Question

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\overrightarrow{c}=\hat{a}+2\hat{b}$ and $\overrightarrow{d}=5\hat{a}-4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (C)

$\frac{\pi }{3}$

Given that $\overrightarrow{c}$ and $\overrightarrow{d}$ are Perpendicular

$\therefore$ $\overrightarrow{c}.\overrightarrow{d}=0$
Putting values
$\Rightarrow 5|\overrightarrow{a}{|}^2+6ab-8|b{|}^2=0$
$\Rightarrow 6\overrightarrow{a}.\overrightarrow{b}=3$
$\Rightarrow \overrightarrow{a}.\overrightarrow{b}=\frac{1}{2}$
$=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta =\frac{1}{2}$

Since $\overrightarrow{a}$ and $\overrightarrow{b}$ are Unit vectors

So, $\cos \theta =\frac{1}{2}$

$\therefore \theta =\frac{\pi }{3}$
Hence, option '$C^{\prime }$ is correct.