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

Answer 1

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

$(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})=\hat{i}-\hat{j}+\hat{k}$
$\Rightarrow$ Unit vector perpendicular to the plane of $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ is $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$
Similarly, other two unit vectors are $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$ and $\frac{1}{\sqrt{3}}(-\hat{i}+\hat{j}+\hat{k})$
$\Rightarrow \text{Volume}=\left[{\hat{n}}_1{\hat{n}}_2{\hat{n}}_3\right]=\frac{1}{3\sqrt{3}}\left|\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\\ -1& 1& 1\end{array}\right|$
$=\frac{4}{3\sqrt{3}}$


Alternatively,
Let $\overrightarrow{a}=\hat{i}+\hat{j};$ $\overrightarrow{b}=\hat{j}+\hat{k}$
and $\overrightarrow{c}=\hat{k}+\hat{i}$
Now, $[\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{b}\times \overrightarrow{c},\overrightarrow{c}\times \overrightarrow{a}]$
$=[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}{]}^2$
$={\left|\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right|}^2$
$={\left[1\right(1)-1(0-1\left)\right]}^2=4$

Hence, actual volume with unit vectors
$=\frac{4}{|\overrightarrow{a}\times \overrightarrow{b}||\overrightarrow{b}\times \overrightarrow{c}||\overrightarrow{c}\times \overrightarrow{a}|}$
Now, $|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{{\overrightarrow{a}}^2{\overrightarrow{b}}^2-(\overrightarrow{a}\cdot \overrightarrow{b}{)}^2}=\sqrt{4-1}=\sqrt{3}$
$\therefore V_{\text{actual}}=\frac{4}{3\sqrt{3}}$