Question

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

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

Answer 1

Step 1: Given data
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors whose sum is also a unit vector $\overrightarrow{c}$.
$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

Also, $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=1$

Step 2: Calculating the magnitude of the sum of the two vectors
Now, the magnitude of the sum of $\overrightarrow{a}$ and $\overrightarrow{b}$ is

${\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\left(ab\cos \theta \right)$
$1=1+1+2\left(ab\cos \theta \right)$
$\left(ab\cos \theta \right)=-\frac{1}{2}$

Step 3: Calculating the magnitude of the difference between the two vectors
The magnitude of their difference is given by
${\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2-2\left(ab\cos \theta \right)$
$1+1-2\times \left(-\frac{1}{2}\right)=3$
$\left|\overrightarrow{a}-\overrightarrow{b}\right|=\sqrt{3}$

Therefore, the magnitude of the difference between the two vectors is $\sqrt{3}$.

Hence, the correct answer is option (B)