2023 ACJC P1 Q4

The points \(P\), \(Q\) and \(R\) have position vectors \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) respectively, with respect to the origin \(O\). The vector \(\mathbf{p}\) is a unit vector and the vectors \(\mathbf{q}\) and \(\mathbf{r}\) are non-zero vectors.

1. Write down the geometrical meaning of \(\left| \mathbf{p}\cdot \mathbf{q} \right|\).[1]

It is given that \(\mathbf{p}\cdot \mathbf{q}=2\) and that the point \(M\) is the foot of the perpendicular from \(Q\) to the line \(OP\).

2. Show that \(\overrightarrow{OM}=2\mathbf{p}\).[1]

It is further given that \(\mathbf{r}=k\left( \mathbf{p}-\mathbf{q} \right)\) for an arbitrary constant \(k\).

3. Show that the area of triangle \(RQM\) is \(a\left| \mathbf{p}\times \mathbf{q} \right|\) units\(^{2}\), where \(a\) is a constant to be determined in terms of \(k\).
   Hence describe the geometrical relationship of points \(R\), \(Q\) and \(M\) when \(k=-2\).[4]