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(112)=1{\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} = 1} and r(230)=2{\mathbf{r} \cdot \begin{pmatrix} 2 \\ - 3 \\ 0 \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
x+y+2z1+k(2x3y+2)=0.{x + y + 2 z - 1} + {k(2 x - 3 y + 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 (1,2,2){\left( - 1, - 2, - 2 \right)} lie.
[5]

Answer