In analytic geometry, a coordinate system is defined as as pair $(O, \mathcal{B})$ consisting of a point $O$ in the space, called the origin, and a basis $\mathcal{B}=(\overrightarrow{b}_1, \overrightarrow{b}_2, \overrightarrow{b}_3)$ for the space of free vectors.
Since $\mathcal{B}$ is a basis for the space of free vector, every free vector $\overrightarrow{v}$ might be written uniquely as:
$$\overrightarrow{v}=v_1\cdot \overrightarrow{b}_1+v_2\cdot \overrightarrow{b}_2+v_3\cdot \overrightarrow{b}_3.$$ In particular, we have coordinates $v_1, v_2$ and $v_3$ for $\overrightarrow{v}$ with respect $\mathcal{B}$, which I write as $$\overrightarrow{v}=(v_1, v_2, v_3)_{\mathcal{B}}.$$
What is really the role of the origin in the coordinate system?
I mean, once we choose a basis we already have a bijection of the space of free vectors to $\mathbb R^3$.
I know that once we fix the origin $O$ we can find the coordinate of any vector $\overrightarrow{AB}$ in terms of the coordinates of $\overrightarrow{OA}$ and $\overrightarrow{OB}$, for:
$$\overrightarrow{OA}=(a_1, a_2, a_3)_{\mathcal{B}}\ \textrm{and}\ \overrightarrow{OB}=(b_1, b_2, b_3)_{\mathcal{B}}\implies \overrightarrow{OA}=\overrightarrow{OB}-\overrightarrow{OA}=(b_1-a_1, b_2-a_2, b_3-a_3)_{\mathcal{B}}.$$ I believe there is a conceptual gap in my understading of the real meaning of a coordinate system. So, what is the theoretical importance of fixing an origin after all?
Maybe what I'm missing is that the idea of the coordinate system is to establish a bijection between $\mathbb E^3$ (the space of euclidean geometry) and $\mathbb R^3$. In order to do that, we could define:
$$\varphi_{O, \mathcal{B}}: \mathbb E^3\rightarrow \mathbb R^3$$ setting
$$\varphi_{O, \mathcal{B}}(P)=(p_1, p_2, p_3)$$ where $\overrightarrow{OP}=(p_1, p_2, p_3)_{\mathcal{B}}$. Is that it? And, in fact, for defining this bijection is necessary to fix the point $O$.
