Question

The three vectors $\hat{i}+\hat{j}$, $\hat{j}+\hat{k}$, $\hat{k}+\hat{i}$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume

• A. $1/3$
• B. $4$
• C. $\frac{3\sqrt{3}}{4}$
• D. $\frac{2}{\sqrt{3}}$

Answer 1

The correct option is D $\frac{2}{\sqrt{3}}$

Unit vectors perpendicular to three planes are

$\overrightarrow{a}=\frac{(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})}{|(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})|}=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$

$\overrightarrow{b}=\frac{(\hat{j}+\hat{k})\times (\hat{k}+\hat{i})}{|(\hat{j}+\hat{k})\times (\hat{k}+\hat{i})|}=\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$

$\overrightarrow{c}=\frac{(\hat{k}+\hat{i})\times (\hat{i}+\hat{j})}{|(\hat{k}+\hat{i})\times (\hat{i}+\hat{j})|}=\frac{-\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

Volume parallelopiped $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\frac{1}{3\sqrt{3}}\left|\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\\ -1& 1& 1\end{array}\right|=\frac{2}{\sqrt{3}}$