Question 1311

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

Question

The plane p{p} passes through the points with coordinates (3,4,1),{\left( 3, 4, - 1 \right),} (10,14,1){\left( 10, 14, - 1 \right)} (1,2,1).{\left( - 1, 2, - 1 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+22=y+75,z=1\frac{x + 2}{-2} = \frac{y + 7}{5}, z = - 1
and the line l2{l_2} has equation
x+125=y+443=z11k,\frac{x + 12}{5} = \frac{y + 44}{3} = \frac{z - 11}{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