Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and $|\overrightarrow{a}+\overrightarrow{b}|=1$ then $|\overrightarrow{a}-\overrightarrow{b}|$ is equal to.

• A. $\sqrt{2}$
• B. $1$
• C. $\sqrt{5}$
• D. $\sqrt{3}$

Answer 1

The correct option is D $\sqrt{3}$

Given $\overrightarrow{a},\overrightarrow{b}$ are unit vectors.

$|a+b|=\sqrt{|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2+2|a||b|\cos \theta }=1$

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

$|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2|a||b|cos\theta }$

$=\sqrt{{(|\overrightarrow{a}+\overrightarrow{b}|)}^2-4|a||b|cos\theta }$

$=\sqrt{1^2-4|a||b|\cos \theta }$

$=\sqrt{1^2-4(-\frac{1}{2})}=\sqrt{3}$

Hence, option D is correct.