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

Answer 1

Answer: (B)

${60}^{\circ }$

We have

$\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $5\overrightarrow{a}-4\overrightarrow{b}$
$\Rightarrow$ their dot product has to be $0$.
$\Rightarrow 5a^2-4\overrightarrow{a}.\overrightarrow{b}+10\overrightarrow{a}.\overrightarrow{b}-8b^2=0$
$\Rightarrow 5+6\cos \theta -8=0$
$\Rightarrow \cos \theta =\frac{1}{2}$
So, $\theta ={60}^o$