Question

If the sum of two unit vectors is a unit vector, then the square of magnitude of their difference is

Answer 1

Let $\overrightarrow{a},\overrightarrow{b}$ are two unit vectors,
$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}|=1$
squaring both sides,
$\Rightarrow |\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2+2|\overrightarrow{a}||\overrightarrow{b}|\cos \theta =1\Rightarrow 2\cos \theta =-1\cdots \left(i\right)\because |\overrightarrow{a}|=|\overrightarrow{b}|=1$
now,
$|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2|\overrightarrow{a}||\overrightarrow{b|}\cos \theta }=\sqrt{1+1+1}=\sqrt{3}$
$\therefore |\overrightarrow{a}-\overrightarrow{b}{|}^2=3$