Question 1315

Vectors II: Lines and Planes
2015 Paper 2 Question 2 Variant

Question

The line L{L} has equation
r=(i+3j+4k)+λ(2i2j+k).\mathbf{r} = \left( \mathbf{i} + 3 \mathbf{j} + 4 \mathbf{k} \right) + \lambda \left( - 2 \mathbf{i} - 2 \mathbf{j} + \mathbf{k} \right).
(i)
Find the acute angle between L{L} and the x{x}-axis.
[2]
The point P{P} has position vector 3k.{3 \mathbf{k}.}
(ii)
Find the points on L{L} which are a distance of 6{\sqrt{6}} from P.{P.}
Hence or otherwise find the point on L{L} which is closest to P.{P.}
[5]
(iii)
Find a cartesian equation of the plane that includes the line L{L} and the point P.{P.}
[3]

Answer