Question

If the sum of two unit vectors is a unit Vector, then find the magnitude of difference.

Answer 1

Let $\overrightarrow{a}$ & $\overrightarrow{b}$ be two unit vector.

Then the resultant sum of two vector is given by $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=1$

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

it given $|\overrightarrow{a}+\overrightarrow{b}|=1$

$1=\sqrt{1+1+2\cos \theta }$

$\cos \theta =-1/2$

$\theta ={120}^0$

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

$=\sqrt{1+1-2\times \left(\frac{-1}{2}\right)}$

$|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{3}$