1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Closest possible points on skew lines

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Find points P,Q which are closest possible with P lying on line:
    x=7-5t, y=-5+11t, z=-3-1t
    and Q lying on line:
    x=-354-8t, y=-194+12t, z=-73+7t

    *the line joining P + Q is perpendicular to the two given lines.

    2. Relevant equations

    Projection formula, cross product....

    3. The attempt at a solution

    So, this is the first time I've seen a problem with skew lines, so am a bit confused how to go about this one.

    I wrote the equations to be :
    X_p = [7,-5,-3] + [-5,11,-1]s
    X_q = [-354,-194,-73] + [-8,12,7]t

    Therefore, the direction vectors are:
    d_p = [-5,11,-1] and
    d_q = [-8,12,7]

    I took the cross product: [-5,11,-1] x [-8,12,7] to get the normal that is perpendicular to both lines, to be:

    I figured I would take a point on Line P, and a point on Line Q, to get an arbitrary vector connecting the two lines; I took the points given in the equation:
    [7,-5,-3] - [-354,-194,-73] to get v = [361,-189,70]

    I then projected this onto the normal, so took:

    Am I correct in doing this? Would this proj_n_v be the distance from P to Q? If it is, what do I do now to get the two points P and Q.

    Am I going about this correctly; I've never encountered this type of problem before, but see it in the practice problems on projections....

    Thanks so much.
  2. jcsd
  3. Nov 24, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Yes, the projection will give you the distance. To get the actual two points, write W=[89,43,28] (your cross product). Then the difference of the two points must be parallel to W. So write X_p(s)-X_q(t)=u*W. That gives you three equations in three unknowns, s,t and u. Solve them. You could also do this more directly by minimizing (X_p(s)-X_q(t))^2. (Take the two partial derivatives wrt s and t and set them to zero.).
  4. Nov 24, 2008 #3
    Thanks for the reply Dick!

    Ahh k, got it now!! )
    Last edited: Nov 24, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook