Question

Let $\overline{a},\overline{b},\overline{c}$ three non-zero vectors such that $\overline{c}$ is a unit vector right angle to both $\overline{a}$ and $\overline{b}$ are inclined at $\frac{\pi }{6}$ then ${\left[\overline{a}\overline{b}\overline{c}\right]}^2$ is

• A. $\frac{3}{4}{\left|\overline{a}\right|}^2{\left|\overline{b}\right|}^2$
• B. $\frac{1}{4}{\left|\overline{a}\right|}^2{\left|\overline{b}\right|}^2$
• C. $1$
• D. $0$

Answer 1

The correct option is B $\frac{1}{4}{\left|\overline{a}\right|}^2{\left|\overline{b}\right|}^2$

We have

${\left[\overline{a}\overline{b}\overline{c}\right]}^2={\left[\overline{c}\overline{a}\overline{b}\right]}^2$

$=(\overline{c}\cdot \overline{a}\times \overline{b}{)}^2$

$=(|\overline{c}||\overline{a}||\overline{b}|\sin {30}^o{)}^2$

$=\frac{1}{4}|\overline{a}{|}^2|\overline{b}{|}^2$