Question

If $\overrightarrow{a}=2\hat{i}-3\hat{j}+5\hat{k},\overrightarrow{b}=3\hat{i}-4\hat{j}+5\hat{k}\mathrm{and}\overrightarrow{c}=5\hat{i}-3\hat{j}-2\hat{k},$ then the volume of the parallelopiped with conterminous edges $\overrightarrow{a}+\overrightarrow{b,}\overrightarrow{b}+\overrightarrow{c,}\overrightarrow{c}+\overrightarrow{a}$is
(a) 2
(b) 1
(c) −1
(d) 0

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \overrightarrow{a}+\overrightarrow{b}=\left(2\hat{i}-3\hat{j}+5\hat{k}\right)+\left(3\hat{i}-4\hat{j}+5\hat{k}\right)=5\hat{i}-7\hat{j}+10\hat{k} \\ \overrightarrow{b}+\mathit{}\overrightarrow{c}=\left(3\hat{i}\mathit{}-4\hat{j}+5\hat{k}\right)+\left(5\hat{i}-3\hat{j}\mathit{}-2\hat{k}\right)=8\hat{i}\mathit{}-7\hat{j}+3\hat{k} \\ \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}=\left(5\hat{\mathrm{i}}-3\hat{\mathrm{j}}-2\hat{\mathrm{k}}\right)+\left(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)=7\hat{\mathrm{i}}-6\hat{\mathrm{j}}+3\hat{\mathrm{k}} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{parallelopiped}\mathrm{whose}\mathrm{three}\mathrm{adjacent}\mathrm{adges}\mathrm{are}\overrightarrow{a}+\mathit{}\overrightarrow{b},\overrightarrow{b}\mathit{}+\mathit{}\overrightarrow{c}\mathit{}\mathrm{and}\mathit{}\overrightarrow{c}\mathit{}+\mathit{}\overrightarrow{a}\mathit{}\mathrm{is}\mathrm{equal}\mathrm{to}\left|\left[\overrightarrow{a}+\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}\right]\right|. \\ \mathrm{We}\mathrm{have} \\ \left[\overrightarrow{a}+\overrightarrow{b}\overrightarrow{b}+\overrightarrow{c}\mathit{}\mathit{}\overrightarrow{c}+\overrightarrow{a}\right]=\left|\begin{array}{ccc}5& -7& 10\\ 8& -7& 3\\ 7& -6& 3\end{array}\right| \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=5\left(-21+18\right)+7\left(24-21\right)+10\left(-48+49\right) \\ =\left(5\times -3\right)+\left(7\times 3\right)+\left(10\times 1\right) \\ =16 \\ \therefore \mathrm{Volume}\mathrm{of}\mathrm{parallelopiped}=\left|16\right|=16 \end{gathered}$$


Disclaimer: None of the given options is correct.