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Writing orthogonal vectors as linear combinations

  1. Nov 5, 2012 #1

    Quick question, not really homework but more of a general inquiry. Take three vectors: a,b and c such that a and c are orthogonal. Is it possible to write c as a linear combination of a and b such that:

    c = ma + nb where m,n are scalars.

    I was thinking not at first glance but reading around has made me think twice.
    Is it possible?

  2. jcsd
  3. Nov 5, 2012 #2
    Vectors a and c define a plane. There is definitely a vector b such that c=ma+nb that lies in that plane. So yes it is possible
  4. Nov 6, 2012 #3


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    Did you mean, s.t. a and b are orthogonal? Yes: c = (a.c/a.a)a + (a.b/b.b)b
    But if you meant a and c are orthogonal, not if a and b are collinear.
    If a and b are neither collinear nor orthogonal, there's a range of solutions for the coefficients.
  5. Nov 6, 2012 #4
    I actually mean a and c are orthogonal.

    I see now, as it is possible to define the vector b in terms of a and c, it is then just rearranging.

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