If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $\bar{a}+2 \bar{b}$ and $5 \bar{a}-4 \bar{b}$ are…
If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $\bar{a}+2 \bar{b}$ and $5 \bar{a}-4 \bar{b}$ are perpendicular to each other, then the angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ is
$\left(\frac{\pi}{4}\right)$
$\left(\frac{\pi}{3}\right)$
$\cos ^{-1}\left(\frac{1}{3}\right)$
$\cos ^{-1}\left(\frac{2}{7}\right)$
Solution
Since $\overline{\mathrm{a}}+2 \overline{\mathrm{b}}$ and $5 \overline{\mathrm{a}}-4 \overline{\mathrm{b}}$ are perpendicular to each other
$\begin{aligned}
& \therefore \quad(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot(5 \overline{\mathrm{a}}-4 \overline{\mathrm{b}})=0 \\
& \Rightarrow 5|\overline{\mathrm{a}}|^2-8|\overline{\mathrm{b}}|^2+6 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=0 \\
& \Rightarrow-3+6|\overline{\mathrm{a}}||\overline{\mathrm{b}}| \cos \theta=0 \quad \ldots[\because|\overline{\mathrm{a}}|=|\overline{\mathrm{b}}|=1] \\
& \Rightarrow \cos \theta=\frac{1}{2} \\
& \Rightarrow \theta=\frac{\pi}{3}
\end{aligned}$