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(233)=3{\mathbf{r} \cdot \begin{pmatrix} - 2 \\ 3 \\ - 3 \end{pmatrix} = - 3} and r(132)=1{\mathbf{r} \cdot \begin{pmatrix} 1 \\ - 3 \\ 2 \end{pmatrix} = - 1} 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+3y3z+3+k(x3y+2z+1)=0.{- 2 x + 3 y - 3 z + 3} + {k(x - 3 y + 2 z + 1)} = 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 (2,1,3){\left( - 2, 1, 3 \right)} lie.
[5]

Answer