2023 NJC Promo Q1

The equations of three planes \({{p}_{1}}\), \({{p}_{2}}\) and \({{p}_{3}}\) are

\(\mathbf{r}\cdot \left( \begin{matrix} 3 \\ 2 \\ -5 \\ \end{matrix} \right)=3\), \(\mathbf{r}\cdot \left( \begin{matrix} -5 \\ 3 \\ 2 \\ \end{matrix} \right)=-5\) and \(\mathbf{r}\cdot \left( \begin{matrix} 17 \\ 5 \\ a \\ \end{matrix} \right)=b\)

respectively, where \(a\) and \(b\) are constants.

If \(a=-20.9\) and \(b=16.6\), find the coordinates of the point at which these planes meet.[2]

The planes \({{p}_{1}}\) and \({{p}_{2}}\) intersect in a line \(l\).

1. Find a vector equation of \(l\).[2]
2. Given that all three planes meet in the line \(l\), find the values of \(a\) and \(b\).[3]