The diagram shows a triangle \(OAB\). The point \(P\) lies on \(AB\). The ratio \(AP:PB\) is \(1:3\).
Given that \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\), find an expression for \(\overrightarrow{OP}\) in terms of \(\mathbf a\) and \(\mathbf b\). Simplify your answer.[2]
Vector \(\mathbf q\) has magnitude \(12\sqrt5\) and direction \(\begin{pmatrix}6\\-3\end{pmatrix}\).
Vector \(\mathbf r\) has magnitude \(15\sqrt2\) and direction \(\begin{pmatrix}-5\\5\end{pmatrix}\).
Find the unit vector in the direction of \(\mathbf q+\mathbf r\).[6]
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