Question 1311

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

Question

The plane p{p} passes through the points with coordinates (14,4,1),{\left( - 14, - 4, - 1 \right),} (11,2,3){\left( - 11, - 2, 3 \right)} (1,4,9).{\left( 1, 4, 9 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x=y1=z+43x = y - 1 = \frac{z + 4}{3}
and the line l2{l_2} has equation
x72=y287=z+23k,\frac{x - 7}{2} = \frac{y - 28}{7} = \frac{z + 23}{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