Question 1314

Vectors II: Lines and Planes

2014 Paper 1 Question 9 Variant

Question

Planes

{p}

and

{q}

are perpendicular. Plane

{p}

has equation

{- y + 4 z = 3.}

Plane

{q}

contains the line

{l}

with equation

x = - 4, \frac{y - 2}{5} = \frac{z - 4}{4}.

The point

{A}

{l}

has coordinates

{\left( - 4, 2, 4 \right).}

(i)

Find a cartesian equation of

{q.}

[4]

(ii)

Find a vector equation of the line

{m}

where

{p}

and

{q}

meet.

[4]

(iii)

{B}

is a general point on

{m.}

Find an expression for the square of the distance

{AB.}

Hence, or otherwise, find the coordinates of the point on

{m}

that is nearest to

{A.}

[5]