrel{^}{j},\stackrel{\to }{c}=13\stackrel{^}{k}$ (iv) $\stackrel{\to }{a}=\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k},\stackrel{\to }{b}=\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k},\stackrel{\to }{c}=\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k}$

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Solution

## $\left(\mathrm{i}\right)\mathrm{Given}:\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{a}=2\stackrel{\mathit{^}}{i}+3\stackrel{\mathit{^}}{j}+4\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{b}=\stackrel{\mathit{^}}{i}+2\stackrel{\mathit{^}}{j}-\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{c}=3\stackrel{\mathit{^}}{i}-\stackrel{\mathit{^}}{j}+2\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{parallelopiped}\mathrm{whose}\mathrm{three}\mathrm{adjacent}\mathrm{edges}\mathrm{are}\mathit{}\stackrel{\mathit{\to }}{a},\stackrel{\to }{b},\stackrel{\to }{c}\mathrm{is}\mathrm{equal}\mathrm{to}\left|\left[\stackrel{\mathit{\to }}{a}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Here},\phantom{\rule{0ex}{0ex}}\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]=\left|\begin{array}{ccc}2& 3& 4\\ 1& 2& -1\\ 3& -1& 2\end{array}\right|=2\left(4-1\right)-3\left(2+3\right)+4\left(-1-6\right)=-37\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}=\left|\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|=\left|-37\right|=37\mathrm{cubic}\mathrm{units}$ $\left(\mathrm{ii}\right)\mathrm{Given}:\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{a}=2\stackrel{\mathit{^}}{i}-3\stackrel{\mathit{^}}{j}+4\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{b}=\stackrel{\mathit{^}}{i}+2\stackrel{\mathit{^}}{j}-\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{c}=3\stackrel{\mathit{^}}{i}-\stackrel{\mathit{^}}{j}-2\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{parallelopiped}\mathrm{whose}\mathrm{three}\mathrm{adjacent}\mathrm{edges}\mathrm{are}\mathit{}\stackrel{\mathit{\to }}{a},\stackrel{\to }{b},\stackrel{\to }{c}\mathrm{is}\mathrm{equal}\mathrm{to}\left|\left[\stackrel{\mathit{\to }}{a}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Here},\phantom{\rule{0ex}{0ex}}\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]=\left|\begin{array}{ccc}2& -3& 4\\ 1& 2& -1\\ 3& -1& -2\end{array}\right|=2\left(-4-1\right)+3\left(-2+3\right)+4\left(-1-6\right)=-35\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}=\left|\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|=\left|-35\right|=35\mathrm{cubic}\mathrm{units}$ $\left(\mathrm{iii}\right)\mathrm{Given}:\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{a}=11\stackrel{\mathit{^}}{i}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{b}=2\stackrel{\mathit{^}}{j}\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{c}=13\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{parallelopiped}\mathrm{whose}\mathrm{three}\mathrm{adjacent}\mathrm{edges}\mathrm{are}\mathit{}\stackrel{\mathit{\to }}{a},\stackrel{\to }{b},\stackrel{\to }{c}\mathrm{is}\mathrm{equal}\mathrm{to}\left|\left[\stackrel{\mathit{\to }}{a}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Here},\phantom{\rule{0ex}{0ex}}\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]=\left|\begin{array}{ccc}11& 0& 0\\ 0& 2& 0\\ 0& 0& 13\end{array}\right|=11\left(26-0\right)-0\left(0-0\right)+0\left(0-0\right)=286\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}=\left|\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|=\left|286\right|=286\mathrm{cubic}\mathrm{units}$ $\left(\mathrm{iv}\right)\mathrm{Given}:\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{a}=\stackrel{\mathit{^}}{i}+\stackrel{\mathit{^}}{j}+\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{b}=\stackrel{\mathit{^}}{i}-\stackrel{\mathit{^}}{j}+k\phantom{\rule{0ex}{0ex}}\stackrel{\mathit{\to }}{c}=\stackrel{\mathit{^}}{i}+2j-\stackrel{\mathit{^}}{k}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{parallelopiped}\mathrm{whose}\mathrm{three}\mathrm{adjacent}\mathrm{edges}\mathrm{are}\mathit{}\stackrel{\mathit{\to }}{a},\stackrel{\to }{b},\stackrel{\to }{c}\mathrm{is}\mathrm{equal}\mathrm{to}\left|\left[\stackrel{\mathit{\to }}{a}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Here},\phantom{\rule{0ex}{0ex}}\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]=\left|\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ 1& 2& -1\end{array}\right|=1\left(1-2\right)-1\left(-1-1\right)+1\left(2+1\right)=4\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}=\left|\left[\stackrel{\mathit{\to }}{\mathrm{a}}\stackrel{\to }{\mathrm{b}}\stackrel{\to }{\mathrm{c}}\right]\right|=\left|4\right|=4\mathrm{cubic}\mathrm{units}$

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