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Homework Help: Using Cross-Product and Vectors to find the distance between parallel lines ?

  1. Sep 11, 2011 #1
    Given the following two lines, prove that they are parallel, then find the distance between them...

    (I have circled in red the two parts of the answer which I don't understand. Namely, why are they using the cross-product here, doesn't that give you a value perpendicular to the lines, hence a new vector that is on the parallel planes of the lines? The second part I highlighted I have no idea why they chose this.)

    Could someone please explain the reasons to me? Something that I could visualize would be helpful.

    Thanks, prior!

    Distancebetweenparallellines.jpg
     
  2. jcsd
  3. Sep 11, 2011 #2

    LCKurtz

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    Remember the formula

    [tex]\| \vec{AP}\times\vec{AB}\| =\|\vec{AP}\|\|\vec{AB}\|\sin(\theta)[/tex]

    so your formula results in
    [tex]\|\vec{AP}\|\sin(\theta)[/tex]

    Draw a picture of two parallel lines and label A,P, and B and the angle θ between the vectors AP and AB. If you draw a perpendicular line between the vectors you will see that is its length by looking at the triangle with θ the angle between the two vectors.
     
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