The limiting ratio of the angle and distance between two vectors Consider two unit vectors $u, v$ and name the angle between them as $\theta$. My claim is that
\[
\lim_{\theta \to 0} \frac{\theta}{|u - v|} = 1.
\]

Let us denote : $|u-v| = p$ . According to the Cosine Law we can write :
$p^2=|u|^2+|v|^2-2\cdot |u|\cdot|v|\cdot \cos \theta \Rightarrow$
$\Rightarrow p^2=2 \cdot(1- \cos \theta)=2\cdot 2 \cdot \sin^2 {\frac{\theta}{2}}=4 \cdot \sin^2 {\frac{\theta}{2}} \Rightarrow$
$ \Rightarrow p=2\cdot \sin {\frac{\theta}{2}}$
So we have that :
$\displaystyle \lim_{\theta \to 0} \frac{\theta}{|u-v|}=\displaystyle \lim_{\theta \to 0} \frac{\theta}{2\cdot \sin {\frac{\theta}{2}}}=\displaystyle \lim_{\theta \to 0} \frac{\frac{\theta}{2}}{ \sin {\frac{\theta}{2}}}=\left(\displaystyle \lim_{\theta \to 0} \frac{ \sin {\frac{\theta}{2}}}{\frac{\theta}{2}}\right)^{-1}=1^{-1}=1$
