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Projecting a vector onto a plane problem

  1. Aug 29, 2004 #1


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    how would one go about spanning a vector say 'u' onto a plane spanned by vectors v1 and v2.

    I have a formula for projecting a vector onto say a subspace w:
    projw(u) = <u,v1>v1 + <u,v2>v2 + .... <u,vn>vn
    But I'm unsure how to use this for when I need to project the vector onto a plane spanned by 2 other vectors.

  2. jcsd
  3. Aug 29, 2004 #2
    dcl : I am confused by the fact that you provide the correct formula !

    Say you have a vector [tex]\vec{V}=\{V_i\}[/tex] with components indiced by [tex]i[/tex] in a general [tex]n[/tex] dimensional linear (vector) space : [tex]i\in \{0,1,2,\cdots ,n\}[/tex]. Say in this [tex]n[/tex] dimensional space you have a plane defined by two vectors [tex]\vec{u}^{(1)} = \{u^{(1)}_i\}[/tex] and [tex]\vec{u}^{(2)} = \{u^{(2)}_i\}[/tex]. Then the straightforward application of your formula leads to the projection [tex]P(\vec{V})[/tex] of the vector [tex]\vec{V}[/tex] onto the plane spanned by [tex]\vec{u}^{(1)}[/tex] and [tex]\vec{u}^{(2)}[/tex] :

    [tex]P(\vec{V}) = \sum_{i=1}^2 \langle \vec{u}^{(i)},\vec{V} \rangle \vec{u}^{(i)} = c_1 \vec{u}^{(1)} + c_2 \vec{u}^{(2)}[/tex]

    with [tex]c_1 = \langle \vec{u}^{(1)},\vec{V} \rangle = \sum_{i=1}^n u^{(1)}_i V_i[/tex] and [tex]c_2 = \langle \vec{u}^{(2)},\vec{V} \rangle = \sum_{i=1}^n u^{(2)}_i V_i[/tex]
  4. Aug 29, 2004 #3


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    Thanks for that, guess it was simpler than I thought. :)
  5. Aug 31, 2004 #4


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    If I'm not mistaken,
    in projw(u) = <u,v1>v1 + <u,v2>v2 + .... <u,vn>vn ,
    the v1,...,vn should be mutually-orthogonal unit vectors
    since a projection must satisfy projw(projw(u))=projw(u).
  6. Oct 19, 2010 #5
    How could you create a matrix that performs this transformation? In other words, what matrix would project any vector V into the plane spanned by v1 and v2?
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