2023 A-Level H2 Mathematics

Vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that \(\mathbf{a}\cdot \mathbf{b}=-1\). It is also given that \(\left( \mathbf{a}\times \mathbf{b}+\mathbf{a} \right)\) is perpendicular to \(\left( \mathbf{a}\times \mathbf{b}+\mathbf{b} \right)\).

1. Show that \(\left| \mathbf{a}\times \mathbf{b} \right|=1\).[3]
2. Hence, find the angle between the direction of \(\mathbf{a}\) and the direction of \(\mathbf{b}\).[3]