11 A 3-D coordinate system, whose units are metres, is set up to model a construction site. The construction site contains four vertical poles \(P _ { 1 } , P _ { 2 } , P _ { 3 }\) and \(P _ { 4 }\). The floor of the construction site is modelled as lying in the \(x - y\) plane and the poles are modelled as vertical line segments. One end of each pole lies on the floor of the construction site, and the other end of each pole is modelled by the points \(( 0,0,18 ) , ( 12,14,20 ) , ( 0,11,7 )\) and \(( 18,2,16 )\) respectively.
A wire, \(S\), runs from the top of \(P _ { 1 }\) to the top of \(P _ { 2 }\). A second wire, \(T\), runs from the top of \(P _ { 3 }\) to the top of \(P _ { 4 }\). The wires are modelled by straight lines segments. The layout of the construction site is illustrated on the diagram below which is not drawn to scale.
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A vector equation of the line segment that represents the wire \(S\) is given by
\(\mathbf { r } = \left( \begin{array} { c } 0 \\ 0 \\ 18 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 7 \\ 1 \end{array} \right) , 0 \leqslant \lambda \leqslant 2\).
- Find, in the same form, a vector equation of the line segment that represents the wire \(T\). The components of the direction vector should be integers whose only positive common factor is 1 .
For the construction site to be considered safe, it must pass two tests.
Test 1: The wires \(S\) and \(T\) need to be at least 5 metres apart at all positions on \(S\) and \(T\). - By using an appropriate formula, determine whether the construction site passes Test 1.
A security camera is placed at a point \(Q\) on wire \(S\).
Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of \(P _ { 3 }\) must be at least 19 m .
- Determine whether it is possible to find point \(Q\) on \(S\) such that the construction site passes Test 2.