2022 TJC J1 MYE Q8

The position vectors of points \(A\), \(B\) and \(C\) relative to \(O\) are \(\mathbf{a}\),\(\mathbf{b}\) and \(\mathbf{c}\) respectively, where \(\mathbf{a}\),\(\mathbf{b}\) and \(\mathbf{c}\) are non-parallel vectors. The point \(M\) is the midpoint of \(AB\) and the point \(N\) lies on \(AC\) produced such that \(AN:CN=4:3\).

1. Find the position vector of \(M\) and \(N\), giving your answers in terms of \(\mathbf{a}\),\(\mathbf{b}\) and/or \(\mathbf{c}\).[3]
2. The point \(D\) is such that \(ABDN\) is a parallelogram. Find the position vector of \(D\) in terms of vectors \(\mathbf{a}\),\(\mathbf{b}\) and \(\mathbf{c}\).[2]
3. The line \(MN\) and \(AD\) intersect at point \(Q\). Find the position vector of \(Q\) in terms of vector of \(\mathbf{a}\),\(\mathbf{b}\) and \(\mathbf{c}\).[5]