Question 1311

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

Question

The plane p{p} passes through the points with coordinates (12,4,8),{\left( - 12, 4, - 8 \right),} (0,2,4){\left( 0, - 2, 4 \right)} (13,4,7).{\left( - 13, 4, - 7 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x+4=z+81,y=2x + 4 = \frac{z + 8}{-1}, y = 2
and the line l2{l_2} has equation
x+61=y62=z+2k,\frac{x + 6}{1} = \frac{y - 6}{2} = \frac{z + 2}{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