Question 1311

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

Question

The plane p{p} passes through the points with coordinates (5,9,14),{\left( 5, - 9, - 14 \right),} (3,0,3){\left( 3, 0, - 3 \right)} (3,3,6).{\left( - 3, 3, 6 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+1=y,z=1x + 1 = y, z = 1
and the line l2{l_2} has equation
x+257=y94=z7k,\frac{x + 25}{- 7} = \frac{y - 9}{4} = \frac{z - 7}{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