Can we show the centre of mass moves in a straight line?

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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.

1. Show that the centre of mass of these three particles moves in a straight line and find the Cartesian equation of this line. Find also the value of \(t\) for which the centre of mass is instantaneously at rest.

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