Question 1316

Vectors II: Lines and Planes
2016 Paper 1 Question 11 Variant

Question

The plane p{p} has equation
r=(130)+λ(534)+μ(a34),\mathbf{r} = \begin{pmatrix} - 1 \\ - 3 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} - 5 \\ - 3 \\ 4 \end{pmatrix}+\mu\begin{pmatrix} a \\ 3 \\ - 4 \end{pmatrix},
and the line l{l} has equation
r=(a7aa9)+t(043),\mathbf{r}=\begin{pmatrix} a - 7 \\ a \\ a - 9 \end{pmatrix}+t\begin{pmatrix} 0 \\ 4 \\ 3 \end{pmatrix},
where a{a} is a constant and λ,μ{\lambda, \mu} and t{t} are parameters.
(i)
In the case where a=4,{a=4,}
(ia)
Show that l{l} is perpendicular to p{p} and find the values of λ,μ{\lambda, \mu} and t{t} which give the coordinates of the point at which l{l} and p{p} intersect,
[5]
(ib)
find the cartesian equations of the planes such that the perpendicular distance from each plane to p{p} is 8.{8.}
[5]
(ii)
Find the value of a{a} such that l{l} and p{p} do not meet in a unique point.
[3]

Answer