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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- Calculate the distance AB .
The section BC is to be drilled in the direction of the vector \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\).
- Find the angle ABC between the sections AB and BC .
The section BC reaches ground level at the point \(\mathrm { C } ( a , b , 0 )\).
- Write down a vector equation of the line BC . Hence find \(a\) and \(b\).
- 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.