Question

If the sum of two unit vectors is also a unit vector, then magnitude of their difference and angle between the two given unit vectors is

• A. $\sqrt{3},60^{∘}$
• B. $\sqrt{3},120^{∘}$
• C. $\sqrt{2},60^{∘}$
• D. $\sqrt{2},120^{∘}$

Answer 1

Answer: (B)

$âˆ\pounds \vec{R}âˆ\pounds =âˆ\pounds \vec{A}+\vec{B}âˆ\pounds =\sqrt{A^2+B^2+2AB\cos θ}$
$âˆ\pounds \vec{A}âˆ\pounds =âˆ\pounds \vec{B}âˆ\pounds =âˆ\pounds \vec{R}âˆ\pounds =1$
$1=1+1+2×1×1×\cos θ$
$\cos θ=−\frac{1}{2}$
$⇒θ=120^{∘}$
$âˆ\pounds \vec{R}âˆ\pounds =âˆ\pounds \vec{A}−\vec{B}âˆ\pounds =\sqrt{A^2+B^2−2AB\cos 120^{∘}}$
$=\sqrt{1^2+1^2−2×1×1×\left(−\frac{1}{2}\right)}$
$=\sqrt{3}=âˆ\pounds \vec{A}−\vec{B}âˆ\pounds$