2010 NJC P1 Q3

The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, relative to the origin \(O\) such that \(\left| \mathbf{a} \right|=\left| \mathbf{b} \right|\). The point \(P\) with position vector \(\mathbf{p}\) lies on \(AB\) such that \(\mathbf{b}\cdot \mathbf{p}=\mathbf{a}\cdot \mathbf{p}\).

1. Show the \(AB\) is perpendicular to \(OP\).[2]
2. Determine the position vector of the point \(D\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\), where \(D\) is the reflection of \(O\) about the line\(AB\).[2]
3. Give the geometrical meaning of \(\left| \mathbf{a}\times \mathbf{b} \right|\).[1]