Question

If $\overrightarrow{x}$ and $ \overrightarrow{y}$ are two unit vectors and $\theta$ is the angle between them, then $\frac{1}{2}|\overrightarrow{x}-\overrightarrow{y}|$ is equal to

• A. 0
• B. $\frac{x}{2}$
• C. $cos\frac{\theta}{2}$
• D. $sin\frac{\theta}{2}$

Answer 1

Answer: (D)

$\frac{1}{4}\left|\vec{x}-\vec{y}\right|^{2}$
$=\frac{1}{4}\left(\bar{x}-\bar{y}\right)^{2}=\frac{1}{4}\left(\vec{x}^{2}+\vec{y}^{2}-2\,\vec{x}\cdot\vec{y}\right)$
$=\frac{1}{4}\left(1+1-2\,cos\,\theta\right)$
$=\frac{2}{4}\left(1-cos\,\theta\right)=sin^{2} \frac{\theta}{2}$
$\therefore \frac{1}{2} \left|\vec{x}-\vec{y}\right|=sin \frac{\theta}{2}$