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Simple Vector Question

  1. Aug 1, 2004 #1
    I'm having trouble with what should be a simple question!

    Let a = (2,4,-2) and b = (4,-2,2)

    I need to be able to express a as the sum of two vectors, one parallel to b and the other perpendicular to b.

    Thing is, I haven't the foggiest idea where to start! Any ideas?

  2. jcsd
  3. Aug 1, 2004 #2
    The vector parallel to b is called the "projection of a onto b". There is a formula for it, and it should be covered in any basic book on linear algebra (at least in the cases of the vectors being in R^2 or R^3). proj(a, b) = (a.b)/(b.b) * b, (but obviously it's no good to just know the formula, so get yourself a book) ;).
  4. Aug 1, 2004 #3
    u mean the component of a parallel to b is "the projection of a onto b" right?

    anyways as muzza said the parallel component of a comes as a projection of a onto b and the entire thing can be written as,
    a = [(a.b)/b^2] b + (a - [(a.b)/b^2] b)

    the first component is parallel to b and the second component is perpendicular to b.

    -- AI
  5. Aug 1, 2004 #4
    Yes, I figured that was understood.
  6. Aug 1, 2004 #5
    Thanks for the speedy replies.

    I get where the parallel component comes from, but I don't understand where the perpendicular component comes from?
  7. Aug 1, 2004 #6
    Let a_p be the aforementioned vector parallel to b, and a_o be the perpendicular vector. Then a = a_p + a_o <=> a_o = a - a_p = a - (a.b)/(b.b) * b.
  8. Aug 1, 2004 #7
    Thanks, that explained it very clearly. Can't believe I didn't notice it was that simple.

  9. Jan 7, 2012 #8
    there should be some another method
    Last edited: Jan 7, 2012
  10. Jan 7, 2012 #9
    misread it
  11. Jan 7, 2012 #10


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    Staff Emeritus
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    aswinsp thank you for your contribution but if you look at the time stamp above people's names you will see that this thread is 8 years old. Posting in such an old thread is called necroposting and is not allowed.
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