The diagram shows a triangle \(O B C\). \(O A: O B=4: 7\) and \(O D: O C=4: 7\). \(\overrightarrow{O B}=\mathbf{b} \text { and } \overrightarrow{O C}=\mathbf{c}\)

The point \(P\) is the point of intersection of \(A C\) and \(B D\) such that \(\overrightarrow{A P}=\lambda \overrightarrow{A C}\) and \(\overrightarrow{B P}=\mu \overrightarrow{B D}\) where \(\lambda\) and \(\mu\) are scalars. (a) Find two expressions for \(\overrightarrow{O P}\), each in terms of \(\mathbf{b}, \mathbf{c}\) and a scalar, and hence show that \(P\) divides both \(A C\) and \(D B\) in the ratio \(4: 7\).

(b) The point \(Q\) is such that \(\overrightarrow{O Q}=\frac{2}{7} \mathbf{b}+\frac{2}{7} \mathbf{c}\).

Use a vector method to show that \(O, Q\) and \(P\) are collinear. Justify your answer.