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 $\bar{a}$ and $\bar{b}$ is
$\frac{\pi}{3}$
$\frac{\pi}{6}$
$\frac{\pi}{4}$
$\frac{2 \pi}{3}$
Solution
Given that, $\overline{\mathrm{a}}+2 \overline{\mathrm{b}}$ and $5 \overline{\mathrm{a}}-4 \overline{\mathrm{b}}$ are perpendicular to each other.
$\begin{array}{ll}
\therefore \quad(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot(5 \overline{\mathrm{a}}-4 \overline{\mathrm{b}})=0 \\
\Rightarrow 5|-\overline{\mathrm{a}}|^2-8|\overline{\mathrm{b}}|^2-4 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+10 \overline{\mathrm{b}} \cdot \overline{\mathrm{a}}=0 \\
\Rightarrow-3+6 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=0 \ldots[|\bar{a}|=|\bar{b}|=1]\\
\Rightarrow 6|\overline{\mathrm{a}}||\overline{\mathrm{b}}| \cos \theta=3 \\
\Rightarrow \cos \theta=\frac{1}{2} \\
\Rightarrow \theta=\frac{\pi}{3}
\end{array}$