Question 1314

Vectors II: Lines and Planes
2014 Paper 1 Question 9 Variant

Question

Planes p{p} and q{q} are perpendicular. Plane p{p} has equation 3x+5z=15.{3 x + 5 z = - 15.} Plane q{q} contains the line l{l} with equation x+23=y+1=z55.{\displaystyle \frac{x + 2}{3} = y + 1 = \frac{z - 5}{5}.} The point A{A} on l{l} has coordinates (2,1,5).{\left( - 2, - 1, 5 \right).}
(i)
Find a cartesian equation of q.{q.}
[4]
(ii)
Find a vector equation of the line m{m} where p{p} and q{q} meet.
[4]
(iii)
B{B} is a general point on m.{m.} Find an expression for the square of the distance AB.{AB.}
Hence, or otherwise, find the coordinates of the point on m{m} that is nearest to A.{A.}
[5]

Answer