Shortest Distance between 2 skew lines (vectors)

  1. 1. The problem statement, all variables and given/known data

    x1 = (0,0,4) + s(2,0,-1)
    x2 = (-4,2,2) + t(-5,1,1)

    2. Relevant equations




    3. The attempt at a solution
    First of all I find the common perpendicular (vector cross product)
    Make it unit
    Find a arbitrary line joining the two 2 lines (set s = t= 0)

    Then scalar produce the unit normal and the line between the vectors to get the minimum distance. I get 2/sqrt(14), while the answers say just sqrt(14). How so?

    Also I am just doing the method, but what does it mean geometrically. Why find a line perpendicular, or use a unit vector?

    Thanks
    Thomas
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,549
    Staff Emeritus
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    The shortest distance from a point to a line is always along the line perpendicular to the given line. That is because the hypotenuse of a right triangle is longer than either leg.

    As a result of that, the shortest distance between two skew lines is along the line perpendicular to both given lines.
     
  4. Ray Vickson

    Ray Vickson 6,190
    Science Advisor
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    I did it another way: I expressed the distance^2 between x1 and x2 as a function of s and t, then minimized it. My distance answer agrees with yours.

    RGV
     
  5. I know that the shortest distance between 2 skew lines is perpendicular, but why does finding the "common perpendicular" help. Let's go to a 2D graph. Let there be 2 parallel lines (2D equiv of skew lines). Taking the vector product produces a vector into or out of the graph (from our view on a 2D graph you cannot see the vector because it goes "into" or "out of" the graph).
    How does finding this vector help. For that matter, how does finding an arbitray line between the 2 vectors help

    Thanks
    Thomas
     
  6. That's how the problem sheet answer is done (I don't like doing it that way... I DON'T understand why you square it) but they get sqrt(14). I get 2/sqrt(7) or sqrt(7/2) . Hmm.
     
    Last edited: Jun 1, 2011
  7. Your answer is correct.
    Geometrically meaning: imagine the 2 parallel planes that each contains one of the lines.
    Easier: imagine 2 parallel planes and on each plane draw one line.
    Connect the 2 lines by a 3rd line AB. In A draw the perpendicular of the 2 planes.
    The length of the perpendicular is the projection of AB onto the perpendicular of the planes in A (or B, the same).
     
  8. Ray Vickson

    Ray Vickson 6,190
    Science Advisor
    Homework Helper

    Minimizing d(s,t)^2 gives the same solution as minimizing d(s,t). What could be simpler?

    RGV
     
  9. why would you want to minimize d(s,t)^2 and not d(s,t). am I missing something stupidly ovbious?
     
  10. SammyS

    SammyS 8,458
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    Gold Member

    The square of the distance does not have square roots, so it's much easier to work with.
     
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