2021 DHS P2 Q5

The points \(A\), \(B\) and \(R\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{r}\) respectively.

1. The point \(C\) has position vector \(\frac{2}{7}\mathbf{a}-\frac{3}{7}\mathbf{b}\) and the point \(D\) is such that the origin \(O\) is the midpoint of the line segment \(CD\). The point \(R\) lies on \(BD\) extended such that the ratio of \(BD\) to \(BR\) is \(4:7\). Show that the points \(A\), \(O\) and \(R\) are collinear and state the ratio of \(OA\) to \(OR\).[4]
2. It is given that the point \(R\) has position vector \(\mathbf{r}=\left( \begin{matrix} x \\ y \\ z \\ \end{matrix} \right)\), and that \(\mathbf{a}=\left( \begin{matrix} 1 \\ 3 \\ 2 \\ \end{matrix} \right)\), and \(\mathbf{b}=\left( \begin{matrix} -1 \\ 5 \\ 3 \\ \end{matrix} \right)\).
   1. Determine the exact area of the triangle \(AOB\).[2]
   2. Give the geometrical interpretation of the point \(R\), given that \(\mathbf{r}\cdot \left( \mathbf{a}\times \mathbf{b} \right)=0\).[2]
   3. Find the shortest distance between the point \(\left( -8,-2,9 \right)\) and the collection of all points \(R\) satisfying
      \(\mathbf{r}\cdot \left( \mathbf{a}\times \mathbf{b} \right)=0\).[2]