Question 1316

Vectors II: Lines and Planes

2016 Paper 1 Question 11 Variant

Question

The plane

{p}

has equation

\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 3 \\ 2 \end{pmatrix}+\mu\begin{pmatrix} a \\ 5 \\ 3 \end{pmatrix},

and the line

{l}

has equation

\mathbf{r}=\begin{pmatrix} a - 8 \\ a - 4 \\ a - 5 \end{pmatrix}+t\begin{pmatrix} 1 \\ 4 \\ - 8 \end{pmatrix},

where

{a}

is a constant and

{\lambda, \mu}

and

{t}

are parameters.

(i)

In the case where

{a=4,}

(ia)

Show that

{l}

is perpendicular to

{p}

and find the values of

{\lambda, \mu}

and

{t}

which give the coordinates of the point at which

{l}

and

{p}

intersect,

[5]

(ib)

find the cartesian equations of the planes such that the perpendicular distance from each plane to

{p}

{3.}

[5]

(ii)

Find the value of

{a}

such that

{l}

and

{p}

do not meet in a unique point.

[3]