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(213)=1{\mathbf{r} \cdot \begin{pmatrix} - 2 \\ 1 \\ 3 \end{pmatrix} = 1} and r(231)=2{\mathbf{r} \cdot \begin{pmatrix} - 2 \\ 3 \\ - 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
2x+y+3z1+k(2x+3yz+2)=0.{- 2 x + y + 3 z - 1} + {k(- 2 x + 3 y - 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 (1,2,3){\left( - 1, - 2, - 3 \right)} lie.
[5]

Answer