Question

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 :

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{2}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (C)

$\frac{\pi}{3}$

Answer 2

$\vec{c} = \hat{a} + 2\hat{b}$ $\vec{d} = 5\hat{a} -4\hat{b}$ $\vec{c}.\vec{d} = 0$ $\Rightarrow\quad\left(\hat{a}+2\hat{b}\right).\left(5\hat{a} -4\hat{b}\right) = 0\quad\quad\Rightarrow\quad5+6\hat{a}.\hat{b}-8 = 0$ $\Rightarrow\quad\hat{a}.\hat{b} = \frac{1}{2}\quad\quad\Rightarrow\quad \theta = \frac{\pi}{3}$