Question

If $\hat{a}$ and $\hat{b}$ are two unit vectors such that $\hat{a}+2\hat{b}$ and $5\hat{a}-4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

• A. ${45}^o$
• B. ${60}^o$
• C. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• D. ${\cos }^{-1}\left(\frac{2}{7}\right)$

Answer 1

The correct option is B ${60}^o$

Let $u=\hat{a}+2\hat{b}$ and $v=5\hat{a}-4\hat{b}$ and let $\theta$ be the angle between $a$ and $b$
It is given that $u$ and $v$ are perpendicular to each other
$\therefore$ $u.v=0$
$\Rightarrow (\hat{a}+2\hat{b}).(5\hat{a}-4\hat{b})=0$
$\Rightarrow 5{\left|a\right|}^2-8{\left|b\right|}^2+10(\hat{a}.\hat{b})-4(\hat{a}.\hat{b})=0$
$\Rightarrow -3+6(\hat{a}.\hat{b})=0[\because \left|a\right|=\left|b\right|=1]$
$\Rightarrow -3+6\cos \theta =0[\because \hat{a}.\hat{b}=\left|a\right|\left|b\right|\cos \theta =\cos \theta ]$
$\Rightarrow \cos \theta =1/2\Rightarrow \theta ={60}^o$