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 $\vec{a \text { and }} \vec{b}$ is

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

Answer 1

Answer: (C)

$\frac {2 \pi}{3}$

Answer 2

The correct option is(C): $\frac {2 \pi}{3}$

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
$\Rightarrow (a+ b)^{2}=(c)^{2}$
$\Rightarrow (a)^{2}+(b)^{2}+2 a \cdot b=(c)^{2}$
$\Rightarrow | a |^{2}+| c |^{2}+2 a \cdot b =| c |^{2}$
$\left[\because(a)^{2}=|a|^{2}\right]$
$\Rightarrow 1+1+2 a \cdot b =1$ [from E (i)]
$\Rightarrow 2 a \cdot b =-1$
$\Rightarrow a \cdot b =-1 / 2=| a || b | \cos \theta$
$\Rightarrow \cos \theta=-1 / 2 =\cos 2 \pi / 3$ [from E (i)]
$\Rightarrow \theta =2 \pi / 3$