<s>[INST] Why are vectors of a regular hexagon cos60 of each other? In a regular hexagon OABCDE $a$ , $b$ denote respectively the position vectors of $A$, $B$ with respect to $O$
I would have assumed that $\overrightarrow {OA} = \overrightarrow {AB} = \overrightarrow {BC}  $ since it's a regular hexagon. And, therefore, $\overrightarrow {OC}$ should equal $\overrightarrow {OA} + \overrightarrow {AB} + \overrightarrow {BC}  $ and therefore  $\overrightarrow {3AB}$. 
However, in the literature I was reading, $\overrightarrow{OC}$ is defined as $\overrightarrow {OA}\cos60 + \overrightarrow {AB} + \overrightarrow {BC} \cos60 $, and thus $\frac12  \overrightarrow{AB} + \overrightarrow {AB} + \frac12 \overrightarrow {AB} $  — ultimately $\overrightarrow{2AB}$. Why is this?
An image is included below.

 [/INST] The actual vectors are:
$$\overrightarrow{OA}=\begin{pmatrix}\frac12\\\frac{\sqrt{3}}2\end{pmatrix}$$
$$\overrightarrow{AB}=\begin{pmatrix}1\\0\end{pmatrix}$$
$$\overrightarrow{BC}=\begin{pmatrix}\frac12\\\frac{\sqrt{3}}2\end{pmatrix}$$
So we have $|\overrightarrow{OA}|=|\overrightarrow {AB}|=|\overrightarrow{BC}|$.
And $\overrightarrow{OA}+\overrightarrow {AB}+\overrightarrow{BC}=\begin{pmatrix}\frac12\\\frac{\sqrt{3}}2\end{pmatrix}+\begin{pmatrix}1\\0\end{pmatrix}+\begin{pmatrix}\frac12\\\frac{-\sqrt{3}}2\end{pmatrix}=\begin{pmatrix}2\\0\end{pmatrix}=2\overrightarrow {AB}$.
The literature uses $OA \cos 60^\circ$ to mean $|\overrightarrow{OA}| \cos 60^\circ$, and not $\overrightarrow{OA} \cos 60^\circ$, to mean the length of $\overrightarrow{OA}$, which is $1$.
</s>