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(211)=1{\mathbf{r} \cdot \begin{pmatrix} 2 \\ - 1 \\ - 1 \end{pmatrix} = 1} and r(322)=3{\mathbf{r} \cdot \begin{pmatrix} - 3 \\ - 2 \\ 2 \end{pmatrix} = - 3} 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
2xyz1+k(3x2y+2z+3)=0.{2 x - y - z - 1} + {k(- 3 x - 2 y + 2 z + 3)} = 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 (2,3,3){\left( - 2, - 3, 3 \right)} lie.
[5]

Answer