Assuming that we have two vectors $\vec{a}$ and $\vec{b}$ of comparable size, forming an angle $\theta$, where $\theta$ is very small.
Is there a small-angle approximation for the expression $(\vec{a}+\vec{b})/|\vec{a}+\vec{b}|$ that is is separable in terms of $\vec{a}$ and $\vec{b}$ (i.e. $F(\vec{a}) + G(\vec{b}) + K(\vec{a})L(\vec{b})$ for some functions F, G, K, L)?
I tried Taylor expanding $|\vec{a}+\vec{b}|$ in terms of the angle $\theta$ but with no success.