- The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } = \left( \begin{array} { l }
5
3
0
\end{array} \right) + s \left( \begin{array} { l }
3
0
1
\end{array} \right) + t \left( \begin{array} { r }
1
- 2
2
\end{array} \right)$$
where \(s\) and \(t\) are scalar parameters.
- Determine a Cartesian equation for \(\Pi _ { 1 }\)
The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } 5
- 2
3 \end{array} \right) = 1\) - Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
Give your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter.
The plane \(\Pi _ { 3 }\) has Cartesian equation \(4 x - 3 y - z = 0\)
- Use the answer to part (b) to determine the coordinates of the point of intersection of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)