Question 1309

Vectors II: Lines and Planes
2009 Paper 1 Question 10 Variant

Question

The planes p1{p_1} and p2{p_2} have equations r(331)=2{\mathbf{r} \cdot \begin{pmatrix} 3 \\ - 3 \\ - 1 \end{pmatrix} = 2} and r(001)=2{\mathbf{r} \cdot \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = 2} respectively, and meet in a line l.{l.}
(i)
Find the acute angle between p1{p_1} and p2.{p_2.}
[3]
(ii)
Find a vector equation of l.{l.}
[4]
(iii)
The plane p3{p_3} has equation
3x3yz2+k(z2)=0.{3 x - 3 y - z - 2} + {k(z - 2)} = 0.
Explain why l{l} lies in p3{p_3} for any constant k.{k.}
Hence, or otherwise, find a cartesian equation of the plane in which both l{l} and the point (1,1,1){\left( 1, - 1, - 1 \right)} lie.
[5]

Answer