Question 1311

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

Question

The plane p{p} passes through the points with coordinates (9,9,9),{\left( - 9, 9, - 9 \right),} (3,5,5){\left( 3, 5, - 5 \right)} (5,4,4).{\left( 5, 4, - 4 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+1=y31=z+3x + 1 = \frac{y - 3}{-1} = z + 3
and the line l2{l_2} has equation
x244=y31=z12k,\frac{x - 24}{4} = \frac{y - 3}{1} = \frac{z - 12}{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