Parametric vector form of the line

  1. 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
     
  2. jcsd
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