Question 1311

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

Question

The plane p{p} passes through the points with coordinates (2,13,7),{\left( - 2, 13, - 7 \right),} (3,1,0){\left( 3, - 1, 0 \right)} (2,5,3).{\left( - 2, 5, - 3 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x3=y32=z+2x - 3 = \frac{y - 3}{-2} = z + 2
and the line l2{l_2} has equation
x+185=y+254=z32k,\frac{x + 18}{- 5} = \frac{y + 25}{- 4} = \frac{z - 32}{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