2025 RVHS P1 Q6

For any vectors \(\mathbf{m}\) and \(\mathbf{n}\), explain why \(\mathbf{m} \cdot (\mathbf{m} \times \mathbf{n}) = 0\).[1]

With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. \(O\), \(A\), \(B\) and \(C\) are non-coplanar. The point \(M\) is the mid-point of \(AC\) and \(p\) denotes the plane \(OAB\).

1. The point \(R\) is such that \(MR\) is perpendicular to \(p\). Show that \(R\) lies on a line \(\ell\) with equation \(\mathbf{r} = k(\mathbf{a}+\mathbf{c}) + \lambda(\mathbf{a} \times \mathbf{b})\), \(\lambda \in \mathbb{R}\), where \(k\) is a constant to be determined.[2]
2. Given that \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors perpendicular to each other, \(\mathbf{a} \cdot \mathbf{c} = -2\) and \(\mathbf{b} \cdot \mathbf{c} = 4\), find in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the position vector of the point of intersection between \(\ell\) and \(p\).[3]