Question

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

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

Answer 1

Answer: (C)

Let $\hat{a}$ and $\hat{b}$ are two unit vectors.
Then, $âˆ\pounds \hat{a}+\hat{b}âˆ\pounds =1$
$⇒âˆ\pounds \hat{a}+\hat{b}{âˆ\pounds }^2=1$
$⇒(\hat{a}+\hat{b})â‹\dots (\hat{a}+\hat{b})=1$
$⇒âˆ\pounds \hat{a}{âˆ\pounds }^2+âˆ\pounds \hat{b}{âˆ\pounds }^2+2\hat{a}â‹\dots \hat{b}=1$
$⇒1+1+2\hat{a}â‹\dots \hat{b}=1$
$⇒2\hat{a}â‹\dots \hat{b}=−1$
Again, $âˆ\pounds \hat{a}−\hat{b}{âˆ\pounds }^2=(\hat{a}−\hat{b})â‹\dots (\hat{a}−\hat{b})$
$=âˆ\pounds \hat{a}{âˆ\pounds }^2+âˆ\pounds b{âˆ\pounds }^2−2\hat{a}â‹\dots \hat{b}$
$=1+1−(−1)$
$=1+1+1=3$
$∴âˆ\pounds \hat{a}−\hat{b}âˆ\pounds =\sqrt{3}$