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Find a point in a line closest to another point

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Given p(4,2,3), Q(5,3,1), R(3,-5,5) and let L be a line that passes through P and Q
    Find the coordinates of that point of the line L which is closest to the point R.


    2. Relevant equations



    3. The attempt at a solution
    I found the equation of the line L= (x=4+t, y=2+t, z=3-2t) then i am unsure on how to find the coordinates closest to R.
    Any hints on solving this will be appreciated.
    Thanks in Advance
     
  2. jcsd
  3. Nov 9, 2009 #2

    lanedance

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    Homework Helper

    the vector from the closest point of the line to the point will be perpindicular to the line (why?)
     
  4. Nov 9, 2009 #3
    (
    I think i got it now...

    So if we let A(x,y,z) be the closest point from L to R. This condition must be satisfy: RA dot PQ= 0. SO RA = (3-x,-5-y,5-z).
    And from the dot product, i get x+y-2z=-12. So, i sub in the L into the equation I just got and solve for t, and equal t=-2, yields x=2
    y=0,z=7. So A is (2,0,7)?
    Can someone correct me if i am wrong

    Thanks
     
    Last edited: Nov 9, 2009
  5. Nov 9, 2009 #4

    lanedance

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    if vectors a & b are perpindicular then:
    a x b is a non-zero vector
    a . b is zero...
     
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