1. Given that \(\overrightarrow{OM}=\mathbf{i}+3\mathbf{j}\) and \(\overrightarrow{ON}=\mathbf{i}+\mathbf{j}\), evaluate \(\overrightarrow{OM}.\overrightarrow{ON}\) and hence calculate \(\widehat{MON}\) to the nearest degree.

  2. The position vectors, relative to an Origin \(O\), of two points \(S\) and \(T\) are \(2\mathbf{p}\) and \(2\mathbf{q}\) respectively. The point \(A\) lies on \(OS\) and is such that \(OA=AS\). The point \(B\) lies on \(OT\) produced and is such that \(OT=2TB\). The lines \(ST\) and \(AB\) intersect at \(R\).

    Given that \(\overrightarrow{AR}=\lambda \overrightarrow{AB}\) and that \(\overrightarrow{SR}=\mu \overrightarrow{ST}\), express \(\overrightarrow{OR}\)

    1. in terms of \(\mathbf{p}\), \(\mathbf{q}\) and \(\lambda\), (ii) in terms of \(\mathbf{p}\), \(\mathbf{q}\) and \(\mu\).

    Hence evaluate \(\lambda\) and \(\mu\) and express \(\overrightarrow{OR}\) in terms of \(\mathbf{p}\) and \(\mathbf{q}\).