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Math Help - vectors proof help

  1. #1
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    vectors proof help

    The non-collinear points A, B, C have position vectors a, b, c with respect to an origin O, which is not in the plane ABC. Show that the position vector of any point in the plane ABC can be expressed in the form
    , where , , are real numbers such that .



    I don't really know how to attempt this, i know the parametric form of a plane is r = a + λb + μc where λ,μ are real numbers and b and c are parallel to the plane
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  2. #2
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    Quote Originally Posted by mephisto50 View Post
    The non-collinear points A, B, C have position vectors a, b, c with respect to an origin O, which is not in the plane ABC. Show that the position vector of any point in the plane ABC can be expressed in the form
    , where , , are real numbers such that .



    I don't really know how to attempt this, i know the parametric form of a plane is r = a + λb + μc where λ,μ are real numbers and b and c are parallel to the plane
    It's an affine subspace (in this case a linear manifold not containing the origin) in some vector space.
    You need to show that the space is closed under affine combinations (i.e., linear combinations with the given restriction on the scalars).
    Another way to look at it is that the space is the affine span of {a,b,c}.
    To get started, you might look at what Wiki has to offer regarding "affine subspaces".
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