Question

$\overline{a},\overline{b},\overline{c}$ are three unit vectors such that $\overline{a}\times (\overline{b}\times \overline{c})=\frac{1}{2}\overline{b}$, then $(\overline{a},\overline{b})=?,(\overline{a},\overline{c})=?,$ [ $\overline{b},\overline{c}$ are non-collinear]

• A. ${90}^0,{60}^0$
• B. ${60}^0,{90}^0$
• C. ${30}^0,{60}^0$
• D. ${45}^0,{30}^0$

Answer 1

The correct option is A ${90}^0,{60}^0$

$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\overrightarrow{b}}{2}$
$\overrightarrow{a}(\overrightarrow{a}\times (\overrightarrow{b}\times \overline{c}\left)\right)=\frac{\overrightarrow{b}-\overrightarrow{a}}{2}$
$0=\overrightarrow{b}-\overrightarrow{a}$
angle between $(\overrightarrow{a},\overrightarrow{b})={90}^0$
$\overrightarrow{c}\left(\right(\overrightarrow{a}-\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}-\overrightarrow{b}\left)\overrightarrow{c}\right)=\frac{\overrightarrow{b}-\overrightarrow{c}}{2}$
$(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}.\overrightarrow{c}=\frac{\overrightarrow{b}.\overrightarrow{c}}{2}$
$(\overrightarrow{a}-\overrightarrow{c})=\frac{1}{2}$
angle between $(\overrightarrow{a},\overrightarrow{c})={60}^0$