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

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