Question 1308

Vectors II: Lines and Planes

2008 Paper 1 Question 11 Variant

Question

The equations of three planes

{p_1, }

{p_2, }

{p_3 }

are

\begin{alignat*}{6} - 3 x & \,+\, & & \,+\, & 3 z & = - 3, \\ - 4 x & \,+\, & 8 y & \,-\, & 4 z & = - 4, \\ 3 x & \,+\, & \lambda y & \,-\, & 5 z & = \mu, \end{alignat*}

respectively, where

{\lambda}

and

{\mu}

are constants.

When

{\lambda=19.3}

and

{\mu = 3,}

find the coordinates of the point at which the three planes meet.

[2]

The planes

{p_1}

and

{p_2}

intersect in a line

{l.}

(i)

Find a vector equation of

{l.}

[4]

(ii)

Given that

{l}

lies in

{p_3,}

find

{\lambda}

and

{\mu.}

[3]

(iii)

Given instead that

{l}

and

{p_3}

have no point in common, what can be said about the values of

{\lambda}

and

{\mu.}

[2]

(iv)

Find the cartesian equation of the plane which contains

{l}

and the point

{\left( 0, 0, - 4 \right).}

[4]