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Projecting a point/vector onto a plane

  1. Mar 8, 2012 #1
    1. The problem statement, all variables and given/known data
    In R4 equipped with the standard inner product find the projection of a=(1,2,0,-1) on the plane
    V spanned by v1=(1,0,0,1) and v2=(1,1,2,0)


    2. Relevant equations



    3. The attempt at a solution
    Firstly I'm not sure if a is a point or a vector, which would surely affect the final answer.
    I think I should find the normal to the plane but I'm not sure what to do with it.
    Does anyone know of a formula for projection onto a plane?
    I find projections really confusing so if anyone can give some guidance on how to approach problems like this one I would be really grateful.

    Many thanks
     
  2. jcsd
  3. Mar 8, 2012 #2

    HallsofIvy

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    Science Advisor

    a is a vector, the vector extending from the point (0, 0, 0, 0) to the point (1, 2, 0, -1). It wouldn't make sense to talk of "projecting" a point onto a plane.
    (The confusion is one reason I prefer to use "< >" for vectors rather than "( )".)

    I think what I would do is find a basis for R4 that contains the two vectors (1,0,0,1) and (1,1,2,0). For example, if we add the vectors (0, 1, 0, 0) and (0, 0, 0, 1), we can see that a(1, 0, 0, 1)+ b(1, 1, 2, 0)+ c(0, 1, 0, 0)+ d(0, 0, 0, 1)= (a+ b, b+ c, 2b, a+ d)= (0, 0, 0, 0) then we have a+ b= 0, b+ c= 0, 2b= 0, a+ d= 0. Obviously b= 0 so a+ b= a= 0, b+ c= c= 0, and a+ d= d= 0. Since we must have a= b= c= d= 0, the four vectors are independent so form a basis for R4. Now, write (1, 2, 0, -1)=a(1, 0, 0, 1)+ b(1, 1, 2, 0)+ c(0, 1, 0, 0)+ d(0, 0, 0, 1) and solve for at least a and b. a(1, 0, 0, 1)+ b(1, 1, 2, 0) will be the projection of v on the given plane.
     
  4. Mar 8, 2012 #3
    Hi HallsofIvy, thanks for your response.

    I'm still quite unsure as to what a projection actually is in this context. Is there any way you could briefly explain the basic idea?

    Thanks
     
  5. Mar 11, 2012 #4
    haha stuck on this one too mate..
     
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