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Math Help - Orthogonal Projection

  1. #1
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    Orthogonal Projection

    Find the orthogonal projection of (x,y,z) onto the subspace of R^3 spanned by vectors (1,2,2),(-2,2,-1).

    what i did was
    [(x,y,z).(1,2,2)](1,2,2) + [(x,y,z).(-2,2,-1)](-2,2,-1)
    which is equals to
    (x+2y+2z,2x+4y+4z,2x+4y+4z) + (4x-4y+2z,-4x+4y-2z,2x-2y+z)
    and then that is equals to
    (5x-2y+4z,-2x+8y+2z,4x+2y+5z)

    but the answer given is 1/9(5x-2y+4z,-2x+8y+2z,4x+2y+5z)
    which is the same as what i got except the 1/9 in front. can anyone tell me what i did wrong?

    another question,
    if i am given 2 vectors X & Y in R^3 and given the inner product of <X,Y>
    how do i calculate the angle between X and Y? i know how to calculate ||X|| and ||Y|| which is just <X,X> and <Y,Y> right?
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  2. #2
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    Quote Originally Posted by yen yen View Post
    Find the orthogonal projection of (x,y,z) onto the subspace of R^3 spanned by vectors (1,2,2),(-2,2,-1).

    what i did was
    [(x,y,z).(1,2,2)](1,2,2) + [(x,y,z).(-2,2,-1)](-2,2,-1)
    which is equals to
    (x+2y+2z,2x+4y+4z,2x+4y+4z) + (4x-4y+2z,-4x+4y-2z,2x-2y+z)
    and then that is equals to
    (5x-2y+4z,-2x+8y+2z,4x+2y+5z)

    but the answer given is 1/9(5x-2y+4z,-2x+8y+2z,4x+2y+5z)
    which is the same as what i got except the 1/9 in front. can anyone tell me what i did wrong?

    another question,
    if i am given 2 vectors X & Y in R^3 and given the inner product of <X,Y>
    how do i calculate the angle between X and Y? i know how to calculate ||X|| and ||Y|| which is just <X,X> and <Y,Y> right?

    What you did is ALMOST correct, but you must work with an orthonormal basis of the subspace! In general this would imply to do Gram-Schmidt on a certain basis of the subspace, but in this particular case it is much easier since the basis elements are orthogonal so we only have to divide each by its length (= its norm), and then the basis we work with should be \frac{(1,2,2)}{\|(1,2,2)\|}=\frac{1}{3}(1,2,2)\,,\  ,\,\frac{(-2,2,-1)}{\|(-2,2,-1)\|}=\frac{1}{3}(-2,2,-1) ...and now do what you did BUT with the two normalized elements of the basis instead (can you see now from where that 1\slash 9 came?)

    Tonio
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