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Determine if the given vectors are orthogonal

  1. Jul 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Screenshot2012-07-25at32835AM.png

    2. Relevant equations
    3. The attempt at a solution

    A set of vectors are orthogonal if any two are perpendicular. the cross product of w1 and w2 is

    -9 + 2 + 3 + 4 = 0

    So the set of vectors is orthogonal. The book says that's false. Why?
     
  2. jcsd
  3. Jul 25, 2012 #2

    ehild

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    Homework Helper
    Gold Member

    The set of vectors are orthogonal if any pair of them is orthogonal. That means not only a single pair being orthogonal, but all pairs, that is

    w1*w2=w1*w3=w1*w4=w2*w3=w2*w4=w3*w4=0

    w2 is orthogonal to w1, but not to w3. Calculate w2*w3, is it zero? What about w3 and w4?

    ehild
     
  4. Jul 25, 2012 #3
    Well, I find the book's use of the English language very unfortunate. If I say a group of people is multi-ethic if any two of its members are of a different race, then it is multiethic. That means only two of its members need be different, not all of them. But if orthogonal means that all pairs must perpendicular then that's the way it is.
     
  5. Jul 25, 2012 #4

    ehild

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    That is the definition of an orthogonal set of vectors. All possible pairs are orthogonal. Choosing
    any two vectors, they are orthogonal. The multi-ethic group means that you can choose at least one pair of people who belong to different races. I would not use "any" in this case.

    ehild
     
  6. Jul 25, 2012 #5
    Like I said, all possible is not equal to any two
     
  7. Jul 25, 2012 #6

    Mark44

    Staff: Mentor

    "multiethic" does not mean multiple races.

    The definition you show for orthogonality is incorrect (translation error?). Here's a simple counterexample.
    Let S = {<1, 0, 0>, <0, 1, 0>, <1, 1, 0>}

    Clearly, the first two vectors in the list above are orthogonal, so by the posted definition, the entire set is orthogonal. However, taking dot products, we see that <1, 0, 0>##\cdot## <1, 1, 0> = 1, so these two vectors aren't orthogonal.

    Likewise, <0, 1, 0>##\cdot## <1, 1, 0> = 1, so these two vectors aren't orthogonal, either.

    For a set of vectors to be orthogonal, every pair of them must be orthogonal.
     
  8. Jul 25, 2012 #7

    Mark44

    Staff: Mentor

    What you've done is the dot product, not the cross product. The result of the cross product of two vectors is another vector, not a scalar.
     
  9. Jul 25, 2012 #8

    I often confuse the two.
     
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