Question

$\overline{a},\overline{b}$ and $\overline{c}$ are three vectors having magnitudes 1,1 and 2 respectively. If $\overline{a}\times (\overline{a}\times \overline{c})+\overline{b}=0$, then acute angle between $\overline{a}$ & $\overline{c}$ is

• A. $\pi /6$
• B. $\pi /4$
• C. $\pi /3$
• D. $5\pi /12$

Answer 1

The correct option is A $\pi /6$

$(\overrightarrow{a}\times \overrightarrow{c})$ $\perp \overrightarrow{a}$ and $\perp \overrightarrow{c}$
$(\overrightarrow{a}\times \overrightarrow{c})=ac\sin \theta \hat{n}$
Hence, angle between $\overrightarrow{a}$ and $(\overrightarrow{a}\times \overrightarrow{c})$ is $\frac{\pi }{2}$
$\therefore \overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{c})=a^{2}c\sin \theta \hat{n}$
$\therefore a^{2}c\sin \theta \hat{n}=-\overrightarrow{b}$
$\therefore \sin \theta =\frac{|-\overrightarrow{b}|}{a^{2}c}=\frac{1}{1^2\times 2}=\frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{6}$