Question 1311

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

Question

The plane p{p} passes through the points with coordinates (0,0,1),{\left( 0, 0, - 1 \right),} (0,4,3){\left( 0, - 4, - 3 \right)} (1,7,4).{\left( 1, - 7, - 4 \right).}
(i)
Find a cartesian equation of p.{p.}
[4]
The line l1{l_1} has equation
x5=y+3=zx - 5 = y + 3 = z
and the line l2{l_2} has equation
x+124=y1=z+9k,\frac{x + 12}{- 4} = \frac{y}{1} = \frac{z + 9}{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