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Math Help - Are the vectors u,v and w are located in the same equation`Linear Alg prob(Basis)

  1. #1
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    Are the vectors u,v and w are located in the same equation`Linear Alg prob(Basis)

    I have done some progress here.

    What we have to get to know is if

    u=(1,1,1) , v(4,5,-3) and w = (7,8,1)
    are located in the same plane
    the coordinates are located in some basis in the room.


    1. Lambda (1,1,1)e + Mu(4,5-3)e + Upsilon(7,8,0)e
    = (Lambda,Lambda,Lambda)e + (4Mu,5mu,-3Mu)e + (7Upsilon,8Upsilon,0Upsilon)e =
    (0,0,0)e

    <-> (Lambda + 4Mu + 7Upsilon , Lambda + 5Mu + 8 Upsilon , Lambda - 3 Mu + 0)e =(0,0,0)e

    2. This gives us an equation system looking like this:

    Lambda + 4mu + 7 upsilon = 0
    Lambda + 5mu + 8upsilon = 0
    Lambda- 3 Mu = 0


    Lambda = 3mu

    And substituting it for the lambdas in the above equations gives us:

    3mu + 4mu + 7upsilon = 0
    3mu + 5mu + 8 upsilon = 0
    Lambda = 3mu


    <->

    7mu + 7upsilon = 0
    8mu + 8upsilon = 0
    Lambda = 3 mu



    I am quite confused on what kind of conlusion I should make of this

    Are they in the same plane or not?

    How can I show it.


    Thanks.
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  2. #2
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    Re: Are the vectors u,v and w are located in the same equation`Linear Alg prob(Basis)

    Quote Originally Posted by Riazy View Post
    u=(1,1,1) , v(4,5,-3) and w = (7,8,1)
    are located in the same plane
    the coordinates are located in some basis in the room.
    How can I show it.
    The vectors \vec{u},~\vec{v},~\&~\vec{w} are co-planar provided \vec{u}\cdot(\vec{v}\times\vec{w})=0.
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