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 parallelopiped of volume

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

Answer 1

The correct option is D $\frac{4}{3\sqrt{3}}$

Let

${\hat{n}}_1=(\hat{i}+\hat{j})\times (\hat{j}\times \hat{k})=\frac{\hat{k}-\hat{j}+\hat{i}}{\sqrt{3}}$
${\hat{n}}_2=(\hat{j}+\hat{k})\times (\hat{k}\times \hat{i})=\frac{\hat{i}-\hat{k}+\hat{j}}{\sqrt{3}}$
${\hat{n}}_3=(\hat{k}+\hat{i})\times (\hat{i}\times \hat{j})=\frac{\hat{j}-\hat{i}+\hat{k}}{\sqrt{3}}$
Volume of parallelopiped=$\left[n_1{n}_2{n}_3\right]$
=$n_1.n_2\times n_3$
=$n_1.(\frac{(i+\hat{j}-\hat{k})}{\sqrt{3}}\times \frac{(-\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}})$
=$\frac{4}{3\sqrt{3}}$