Three particles \(A, B\) and \(C\), each of mass \(m\), are moving in a plane such that at time \(t\) their position vectors with respect to the origin \(O\) are
\[\begin{align*}
(2t+1)\mathbf{i}&+(2t+3)\mathbf{j} \\
(10-t)\mathbf{i}&+(12-t)\mathbf{j} \\
(3t^2-4t+1)\mathbf{i}&+ (-3t^2+2t)\mathbf{j}
\end{align*}\]

respectively.

- Show that the centre of mass of these three particles moves in a straight line and find the Cartesian equation of this line.

If we know the positions of the masses, and we know the masses are equal, what is the position vector of the centre of mass?

Find also the value of \(t\) for which the centre of mass is instantaneously at rest.

How does the \(x\)-coordinate of the centre of mass vary with time? When is it not changing? And what about \(y\)?

- Verify that the particles \(A\) and \(B\) are both moving along the straight line with equation \(y=x+2\) and that they collide when \(t = 3\).

How do we find the locus of \(A\) in terms of \(x\) and \(y\)?