2023 HCI P2 Q5

The equations of two planes\({{p}_{1}}\) and \({{p}_{2}}\) are

\(2x+sy+z=-3\),

\(x-2y+3z=1\),

respectively, where \(s\) is a negative real constant.

1. Find a vector parallel to both \({{p}_{1}}\) and \({{p}_{2}}\), giving your answer in terms of \(s\).[2]
2. It is given that \({{p}_{1}}\) and \({{p}_{2}}\) intersect in a line \(l\), and \(l\) meets the \(~xz\)-plane at a point \(A\). Find the coordinates of \(A\).[3]
3. The shortest distance between the origin \(O\) and \({{p}_{1}}\) is \(\frac{3}{\sqrt{6}}\). Find the value of \(s\).[2]

It is now given that \(s=-2\), with the normal vectors to \({{p}_{1}}\) and \({{p}_{2}}\) denoted as \({{\mathbf{n}}_{1}}=\left( \begin{matrix} 2 \\ -2 \\ 1 \\ \end{matrix} \right)\) and \({{\mathbf{n}}_{2}}=\left( \begin{matrix} 1 \\ -2 \\ 3 \\ \end{matrix} \right)\) respectively. It is given that \({{p}_{1}}\) and \({{p}_{2}}\) divide the real space into four regions \({{R}_{1}}\), , and as shown in the following two-dimensional diagram (not drawn to scale).

4. Relative to the origin \(O\), the point \(B\) has position vector \(\mathbf{k}\). Determine whether \(B\) lies in \({{R}_{1}}\), \({{R}_{2}}\), \({{R}_{3}}\) or \({{R}_{4}}\) , justifying your answer.[2]