Let $\vec{a}$ and $\vec{b}$ be two unit vectors. If the vectors $\vec{c}=5 \vec{a}-4 \vec{b}$ and $\vec{d}=\vec{a}+2 \vec{b}$ are perpendicular to each other, then the angle between $\vec{a}$ and $\vec{b}$ is:

$\frac{\pi}{3}$

The correct answer is Option (2) - $\frac{\pi}{3}$

$\vec c.\vec d=0$

$⇒(5\vec a-4\vec b)(\vec a+2\vec b)=0$

so $5+10\vec a.\vec b-4\vec a.\vec b-8=0$

$6\vec a.\vec b=3$

$\vec a.\vec b=\frac{1}{2}$

so $\cos θ=\frac{1}{2}⇒θ=\frac{\pi}{3}$