Question 1311

Vectors II: Lines and Planes
2011 Paper 1 Question 11 Variant

Question

The plane p{p} passes through the points with coordinates (0,2,3),{\left( 0, - 2, 3 \right),} (4,4,6){\left( 4, - 4, 6 \right)} (6,1,1).{\left( - 6, 1, - 1 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+22=y+11=z32\frac{x + 2}{2} = \frac{y + 1}{-1} = \frac{z - 3}{2}
and the line l2{l_2} has equation
x+125=y145=z+3k,\frac{x + 12}{5} = \frac{y - 14}{- 5} = \frac{z + 3}{k},
where k{k} is a constant. It is given that l1{l_1} and l2{l_2} intersect.
(ii)
Find the value of k.{k.}
[4]
(iii)
Show that l1{l_1} lies in p{p} and find the coordinates of the point at which l2{l_2} intersects p.{p.}
[4]
(iv)
Find the acute angle between l2{l_2} and p.{p.}
[3]

Answer