Vectors

A subterranean survey records the following data relative to a fixed origin $O$ :

The path of a straight tunnel $l$ starts at the point $A$ with position vector $i+4j−2k\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$
The tunnel passes through the point $B$ with position vector $3i+2j+2k3\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$
A significant mineral deposit is located at the point $P$ with position vector $−4i+j+2k-4\mathbf{i} + \mathbf{j} + 2\mathbf{k}$

Units are measured in decametres.

Find a vector equation for the line $l$ representing the tunnel.

[2]

A side-shaft is planned to run from $P$ to a point $C$ on $l$ such that $P C$ is perpendicular to $l$ . Determine the coordinates of $C$ .

[4]

A pressure sensor is placed at the point $P^{'}$ , where $P^{'}$ is the reflection of $P$ in the line $l$ . Find the coordinates of $P^{'}$ .

[2]

Hence, calculate the distance between the mineral deposit and the sensor, $∣PP′⃗∣|\vec{PP'}|$ , giving your answer as a simplified surd.

[2]