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

Since \(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}\) and \(5 \overline{\mathrm{a}}-4 \overline{\mathrm{b}}\) are perpendicular to each other
\(\therefore(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot(5 \overline{\mathrm{a}}-4 \overline{\mathrm{b}})=0 \)
\( \Rightarrow 5|\overline{\mathrm{a}}|^2-8|\overline{\mathrm{b}}|^2+6 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=0 \)
\( \Rightarrow-3+6|\overline{\mathrm{a}}||\overline{\mathrm{b}}| \cos \theta=0 \quad \ldots[\because|\overline{\mathrm{a}}|=|\overline{\mathrm{b}}|=1] \)
\( \Rightarrow \cos \theta=\frac{1}{2} \)
\( \Rightarrow \theta=\frac{\pi}{3}\)