Question 1311

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

Question

The plane p{p} passes through the points with coordinates (13,2,12),{\left( 13, 2, - 12 \right),} (9,3,6){\left( 9, 3, - 6 \right)} (7,4,8).{\left( 7, - 4, - 8 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+13=z24,y=5\frac{x + 1}{3} = \frac{z - 2}{-4}, y = - 5
and the line l2{l_2} has equation
x102=y+41=z+5k,\frac{x - 10}{2} = \frac{y + 4}{1} = \frac{z + 5}{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