8 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point 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]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-4_563_716_413_635}
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\caption{Fig. 8}
\end{figure}
- 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).