# June 2009 Further maths

#### anillatoo

Having a problem with this...

The line (l1) is parallel to the vector 4j − k and passes through the point A whose position vector is
2i + j + 4k. The variable line (l2) is parallel to the vector i − (2 sin t)j, where 0 ≤ t < 2p, and passes
through the point B whose position vector is i + 2j + 4k. The points P and Q are on (l1)
and (l2), respectively, and PQ is perpendicular to both (l1)and (l2).
(i) Find the length of PQ in terms of t.
(ii) Hence find the values of t for which (l1)and (l2)intersect.
(iii) For the case t = pi/4, find the perpendicular distance from A to the plane BPQ, giving your answer
correct to 3 decimal places. (Answer = 0.219)

I managed to do the first two parts but can someone help me how to find the parametric equation of plane BPQ, so that I can then use the distance formula to find the perpendicular distance.(No need to do the first 2 parts to be able to do the final part)

#### TheQuantiser

(iii)
Find the normal to the plane BPQ using these properties:

* B and Q both lie on l2, so BQ is parallel to the direction vector of l2
* PQ is parallel to the cross product of the direction vectors of the two lines, l1 and l2

Find the cross product of BQ and PQ. This is the normal to the plane BPQ. Since you have the coordinates of B, you can find the equation of this plane.

The distance from A to this plane is then found by using the formula
b.p - a.p
where p is the unit normal to the plane, and a and b are the position vectors of A and B.