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Math Help - vector proof using cross product

  1. #1
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    vector proof using cross product

    Let A and B be 2 points on a line and P be a point off the line. Prove that the shortest distance from P to AB is given by the following equation:

    d=\frac{\parallel\vec{AP}\times\vec{AB}\parallel}{  \parallel\vec{AB}\parallel}

    I've tried to solve for d using the Pythagorean theorem and the projection of \vec{AP} onto \vec{AB} for one side and \vec{AP} and d for the other sides, but I'm stumped.

    I can't relate it to the cross product. Can anyone help? Thanks!
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  2. #2
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    Quote Originally Posted by duaneg37 View Post
    Let A and B be 2 points on a line and P be a point off the line. Prove that the shortest distance from P to AB is given by the following equation:

    d=\frac{\parallel\vec{AP}\times\vec{AB}\parallel}{  \parallel\vec{AB}\parallel}
    You need to realize that \left\|\vec{AP}\times\vec{AB}\right\|=\left\|\vec{  AB} \right\|\left\|\vec {AP} \right\|\sin(\theta) where \theta is the angle between the vectors.
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  3. #3
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    I feel stupid for not remembering that! It's plain as day now. Thank you so much!
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