2018 CJC Promo Q5

Relative to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel. It is given that \(\left| \mathbf{a} \right|=5\), \(\left| \mathbf{b} \right|=10\) and \(\left| 2\mathbf{a}+\mathbf{b} \right|=18\).

1. By using the result \(\mathbf{v}\cdot \mathbf{v}={{\left| \mathbf{v} \right|}^{2}}\), or otherwise, show that \(\mathbf{a}\cdot \mathbf{b}=31\).[2]
2. State the geometrical meaning of \(\left| \mathbf{a}\cdot \frac{\mathbf{b}}{\left| \mathbf{b} \right|} \right|\) and find its numerical value.[2]
3. The points \(P\), \(Q\) and \(R\) have position vectors \(3\mathbf{a}-2\mathbf{b}\), \(4\mathbf{a}+3\mathbf{b}\) and \(7\mathbf{a}+\lambda \mathbf{b}\) respectively. Given that these three points are collinear, find the value of \(\lambda \).[3]