2024 NYJC P2 Q2

With reference to the origin \({O}\), the points \({A}\), \({B}\) and \({R}\) have position vectors \({\mathbf{a}}\), \({\mathbf{b}}\) and \({\mathbf{r}}\) respectively.

The points \({A}\) and \({B}\) are fixed and \({R}\) varies.

1. Interpret geometrically the vector equation \({\mathbf{r} = \lambda \mathbf{a}}\) where \({0 \leq \lambda \leq 1}\).[1]
2. Interpret geometrically the vector equation \({\mathbf{r} \times (\mathbf{a} - \mathbf{b}) = 0}\).[2]
3. Interpret geometrically if \({\mathbf{r} \cdot (\mathbf{r} - \mathbf{a}) = 0}\).[2]
4. Given that \({(\mathbf{r} - \mathbf{a}) \times (\mathbf{r} - \mathbf{b}) = 0}\), find \({\mathbf{r}}\) in terms of \({\mathbf{a}}\) and \({\mathbf{b}}\), showing your working clearly.[2]