Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors , $\overrightarrow{b}$ and $\overrightarrow{c}$ are inclined to $\overrightarrow{a}$ at an angle of $\frac{\pi }{6}$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}$ then $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})=$

• A. $0$
• B. $1$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{-1}{4}$

Answer 1

The correct option is D $\frac{-1}{4}$

$\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}=\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}$ and $\overrightarrow{b}.\overrightarrow{c}=\cos \frac{\pi }{3}=\frac{1}{2}$
$(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})=\left|\begin{array}{cc}\overrightarrow{a}.\overrightarrow{a}& \overrightarrow{a}.\overrightarrow{c}\\ \overrightarrow{b}.\overrightarrow{a}& \overrightarrow{b}.\overrightarrow{c}\end{array}\right|$
$=\left|\begin{array}{cc}1& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right|$
$=\frac{1}{2}-\frac{3}{4}=\frac{-1}{4}$on simplification.