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Math Help - Vectors - Scalar equation of a plane/symmetric equation of a plane problem

  1. #1
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    Vectors - Scalar equation of a plane/symmetric equation of a plane problem

    Hey, thanks for coming into my thread!

    I have a problem involving vectors that I do not know how to start solving.

    Question
    Find the value of k for which the plane  kx + 4y + 2z - 6 = 0 is parallel to the line  \frac{x - 3}{5} = \frac{y}{1} = \frac {z}{-3} .

    Solution
    I know that (k, 4, 2) will be perpendicular to a direction vector on the plane given in the symmetric equation.
    Rearranging the symmetric equation gives me:
     x = 5t + 3
     y = t
     z = -3t

    Which I believe gives me a direction vector on that plane of: (5, 1, -3).

    With this information, how can I find the value of k? If I had a second direction vector of the parallel plane I could do a cross product, so is it possible to find that?

    Thanks!
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  2. #2
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    Quote Originally Posted by Kakariki View Post
    Hey, thanks for coming into my thread!

    I have a problem involving vectors that I do not know how to start solving.

    Question
    Find the value of k for which the plane  kx + 4y + 2z - 6 = 0 is parallel to the line  \frac{x - 3}{5} = \frac{y}{1} = \frac {z}{-3} .

    Solution
    I know that (k, 4, 2) will be perpendicular to a direction vector on the plane given in the symmetric equation.
    Rearranging the symmetric equation gives me:
     x = 5t + 3
     y = t
     z = -3t

    Which I believe gives me a direction vector on that plane of: (5, 1, -3).

    With this information, how can I find the value of k? If I had a second direction vector of the parallel plane I could do a cross product, so is it possible to find that?

    Thanks!
    All your considerations and calculations are OK.

    If the direction vector of the line is perpendicular to the normal vector of the plane then the plane must be parallel to the line.
    Attached Thumbnails Attached Thumbnails Vectors - Scalar equation of a plane/symmetric equation of a plane problem-geradparallelebene.png  
    Last edited by earboth; May 26th 2010 at 10:41 AM. Reason: sketch added
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  3. #3
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    Quote Originally Posted by earboth View Post
    All your considerations and calculations are OK.

    If the direction vector of the line is perpendicular to the normal vector of the plane then the plane must be parallel to the line.
    Would doing a dot product help? setting it equal to zero and then solving for k?
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  4. #4
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    Quote Originally Posted by Kakariki View Post
    Would doing a dot product help? setting it equal to zero and then solving for k?
    Correct!
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  5. #5
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    Quote Originally Posted by earboth View Post
    Correct!
    SWEET! Thank you very much for your help.
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