1 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point \(\mathrm { A } ( 1,2,4 )\), and is reflected at point B off the plane with equation \(x + 2 y - 3 z = 0\), as shown in Fig. 8. \(\mathrm { A } ^ { \prime }\) is the point \(( 2,4,1 )\), and \(M\) is the midpoint of \(\mathrm { AA } ^ { \prime }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b46db958-aa88-47fb-8db3-786472791577-1_562_716_464_650}
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\caption{Fig. 8}
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- Show that \(\mathrm { AA } ^ { \prime }\) is perpendicular to the plane \(x + 2 y - 3 z = 0\), and that M lies in the plane.
The vector equation of the line AB is \(\mathbf { r } = \left( \begin{array} { l } 1
2
4 \end{array} \right) + \lambda \left( \begin{array} { r } 1
- 1
2 \end{array} \right)\). - Find the coordinates of B , and a vector equation of the line \(\mathrm { A } ^ { \prime } \mathrm { B }\).
- Given that \(\mathrm { A } ^ { \prime } \mathrm { BC }\) is a straight line, find the angle \(\theta\).
- Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the \(x\) - and \(z\)-axes)