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Math Help - Help with vectors

  1. #1
    Member Ranger SVO's Avatar
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    Help with vectors

    I was doing good until I got to the last two problems.

    14. Give a unit vector that points in the opposite direction to the vector
    -i + 2k

    16. Find two nonparallel vectors that are perpendicular to i - j + k

    I would like an explination alot more than an answer
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  2. #2
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    Quote Originally Posted by Ranger SVO

    14. Give a unit vector that points in the opposite direction to the vector
    -i + 2k
    To find the unit vector divide by the norm (magnitude) which is,
    ||\bold{v}||=\sqrt{1^2+2^2}=\sqrt{5}
    Thus,
    -\frac{1}{\sqrt{5}}\bold{i}+\frac{2}{\sqrt{5}}\bold  {k}
    In the opposite direction means that is a negative of the vector.
    Multiply by (-1) to get,
    \frac{1}{\sqrt{5}}\bold{i}-\frac{2}{\sqrt{5}}\bold{k}.
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    Quote Originally Posted by Ranger SVO
    16. Find two nonparallel vectors that are perpendicular to i - j + k
    Any vector perpendicular (orthogonal) to another has its dot porduct zero.
    Thus, you want to find a vector perpendicular to the given one.
    If the components of the first vector are v_1,v_2,v_3
    Then, you need that, (dot product)
    v_1-2v_2+v_3=0

    The second vector has components of w_1,w_2,w_3
    Then you need that, (dot product)
    w_1-2w_2+w_3=0

    And they are not parrallet meaning v_k and w_k are not (scalar) multiples of each other.

    There a infinitely many solutions to the first and second equation.
    One such solution is,
    v_1=1,v_2=0,v_3=-1
    v_2=2,v_1=1,w_3=0
    Note they are not multiples.
    Thus,
    \bold{v}=\bold{i}-\bold{k}
    \bold{w}=2\bold{i}-2\bold{k}
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    Member Ranger SVO's Avatar
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    Quote Originally Posted by ThePerfectHacker
    To find the unit vector divide by the norm (magnitude) which is,
    ||\bold{v}||=\sqrt{1^2+2^2}=\sqrt{5}
    Thus,
    -\frac{1}{\sqrt{5}}\bold{i}+\frac{2}{\sqrt{5}}\bold  {k}
    In the opposite direction means that is a negative of the vector.
    Multiply by (-1) to get,
    \frac{1}{\sqrt{5}}\bold{i}-\frac{2}{\sqrt{5}}\bold{k}.
    I should have seen that, thank you very much for your time.
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  5. #5
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    Quote Originally Posted by ImPerfectHacker
    ...
    There a infinitely many solutions to the first and second equation.
    One such solution is,
    v_1=1,v_2=0,v_3=-1
    v_2=2,v_1=1,w_3=0
    Note they are not multiples.
    Thus,
    \bold{v}=\bold{i}-\bold{k}
    \bold{w}=2\bold{i}-2\bold{k}
    Typos here, presumably we should have:

    One such solution is,
    v_1=1,v_2=0,v_3=-1
    w_1=2,w_2=1,w_3=0
    Note they are not multiples.
    Thus,
    \bold{v}=\bold{i}-\bold{k}
    \bold{w}=2\bold{i}+1\bold{j}?

    RonL
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