Question 1317

Vectors II: Lines and Planes

2017 Paper 1 Question 10 Variant

Question

Some engineers are installing water pipes at a site. Points

{(x,y,z)}

are defined relative to a main outlet at

{(0,0,0), }

where units are kilometres. Pipes are laid in straight lines and the width of the pipes can be neglected.

An existing water pipe

{W}

starts at the main outlet and goes in the direction

{\begin{pmatrix} 1 \\ - 2 \\ - 1 \end{pmatrix}.}

A new water pipe is installed which passes through points

{P \left( 4, 3, 4 \right)}

and

{Q\left( 6, - 5, a \right).}

(i)

Find the value of

{a}

for which

{W}

and the new pipe will meet.

[4]

To ensure that the pipes do not meet, the engineers use

{a=4.}

The engineers wish to connect each of the points

{P}

and

{Q}

to a point

{R}

{W.}

(ii)

The engineers wish to reduce the length of the pipes required and believe in order to do this that angle

{PRQ}

should be

{90^\circ.}

Show that this is not possible.

[4]

(iii)

The engineers discover that the ground between

{P}

and

{R}

is difficult to drill through and now decide to make the length of

{PR}

as small as possible. Find the coordinates of

{R}

in this case and the exact minimum length.

[5]