Relative to an origin \(O\), the position vectors of two points, \(A\) and \(B\), are \(\begin{pmatrix}2\\5\end{pmatrix}\) and \(\begin{pmatrix}4\\1\end{pmatrix}\) respectively. The point \(C\) is on \(AB\). Given that the position of \(C\) is \(\begin{pmatrix}2t\\t\end{pmatrix}\), find the ratio \(AC:CB\).

A diagram will help. How can we express the fact that \(C\) is on \(AB\)?

The point \(D\) is on \(OC\), between \(O\) and \(C\), and \(\widehat{ADB}\) is a right angle. Given that the position vector of \(D\) is \(\begin{pmatrix}2s\\s\end{pmatrix}\), find the value of \(s\).

If \(AD\) is perpendicular to \(DB\), what can we say about the gradients of \(AD\) and \(DB\)?