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Distance Formula in 3 Dimensions

  1. Oct 25, 2009 #1
    Is it possible to change:
    _______
    distance = \/X2 + Y2

    To:
    _______________
    distance = \/X^2 + Y^2 + Z^2

    And get the distance between a point in 3 dimensional space and a the point of origin, just as the first equation does in 2 dimensional space?

    Thanks!
     
  2. jcsd
  3. Oct 25, 2009 #2

    HallsofIvy

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    Yes, that's correct. On way to see that is to "drop" a perpendicular from (x, y, z) to (x, y, 0). The x-axis, from (0,0,0) to (x,0,0), the line from (x,0,0) to (x,y,0), and the direct line from (0,0,0) to (x, y, z) form a right triangle with legs of length x and y and hypotenuse of length [itex]\sqrt{x^2+ y^2}[/itex]. Now that line from (0,0,0) to (x,y,0), the line from (x,y,0) to (x,y,z), and the line from (0,0,0) to (x,y,z) form a right triangle with legs of length [itex]\sqrt{x^2+ y^2}[/itex] and z.

    So you can use the Pythagorean theorem again to get that the distance from (0,0,0) to (x,y,z), the hypotenuse is [itex]\sqrt{(\sqrt{x^2+y^2}^2+ z^2}= \sqrt{x^2+ y^2+ z^2}[/itex].
     
  4. Oct 25, 2009 #3
    Ok cool!
     
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