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

Solution :
Let $$\vec c = \hat a + 2\hat b$$   and $$\vec d = 5\hat a - 4\hat b$$
Since $${\vec c}$$ and $${\vec d}$$ are perpendicular to each other
$$\eqalign{ & \therefore \vec c.\vec d = 0 \Rightarrow \left( {\hat a + 2\hat b} \right).\left( {5\hat a - 4\hat b} \right) = 0 \cr & \Rightarrow 5 + 6\hat a.\hat b - 8 = 0\,\,\left( {\because \vec a.\vec a = 1} \right) \cr & \Rightarrow \hat a.\hat b = \frac{1}{2} \Rightarrow \theta = \frac{\pi }{3} \cr} $$