Vectors - Collinearity Using Vectors

The position vectors of points \(A\) and \(B\) with respect to an origin \(O\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. Show that, for all values of the real constant \(\lambda\), the point \(P\) with position vector \(\mathbf{p}\) given by \(\mathbf{p}=\lambda\mathbf{a}+(1-\lambda)\mathbf{b}\) is collinear with \(A\) and \(B\). Show that the point \(Q\) with position vector \(\mathbf{q}\) given by \(\mathbf{q}=3\mathbf{a}-2\mathbf{b}\) lies on \(BA\) produced, and find the ratio \(QA:QB\).