The position vectors of three points \(O, A\) and \(B\) are \(\begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}3\\3.5\end{pmatrix}\) and \(\begin{pmatrix}6\\-1.5\end{pmatrix}\) respectively. Given that \(\mathbf{OD} = \dfrac{1}{3}\mathbf{OB}\) and \(\mathbf{AE} = \dfrac{1}{4}\mathbf{AB}\) write down the position vectors of \(D\) and \(E\).
How can we express the position vector of \(E\) in terms of things we already know? (A diagram might help.)
Given also that \(OE\) and \(AD\) intersect at \(X\), and that \(\mathbf{OX} = p\:\mathbf{OE}\), and that \(\mathbf{XD} = q\:\mathbf{AD}\), find the position vector of \(X\) in terms of (i) \(p\) (ii) \(q\).
What other ways can we express \(\mathbf{XD}\)? Which of these use the position vector of \(X\)? Can we rearrange this in a helpful way?
Hence calculate \(p\) and \(q\).
How can we derive two simultaneous equations in \(p\) and \(q\) from our work above?
