Question 1311

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

Question

The plane p{p} passes through the points with coordinates (10,5,8),{\left( - 10, 5, - 8 \right),} (2,0,2){\left( - 2, 0, 2 \right)} (1,4,6).{\left( - 1, 4, - 6 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+2=y61=z+102x + 2 = \frac{y - 6}{-1} = \frac{z + 10}{2}
and the line l2{l_2} has equation
x53=y+177=z8k,\frac{x - 5}{- 3} = \frac{y + 17}{7} = \frac{z - 8}{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