Question

Let $\vec{a}$ and $\vec{b}$ be two equal vectors inclined at an angle $\theta$ , then $asin\frac{\theta }{2}$ is equal to

• A. $\frac{|\vec{a}-\vec{b}|}{2}$
• B. $\frac{|\vec{a}+\vec{b}|}{2}$
• C. $\left|\vec{a}-\vec{b}\right|$
• D. $\left|\vec{a}+\vec{b}\right|$

Answer 1

Answer: (A)

$\left|\vec{a}-\vec{b}\right|=\sqrt{{\vec{a}}^2+{\vec{b}}^2-2\left|\vec{a}\right|\cdot \left|\vec{b}\right|cos\theta }$
Here, $\vec{a}=\vec{b}$
$\therefore \left|\vec{a}-\vec{b}\right|=\sqrt{{\vec{a}}^2+{\vec{a}}^2-2{\vec{a}}^{2}cos\theta }$
$=\sqrt{2{\vec{a}}^2\left(1-cos\theta \right)}=\sqrt{2a^2\cdot 2si{n}^2\frac{\theta }{2}}$
$=2asin\frac{\theta }{2}$
$\Rightarrow \left|\vec{a}-\vec{b}\right|=2asin\frac{\theta }{2}$
$\Rightarrow asin\frac{\theta }{2}=\frac{|\vec{a}-\vec{b}|}{2}$