If $\vec a$ and $\vec b$ are two unit vectors such that $\vec a +2\vec b$ and $5\vec a -4\vec b$ are perpendicular to each other, then the angle between $\vec a$ and $\vec b$ is
Answer & explanation
Correct answer: option 2
Let $\vec u= \vec a + 2\vec b$ and $\vec v=5\vec a-4\vec b$ and let θ be the angle between $\vec a$ and $\vec b$. It is given that $\vec u$ and $\vec v$ are perpendicular to each other. Therefore,
$\vec u.\vec v=0$
$⇒(\vec a+2\vec b).(5\vec a-4\vec b)=0$
$⇒5|\vec a|^2-8|\vec b|^2+10 (\vec a.\vec b)-4(\vec a.\vec b)=0$
$⇒-3+6(\vec a.\vec b)=0$ $[∵|\vec a|=|\vec b|=1]$
$⇒-3+6\cos θ=0$ $[∵\vec a.\vec b=|\vec a||\vec b|\cos θ=\cos θ]$
$⇒\cos θ=\frac{1}{2}⇒ θ=60°$