9 A laser beam is aimed from a point ( \(12,10,10\) ) in the direction \(- 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) towards a plane surface.
- Give the equation of the path of the laser beam in vector form.
The points \(\mathrm { A } ( 1,1,1 ) , \mathrm { B } ( 1,4,2 )\) and \(\mathrm { C } ( 6,1,0 )\) lie on the plane.
- Show that the vector \(3 \mathbf { i } - 5 \mathbf { j } + 15 \mathbf { k }\) is perpendicular to the plane and hence find the cartesian equation of the plane.
- Find the coordinate of the point where the laser beam hits the surface of the plane.
- Find the angle between the laser beam and the plane.
\section*{Insert for question 6.}
The graph of \(y = \tan x\) is given below.
On this graph sketch the graph of \(y = \cot x\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes. [4]
\includegraphics[max width=\textwidth, alt={}, center]{23771896-942c-4a1d-ab95-6b6d3cc5643c-5_853_1555_703_262}