$\begin{aligned}|\hat{a}-\hat{b}|^{2} &=(\hat{a}-\hat{b}) \cdot(\hat{a}-\hat{b}) \\ &=\hat{a} \cdot \hat{a}-\hat{b} \cdot \hat{a}-\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{b} \\ &=|\hat{a}|^{2}-2|\hat{a}||\hat{b}| \cos \theta+|\hat{b}|^{2} \\ &=2-2 \cos \theta \\ &=2(1-\cos \theta) \\|\hat{a}-\hat{b}|^{2} &=2 \cdot 2 \sin ^{2} \frac{\theta}{2} \\ \sin \frac{\theta}{2}=\frac{|\hat{a}-\hat{b}|}{2} \end{aligned}$