Points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively relative to a point \(O\). Given that \(L\) is the midpoint of \(OA\), and that \(M\) is the point on \(OB\) produced such that \(OM = 3OB\), express \(\overrightarrow{LM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

Given further that \(P\) is the point on \(LM\) such that \(LP = \lambda LM\), express \(\overrightarrow{AP}\) in terms of \(\mathbf{a, b}\) and \(\lambda\).

In the case where \(A, P\) and \(B\) are collinear, calculate the value of

  1. \(\lambda\)
  2. \(\dfrac{AP}{PB}\).