Question

Let $\overline{a}={\alpha }_1\hat{i}+{\alpha }_2\hat{j}+{\alpha }_3\hat{k}$, $\overline{b}={\beta }_1\hat{i}+{\beta }_2\hat{j}+{\beta }_3\hat{k}$ and $\overline{c}={\gamma }_1\hat{i}+{\gamma }_2\hat{j}+{\gamma }_3\hat{k}$, $\left|\overline{a}\right|=2\sqrt{2}$. $\overline{a}$ makes angle $\frac{\pi }{3}$ with plane of $\overline{b}$ and $\overline{c}$ and angle between $\overline{b}$ and $\overline{c}$ is $\frac{\pi }{3}$,

then ${\left|\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right|}^n$ is equal to (n is even natural number).

• A. ${\left(\frac{\left|\overline{b}\right|\left|\overline{a}\right|}{2\times 3}\right)}^{n/2}$
• B. ${\left(\frac{3\left|\overline{b}\right|\left|\overline{c}\right|}{\sqrt{2}}\right)}^n$
• C. $\frac{{\left(\left|\overline{b}\right|\left|\overline{c}\right|\right)}^n}{\sqrt{3}{2}^n}$
• D. None of these

Answer 1

The correct option is B ${\left(\frac{3\left|\overline{b}\right|\left|\overline{c}\right|}{\sqrt{2}}\right)}^n$

Given determinant shows the volume of perellelopiped made by vectors $\overline{a}$, $\overline{b}$ and $\overline{c}$
$\therefore$ $\left|\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right|=|\overline{a}.(\overline{b}\times \overline{c})|$

$=\left|\overline{a}\right||\overline{b}\times \overline{c}|\cos \frac{\pi }{6}=\left|\overline{a}\right|\left|\overline{b}\right|\left|\overline{c}\right|\sin \frac{\pi }{3}\cos \frac{\pi }{6}$

$=2\sqrt{2}\left|\overline{b}\right|\left|\overline{c}\right|\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}$ ..($\because \left|\overline{a}\right|=2\sqrt{2})$

$=\frac{3\left|\overline{b}\right|\left|\overline{c}\right|}{\sqrt{2}}$