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Math Help - Finding decomposition of vectors

  1. #1
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    Finding decomposition of vectors

    Hi guys, this is part c of a series of problems, where I'm now stuck.

    Let P be the plane 2x + y - z = 1

    Point Q is at (0, 0, -1)

    Take n = 2i + j - k as a normal vector of the plane P.


    Decompose the vector QO into the sum of two vectors; one of them is parallel to n and the other one is orthogonal to n.

    Any help greatly appreciated!
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  2. #2
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    Re: Finding decomposition of vectors

    Well QO = -OQ = (0, 0, 1). If you want to write this as the sum of two vectors (one of which you know, n, and one of which you don't, a), you have n + a = -OQ. Surely you can figure out what a has to be.
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  3. #3
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    Re: Finding decomposition of vectors

    Of course you should also check that the two vectors are orthogonal too...
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    Re: Finding decomposition of vectors

    Quote Originally Posted by SydneyGuy View Post
    Decompose the vector QO into the sum of two vectors; one of them is parallel to n and the other one is orthogonal to n.
    Given two non-zero, non-parallel vectors it possible to do such a decomposition:

    $\overrightarrow {{B_\parallel }} = \dfrac{{\overrightarrow A \cdot \overrightarrow B }}{{\overrightarrow B \cdot \overrightarrow B }}\overrightarrow B \,\& \,\overrightarrow {{B_ \bot }} = \overrightarrow A - \overrightarrow {{B_\parallel }} $
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  5. #5
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    Re: Finding decomposition of vectors

    is the answer 1/6(2i+j-k) for parallel and -1/6(2i+j+5k) for orthogonal?. If so, do we get the 2i+j+5k with trial and error? how would we explain in our answer how we got 2i+j+5k?
    Thanks
    Last edited by selsunblue; April 27th 2014 at 05:27 PM.
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