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Point-Plane Distance?

  1. Aug 26, 2010 #1
    Hey I just have a little concern about the distance between a plane & a point not on the plane.

    In the picture here:
    [PLAIN]http://img838.imageshack.us/img838/4197/normalyy.jpg [Broken]

    we can define the equation of a line passing through B down to the plane and beyond as

    X = B + tN

    If we want to find the equation of the line passing through the point P in the plane
    through to A we'll use the idea of the dot product due to it's orthogonal characteristic.

    (X - A)•N = 0

    I want to point out that X is going to have to be equal to the point P in the picture so

    X = P = B + tN

    and

    (X - A)•N = (P - A)•N = (B + tN - A)•N = 0

    We'll fineagle it to get

    (B + tN - A)•N = 0

    (B - A)•N + tNN = 0

    tNN = - (B - A)•N

    tNN = (A - B)•N

    [tex] t \ = \ \frac{ ( \overline{A} \ - \ \overline{B}) \cdot \overline{N}}{ \overline{N} \cdot \overline{N}} [/tex]

    Then you can sub this into your original equation

    X = B + tN

    To find the actual value at the point P.

    Using this couldn't you just find the norm of this numerical value to find the length of
    the line connecting B to P?


    Then you could go on to use the General Pythagoras Theorem and get

    [tex] || \overline{A} \ + \ \overline{B} || \ = \ || \overline{AP} || \ + \ || \overline{PB} || [/tex]
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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