2023 RI P2 Q4

With reference to the origin \(O\), the points \(A\) and \(B\) are such that \(\mathbf{a}=\overrightarrow{OA}\) and \(\mathbf{b}=\overrightarrow{OB}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are two non-zero and non-parallel vectors.

1. Show that the vector \(\mathbf{n}=\left( \mathbf{a}\cdot \mathbf{b} \right)\mathbf{a}-\left( \mathbf{a}\cdot \mathbf{a} \right)\mathbf{b}\) is perpendicular to \(\mathbf{a}\) and is parallel to the plane \(OAB\).[4]

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are now given by \(\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k}\) and \(\mathbf{b}=2\mathbf{i}+\mathbf{j}+3\mathbf{k}\).

2. Find a unit vector \(\mathbf{m}\) which is perpendicular to \(\mathbf{a}\) and is parallel to plane \(OAB\).[2]
3. Find the coordinates of the point of intersection between the line \(OA\) and the line which passes through point \(B\) and is parallel to \(\mathbf{m}\).[2]
4. Hence, or otherwise, find an equation of the line which is a reflection of the line \(OB\) in the line \(OA\).[3]