Question

If the sum of two unit vectors is also a unit vector, then the angle between the two vectors is

• A. $\frac{\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. $\frac{\pi }{4}$
• D. None of these

Answer 1

Answer: (B)

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

Given condition

$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

where

$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=1$

Squaring both sides

${\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2.\left|\overrightarrow{a}\right|.\left|\overrightarrow{b}\right|.\cos \theta ={\left|\overrightarrow{c}\right|}^2$

$1+1+2\cos \theta ={\left|\overrightarrow{c}\right|}^2=1$

$2\cos \theta =-1$

$\cos \theta =\frac{-1}{2}$

$\cos \theta =\cos (\pi -\frac{\pi }{3})$

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