Question

Unit vectors $\hat{P}$ and $\hat{Q}$ are inclined at an angle $\theta$, Prove that $|\hat{P}-\hat{Q}|=2\sin \left(\theta /2\right)$ is?

Answer 1

$|\hat{P}|=|\hat{Q}|=1$, because they are unit vectors.

$LHS=|\hat{P}-\hat{Q}|$

$=\sqrt{(\hat{P}-\hat{Q}).(\hat{P}-\hat{Q})}=\sqrt{(P^2+Q^2-2\hat{P}.\hat{Q})}$

$=\sqrt{1^2+1^2-2PQ\cos \theta }=\sqrt{2-2\times 1\times 1\cos \theta }$

$=\sqrt{2(1-\cos \theta )}$

$=\sqrt{2\left[2{\sin }^2\right(\theta /2\left)\right]}=2\sin \left(\theta /2\right)=RHS$