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Math Help - Vector problem involving orthogonal

  1. #1
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    Vector problem involving orthogonal

    Hi, I'm struggling with this problem:

    For \vec{u} = [-4, 1, 10]^T and \vec{v} = [−12, −6, 8]^T find the vectors  \vec{u1} and \vec{u2} such that:

    (i) \vec{u1} is parallel to \vec{v}
    (ii) \vec{u2} is orthogonal to \vec{v}
    (iii) \vec{u} = \vec{u1} + \vec{u2}

    I figured I should firstly try and find u2 by (ii), and then after I found that I would be able to use (iii) to get u1. This approach didn't work out too well for me, heh. Basically I tried to set it up with dot product, and solve for u2:

     \vec{v} * \vec{u2} = 0

    Didn't work out, just ended up with something like: -12a - 6b + 8c = 0
    Where a, b, c are the numbers in u2.

    From there I couldn't see what more I could do to find a, b, c. Basically, I don't really know how to approach this problem .

    Anyone mind helping out a math newbie? Thanks in advance!
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  2. #2
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    Quote Originally Posted by cb220 View Post
    Hi, I'm struggling with this problem:

    For \vec{u} = [-4, 1, 10]^T and \vec{v} = [−12, −6, 8]^T find the vectors  \vec{u1} and \vec{u2} such that:

    (i) \vec{u1} is parallel to \vec{v}
    (ii) \vec{u2} is orthogonal to \vec{v}
    (iii) \vec{u} = \vec{u1} + \vec{u2}

    I figured I should firstly try and find u2 by (ii), and then after I found that I would be able to use (iii) to get u1. This approach didn't work out too well for me, heh. Basically I tried to set it up with dot product, and solve for u2:

     \vec{v} * \vec{u2} = 0

    Didn't work out, just ended up with something like: -12a - 6b + 8c = 0
    Where a, b, c are the numbers in u2.

    From there I couldn't see what more I could do to find a, b, c. Basically, I don't really know how to approach this problem .

    Anyone mind helping out a math newbie? Thanks in advance!
    Let

    \vec{u}_1=\text{Proj}_{\vec{v}}\vec{u}=\frac{\vec{  u}\cdot \vec{v}}{||v||^2}\vec{v}

    Then

    \vec{u}_2=\vec{u}-\vec{u}_1
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  3. #3
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    Quote Originally Posted by cb220 View Post
    Hi, I'm struggling with this problem:

    For \vec{u} = [-4, 1, 10]^T and \vec{v} = [−12, −6, 8]^T find the vectors  \vec{u1} and \vec{u2} such that:

    (i) \vec{u1} is parallel to \vec{v}
    (ii) \vec{u2} is orthogonal to \vec{v}
    (iii) \vec{u} = \vec{u1} + \vec{u2}

    I figured I should firstly try and find u2 by (ii), and then after I found that I would be able to use (iii) to get u1. This approach didn't work out too well for me, heh. Basically I tried to set it up with dot product, and solve for u2:

     \vec{v} * \vec{u2} = 0

    Didn't work out, just ended up with something like: -12a - 6b + 8c = 0
    Where a, b, c are the numbers in u2.

    From there I couldn't see what more I could do to find a, b, c. Basically, I don't really know how to approach this problem .

    Anyone mind helping out a math newbie? Thanks in advance!
    That's not a bad start! Yes, to satisfy (ii) you want -12a- 6b+ 8c= 0.
    And to satisfy (iii) you want u1= <-4- a, 1- b, 10- c>.
    And, then, to satisfy (i) you want u1 to be a multiple of v: u1= <-4-a, 1- b, 10- c>= d<12, 6, 8>. That is you have four equations, -12a- 6b+ 8c= 0, -4-a= 12d, 1- b= 6d, and 10- c= 8d, for the four numbers, a, b, c, and d.
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