2026 EJC P2 Q4

Referred to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf a\) and \(\mathbf b\) respectively. \(O\), \(A\) and \(B\) are not collinear. Point \(C\) has position vector \(2\mathbf a+\mathbf b\) and point \(D\) lies on \(AB\) extended such that \(AB=BD\).

1. Show that the position vector of \(D\) is \(2\mathbf b-\mathbf a\).[2]
2. Given that \(OC=OD\), use suitable scalar products to show that \(8\mathbf a\cdot\mathbf b=3|\mathbf b|^2-3|\mathbf a|^2\).[2]
3. Given also that \(|\mathbf b|=2|\mathbf a|\), show that \(\cos\theta=\dfrac9{16}\), where \(\theta\) is the angle between \(\mathbf a\) and \(\mathbf b\).[2]
4. Given further that \(\mathbf a\) is a unit vector, find the exact area of triangle \(OAC\).[4]