Question

if the sum of two vectors is a unit vector , prove that the magnitude of their difference $\sqrt{3}$.

Answer 1

Given, if the sum of teo unit vector is a unit vector

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors

i.e,$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

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

$=1+2.\left(1\right).\left(1\right).\cos \theta +1=1$

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

Now${|\overrightarrow{a}-\overrightarrow{b}|}^2={\overrightarrow{a}}^2+2\overrightarrow{a}\overrightarrow{b}+{\overrightarrow{b}}^2$

$=1-2\left|a\right|.\left|b\right|\cos \theta +1$

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

$=1+1+1=3$

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