Question

Three vectors $\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{k}+\hat{i}$, when taken two at a time, form three planes. The three unit vectors drawn perpendicular to these three planes form a parellelopiped of volume $V$, then the value of $V$ is

• A. $\frac{1}{3}$
• B. $4$
• C. $\frac{3\sqrt{3}}{4}$
• D. $\frac{4}{3\sqrt{3}}$

Answer 1

The correct option is B $4$

Two of the three vectors given $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c})$ are taken at a time to form a plane.

The normal vectors to the plane will be given by

$\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{b}\times \overrightarrow{c},\overrightarrow{c}\times \overrightarrow{a}$

Now, the normal to that planes are forming a parallelopiped so the volume is given by $[\overrightarrow{a}\times \overrightarrow{b}\overrightarrow{b}\times \overrightarrow{c}\overrightarrow{c}\times \overrightarrow{a}]$

$\overrightarrow{a}\times \overrightarrow{b}=(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})=\hat{k}-\hat{j}+\hat{i}$

$\overrightarrow{b}\times \overrightarrow{c}=(\hat{j}+\hat{k})\times (\hat{k}+\hat{i})=\hat{i}-\hat{k}+\hat{j}$

$\overrightarrow{c}\times \overrightarrow{a}=(\overrightarrow{k}+\overrightarrow{i})\times (\overrightarrow{i}+\overrightarrow{j})=\overrightarrow{j}-\overrightarrow{i}+\overrightarrow{k}$

Volume, $V=[\hat{k}-\hat{j}+\hat{i}\hat{i}-\hat{k}+\hat{j}\hat{j}-\hat{i}+\hat{k}]$

$=\left|\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\\ -1& 1& 1\end{array}\right|$

$=1(1+1)+1(1-1)+1(1+1)=4$

Hence, option $B$.