Definition

Coordinates are the projection of a vector, $\mathbf{r}_{a}=\begin{bmatrix}r_{1} \\ r_{2} \\ r_{3} \end{bmatrix}$, into a specific reference frame, $\vec{\mathcal{F}}_{a}$, resulting in a $3\times1$ column vector of scalars: $$\vec{r}=\vec{\mathcal{F}}_{a}^{\top}\mathbf{r}_{a}$$