Question

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

• A. $\sqrt { 2 }$
• B. $\sqrt { 3 }$
• C. $1 / \sqrt { 2 }$
• D. $\sqrt { 5 }$

Answer 1

Answer: (B)

$\sqrt { 3 }$

Answer 2

Let $\hat { n } _ { 1 }$ and $\hat { n } _ { 2 }$ are the two unit vectors, then the sum is

$n _ { s } ^ { 2 } = n _ { 1 } ^ { 2 } + n _ { 2 } ^ { 2 } + 2 n _ { 1 } n _ { 2 } \cos \theta$

$= 1 + 1 + 2 \cos \theta$

Since it is given that $1 = 1 + 1 + 2 \cos \theta$

or $\theta = 120 ^ { \circ }$

Now the difference vector is $n _ { d } = n _ { 1 } - n _ { 2 }$ or

$n _ { d } ^ { 2 } = n _ { 1 } ^ { 2 } + n _ { 2 } ^ { 2 } - 2 n _ { 1 } n _ { 2 } \cos \theta$ $= 1 + 1 - 2 \cos \left( 120 ^ { \circ } \right)$

∴ $\Rightarrow n _ { d } = \sqrt { 3 }$