2022 NYJC J1 CT P1 Q5

The two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are related by \(\mathbf{a}=\left( \mathbf{a}\cdot \mathbf{b} \right)\mathbf{b}\). State the relationship between \(\mathbf{a}\) and \(\mathbf{b}\), and find \(\left| \mathbf{b} \right|\).[3]

The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. Show that the point \(Q\) with position vector \(3\mathbf{a}-2\mathbf{b}\) lies on \(BA\) produced. State the ratio \(QA:QB\).

The point \(R\) has position vector \(3\mathbf{a}+5\mathbf{b}\). By finding the area of triangle \(OQR\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\), deduce the ratio of the areas of triangles \(OQR\) to \(OQB\).[6]