2015 HCI P2 Q1

The position vectors of the fixed points \(A\), \(B\) and \(C\) relative to the origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively.

1. A variable point \(R\) has position vector \(\mathbf{r}\). Describe the locus of \(R\) if \(\mathbf{r}\times (\mathbf{b}-\mathbf{a})=\mathbf{0}\).[2]
2. Given that \(\mathbf{a}=\left( 2\mathbf{a}\cdot \mathbf{b} \right)\mathbf{b}\), state the geometrical relation between \(\mathbf{a}\) and \(\mathbf{b}\), and find \(\left| \mathbf{b} \right|\).[3]
3. A point \(N\) divides \(AB\) in the ratio \(2:1\) and \(M\) is the mid-point of \(BC\). Given that \(O\) is the mid-point of \(MN\), show that \(2\mathbf{a}+7\mathbf{b}+3\mathbf{c}=\mathbf{0}\).[3]