Question

If $\overrightarrow{a},\overrightarrow{b}$ are two unit vectors inclined at an angle $\frac{\pi }{3}$, then $\{\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}\left)\right\}.\overrightarrow{b}=$

• A. $\frac{-3}{4}$
• B. $\frac{1}{4}$
• C. $\frac{3}{4}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{-3}{4}$

We have

$\{\overline{a}\times (\overline{b}+\overline{a}\times \overline{b}\left)\right\}.\overline{b}$
$=[\overline{a}\times \overline{b}].\overline{b}+[\overline{a}\times (\overline{a}\times \overline{b}\left)\right].\overline{b}$
$=0+\left[\right(\overline{a}.\overline{b})\overline{a}-(\overline{a}.\overline{a}\left)\overline{b}\right].\overline{b}$
$=\frac{1}{4}-1$

$=\frac{-3}{4}$