4.01: Particle motion and position vectors

At time \(t=0\), particle \(A\) starts at the origin and moves with velocity \(\begin{pmatrix}2\\1\end{pmatrix}\). At the same time, particle \(B\) starts from the point with position vector \(\begin{pmatrix}-3\\4\end{pmatrix}\) and moves with velocity \(\begin{pmatrix}4\\-2\end{pmatrix}\). Write down the position vector of \(B\) at time \(t\).

Determine whether or not the particles collide.

The point \(Q\) has position vector \(3\mathbf{i}+2\mathbf{j}\). The point \(R\) has position vector \(-\mathbf{i}+4\mathbf{j}\). The point \(S\) has position vector \(a\mathbf{i}+b\mathbf{j}\), where \(a\) and \(b\) are integers. It is given that \(S\) lies on the straight line \(QR\) and \(|\overrightarrow{RS}|=\sqrt5\). Find the possible values of \(a\) and \(b\).