Question

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

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

Answer 1

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

Given that $(\overrightarrow{a}+2\overrightarrow{b})\cdot (5\overrightarrow{a}-4\overrightarrow{b})=0$
$\Rightarrow 5{\left|\overrightarrow{a}\right|}^2-8{\left|\overrightarrow{b}\right|}^2-4\overrightarrow{a}\cdot \overrightarrow{b}+10\overrightarrow{b}\cdot \overrightarrow{a}=0$
$\Rightarrow -3{\left|\overrightarrow{a}\right|}^2+6\overrightarrow{a}\cdot \overrightarrow{b}=0$
$\Rightarrow 6\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =3$
$\Rightarrow \cos \theta =\frac{1}{2}\Rightarrow \theta ={60}^{\circ }$