2021 TJC Promo Q2

The position vectors of the points \(A\) and \(B\) relative to \(O\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero and non-parallel vectors. The point \(C\), with position vector \(\mathbf{c}\), is the reflection of \(A\) in the line \(OB\). Given that \(\mathbf{b}\) is a unit vector,

1. Show that \(\mathbf{c}=2\left( \mathbf{a}\cdot \mathbf{b} \right)\mathbf{b}-\mathbf{a}\),[2]
2. Show that \(\mathbf{c}\times \mathbf{a}=2\left( \mathbf{a}\cdot \mathbf{b} \right)\left( \mathbf{c}\times \mathbf{b} \right)\).[2]