Obtain an expression for the position vector of the centre of mass of a system of $n$ particles in two dimensions.

(N/A) For a system of $n$ particles with masses $m_{1}, m_{2}, ..., m_{n}$ located at position vectors $\vec{r}_{1}, \vec{r}_{2}, ..., \vec{r}_{n}$ in a two-dimensional plane,the position vector of the centre of mass $\vec{R}$ is defined as:
$\vec{R} = \frac{\sum_{i=1}^{n} m_{i} \vec{r}_{i}}{\sum_{i=1}^{n} m_{i}}$
In terms of Cartesian coordinates $(x, y)$,where $\vec{r}_{i} = (x_{i}, y_{i})$ and $\vec{R} = (X, Y)$,the components are:
$X = \frac{\sum_{i=1}^{n} m_{i} x_{i}}{\sum_{i=1}^{n} m_{i}}$
$Y = \frac{\sum_{i=1}^{n} m_{i} y_{i}}{\sum_{i=1}^{n} m_{i}}$
Thus,the position vector of the centre of mass is given by $\vec{R} = (X, Y) = \left( \frac{\sum m_{i} x_{i}}{\sum m_{i}}, \frac{\sum m_{i} y_{i}}{\sum m_{i}} \right)$.