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Math Help - Find vectors that satisfy the following conditions

  1. #1
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    Find vectors that satisfy the following conditions

    Find vectors \vec{u} and \vec{v} \in \mathbb{R}^3 that simultaneously satisfy all of the following:

    - \vec{u}-\vec{v} = (3,4,10)
    - \vec{u} is parallel to (2,1,5)
    - \vec{v} is perpendicular to (2,1,5)

    I understand what each condition means in terms of how it affects the vectors I am trying to find; however, I can't seem to relate them enough to derive the actual vectors. I know that:

    u_1 - v_1 = 3
    u_2 - v_2 = 4
    u_3 - v_3 = 10

    and

    \vec{u} = (2m,1m,5m)

    also that

    \vec{v}.(2,1,5) = 0

    How do I go about finding the two vectors that satisfy these conditions other than trial and error?

    Thanks.
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  2. #2
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    Re: Find vectors that satisfy the following conditions

    from u1 = 2m, u2 = m, u3 = 5m, and u1-v1 = 3, u2-v2 = 4, u3-v3 = 10 we have:

    v1 = u1 - 3 = 2m - 3
    v2 = u2 - 4 = m - 4
    v3 = u3 - 10 = 5m - 10.

    from 2v1 + v2 + 5v3 = 0, we have:

    2(2m-3) + m-4 + 5(5m-10) = 0, so

    4m - 6 + m - 4 + 25m - 50 = 0
    30m = 60
    m = 2.

    hence u = (4,2,10) and v = (1,-2,0).
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