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Basis vectors

  1. Jan 7, 2007 #1
    1. The problem statement, all variables and given/known data
    Determine whether the the vectors a = (2, -3,2), b = (1, 1, -1) and
    c = (8, 5, -2) can be used as a basis for vectors in R^3 (3D space)


    2. Relevant equations



    3. The attempt at a solution
    I really have no clue, I think maybe you use either cross product, dot product or triple scalar product...?
     
  2. jcsd
  3. Jan 7, 2007 #2

    cristo

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    Why don't you try one of these? I'd use the dot product first, to show whether or not the vectors are mutually orthogonal.
     
  4. Jan 7, 2007 #3

    radou

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    What's the definition of a basis?
     
  5. Jan 7, 2007 #4

    LeonhardEuler

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    They don't have to be mutually orthogonal to be linearly independant, and it is unlikely that they will be. To kevykevy: You were on the right track with the scalar triple product. What properties of this product do you know?
     
  6. Jan 7, 2007 #5

    cristo

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    Sorry, I read "orthogonal" that wasn't in the question!
     
  7. Jan 7, 2007 #6

    LeonhardEuler

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    I know what you're talking about. I've been there more than a few times myself. :redface:
     
  8. Jan 7, 2007 #7
    Cross Product
    a x b = (1, 4, 5)

    Dot Product
    (1, 4, 5) x (8, 5, -2) = 18

    Since 18 doesn't equal 0, then the vectors cannot be used as basis vectors

    is that right?
     
  9. Jan 7, 2007 #8
    to radou - basis vectors, example i, j, and k with the carot(^) on top
     
  10. Jan 7, 2007 #9

    radou

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    Ok, that's an example of a basis. We can add that every set consisting of three linearly independent vectors forms a basis for R^3. All you have to do is check if your vectors are linearly independent.
     
  11. Jan 8, 2007 #10

    HallsofIvy

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    And if you are going to be doing problems like this it would be a really good idea for you to look at the definition of "basis" in your textbook.
     
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