Question

If $a,b,c$ are unit vectors satisfying the relation $a+b+\sqrt{3}c=0$, then the angle between $a$ and $b$ is

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

Answer 1

Answer: (C)

Given, $a+b+\sqrt{3}c=0$
$⇒a+b=−\sqrt{3}c...(i)$
$⇒(a+b)â‹\dots (a+b)=(−\sqrt{3}c)(−\sqrt{3}c)$
$⇒âˆ\pounds a{âˆ\pounds }^2+2aâ‹\dots b+âˆ\pounds b{âˆ\pounds }^2=3âˆ\pounds c{âˆ\pounds }^2$
Since, $âˆ\pounds aâˆ\pounds ,âˆ\pounds bâˆ\pounds$ and $âˆ\pounds câˆ\pounds$ are unit vectors.
$∴(1{)}^2+2(1)(1)\cos θ+1=3(1)$
$⇒1+2\cos θ+1=3$
$⇒2\cos θ=3−2$
$⇒2\cos θ=1$
$⇒\cos θ=\frac{1}{2}=\cos \frac{π}{3}$
$⇒θ=\frac{π}{3}$