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Question 38

A subterranean tunnel follows a linear path $l\,$ connecting two access shafts. Relative to a fixed surveying origin $O$ , where coordinates are measured in metres:

shaft $A\,$ has position vector $i+2j−3k\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$
shaft $B\,$ has position vector $7i−j+0k7\mathbf{i} - \mathbf{j} + 0\mathbf{k}$

(i) Find $AB→\overrightarrow{AB}$ . (ii) Find a vector equation for the line $l\,$ representing the tunnel path.

[3]

A utility sensor is located at point $C\,$ with position vector $5i+3j+2k5\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$ . A maintenance point $P\,$ is located on $l$ .

Given that $CP→\overrightarrow{CP}$ is perpendicular to $l$ :

Find the position vector of the point $P$ .

[4]

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