In this question, time is in seconds and distances are in metres.
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 \(t\).[1]
Determine whether or not the particles collide.[4]
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 \(QR\)
\(\left|\overrightarrow{RS}\right|=\sqrt5\).
Find the possible values of \(a\) and \(b\).[5]
Solution:
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Answer:(a)(i) \(\begin{pmatrix}-3+4t\\4-2t\end{pmatrix}\) (ii) The particles do not collide. (b) \((a,b)=(1,3)\) or \((-3,5)\)