Question

If $\vec{a},\vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$ then angle between $\overrightarrow{a\text{ and }}\vec{b}$ is

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

Answer 1

Answer: (C)

Given, $a,b,c$ are unit vectors.
$\Rightarrow |a|=|b|=|c|=1$...(i)
Also, given $a+b+c=0$
$\Rightarrow (a+b)=-c$
Squaring on both sides, we get
$\Rightarrow (a+b{)}^2=(c{)}^2$
$\Rightarrow (a{)}^2+(b{)}^2+2a\cdot b=(c{)}^2$
$\Rightarrow |a{|}^2+|c{|}^2+2a\cdot b=|c{|}^2$
$\left[\because (a{)}^2=|a{|}^2\right]$
$\Rightarrow 1+1+2a\cdot b=1$ [from Eq. (i)]
$\Rightarrow 2a\cdot b=-1$
$\Rightarrow a\cdot b=-1/2=|a||b|\cos \theta$
$\Rightarrow \cos \theta =-1/2=\cos 2\pi /3$ [from Eq. (i)]
$\Rightarrow \theta =2\pi /3$