# Parametric vector form of the line

1. ### ronho1234

34
The parametric vector form of the line 1
 is given as r1 = u1 + rv1 (r element of real field)
where u1 is the position vector of P1 = (1,1,−3) and v1=vectorP1P2

where
P2 = (3,3,−2) .
The parametric vector form of the line 2
 is given as r2 = u2 + sv2 (s element of real field)
where 2 u is the position vector of P3 = (−2,0,2) and
v2= −j− k .
(a) Give the parametric scalar equations of the lines l1
 and l2
 .
(b) Find the unit vector ˆn with negative i component which is perpendicular to
both l1
 and l2

(c) The shortest distance between two lines is the length of a vector that
connects the two lines and is perpendicular to both lines. For l1
 and l2

this is expressed in the vector equation 2 1
r −r = tnˆ where t element of real field is a
parameter. Write this equation as 3 scalar equations and hence obtain a
system of three linear equations for the three parameters r, s and t .
(d) Solve this system of equations for r, s and t and hence find the shortest
distance between the two lines 1
 and 2
 .
(e) Find the point Q on line 1
 which is closest to line 2

I've done bits and pieces of the question bu i'm especially stuck on parts c d and e. please help thank you