2025 SAJC Promo Q8

Relative to the origin \(O\), two points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. It is given that \(|\mathbf{a}|=2\), \(|\mathbf{b}|=1\) and \(|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}\).

1. By considering the scalar product \((3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})\), show that \(\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}\), and give the geometrical meaning of \(|\mathbf{a}\cdot\mathbf{b}|\).[4]
2. By finding the cosine of the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\), deduce the exact value of \(|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|\).[3]
3. Interpret geometrically the value of \(|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|\).[1]
4. Write down, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), a vector equation of the line that passes through \(O\) and bisects the angle \(AOB\).[2]