8 A pipeline is to be drilled under a river (see Fig. 8). With respect to axes Oxyz, with the $x$-axis pointing East, the $y$-axis North and the $z$-axis vertical, the pipeline is to consist of a straight section AB from the point $\mathrm { A } ( 0 , - 40,0 )$ to the point $\mathrm { B } ( 40,0 , - 20 )$ directly under the river, and another straight section BC . All lengths are in metres.

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\caption{Fig. 8}
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(i) Calculate the distance AB .

The section BC is to be drilled in the direction of the vector $3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }$.\\
(ii) Find the angle ABC between the sections AB and BC .

The section BC reaches ground level at the point $\mathrm { C } ( a , b , 0 )$.\\
(iii) Write down a vector equation of the line BC . Hence find $a$ and $b$.\\
(iv) Show that the vector $6 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k }$ is perpendicular to the plane ABC . Hence find the cartesian equation of this plane.

\hfill \mbox{\textit{OCR MEI C4 2007 Q8 [16]}}