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

Answer 1

Answer: (B)

$|\vec{R}|=|\vec{A}+\vec{B}|=\sqrt{A^{2}+B^{2}+2 A B \cos \theta}$
$|\vec{A}|=|\vec{B}|=|\vec{R}|=1$
$1=1+1+2 \times 1 \times 1 \times \cos \theta$
$\cos \theta=-\frac{1}{2}$
$\Rightarrow \theta=120^{\circ}$
$|\vec{R}|=|\vec{A}-\vec{B}|=\sqrt{A^{2}+B^{2}-2 A B \cos 120^{\circ}}$
$=\sqrt{1^{2}+1^{2}-2 \times 1 \times 1 \times\left(-\frac{1}{2}\right)}$
$=\sqrt{3}=|\vec{A}-\vec{B}|$