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Math Help - Having difficulty with a proof - distance from a point to a line

  1. #1
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    Having difficulty with a proof - distance from a point to a line

    Hey!

    I am having trouble proving something, and I was hoping someone here could help me out!

    Question
    Prove that the distance from a point Q in space to a line through a point P with direction vector d is equal to  \frac{|PQ \times d|}{|d|} .

    Solution
    What I have learned about this topic so far is: to find the smallest distance from a point to a line you find the equation of a line that passes through the point and is perpendicular to the line.
    I really have no idea where to start with this proof, any help is greatly appreciated!
    Last edited by Kakariki; June 6th 2010 at 07:33 AM.
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  2. #2
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    To get the cross-product symbol:
    [tex]PQ\times d[/tex] gives PQ\times d
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  3. #3
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    Quote Originally Posted by Plato View Post
    To get the cross-product symbol:
    [tex]PQ\times d[/tex] gives PQ\times d
    Thanks for the info! Any advice you can give with regards to the question I am having difficulty with?
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  4. #4
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    And does P lies on the line with direction d?
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    Make a drawing, you will find that the distance is \left| PQ sin(\theta) \right|.
    Hence, play with PQ\times d
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  6. #6
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    It should be obvious if you notice that |PQ \times d| is the area of a parallelogram and |d| is the length of one of its sides. A plot will help to see the solution.
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    Quote Originally Posted by Kakariki View Post
    Thanks for the info! Any advice you can give with regards to the question I am having difficulty with?
    We know that \sin (\phi ) = \frac{{\left\| {\overrightarrow {QP}  \times d} \right\|}}{{\left\| {\overrightarrow {QP} } \right\|\left\| d \right\|}} where \phi is the angle between  \overrightarrow {QP} and the line.
    From that we see  D(P;\ell ) = \left\| {\overrightarrow {QP} } \right\|\sin (\phi ) = \frac{{\left\| {\overrightarrow {QP}  \times d} \right\|}}{{\left\| d \right\|}}
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