2019 TJC P1 Q9

Vectors \(\mathbf{u}\) and \(\mathbf{v}\) are such that \(\mathbf{u}\cdot \mathbf{v}=-1\) and \(\left( \mathbf{u}\times \mathbf{v} \right)+\mathbf{u}\) is perpendicular to \(\left( \mathbf{u}\times \mathbf{v} \right)+\mathbf{v}\).

Show that \(\left| \mathbf{u}\times \mathbf{v} \right|=1\) .[3]

Hence find the angle between \(\mathbf{u}\) and \(\mathbf{v}\) .[3]

The figure shows a regular hexagon \(ABCDEF\) with \(O\) at the centre of the hexagon.

\(X\) is the midpoint of \(BC\).

Given that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\), find \(\overrightarrow{OF}\) and \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[2]

Line segments \(AC\) and \(FX\) intersect at the point \(Y\). Determine the ratio \(AY:YC\).[4]