2022 RI Promo Q3

The origin \(O\) and the points \(A\), \(B\) and \(C\) lie in the same plane, where \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and \(\overrightarrow{OC}=\mathbf{c}\).

1. Show that if \(\mathbf{c}=\left| \mathbf{a} \right|\mathbf{b}+\left| \mathbf{b} \right|\mathbf{a}\), then \(OC\) bisects the angle \(AOB\).[3]
2. Given that \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\), show that \(\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}\).[2]