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Homework Help: Vectors (incl. axial/psuedovectors)/Scalars

  1. Jan 20, 2010 #1

    bon

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    1. The problem statement, all variables and given/known data

    Basically, trying to determine whether the following are scalars of vectors: area and angle of polarization.

    2. Relevant equations

    None

    3. The attempt at a solution

    My guess was that area would be a scalar rather than a vector, but my notes say that it is a psuedovector - is this true? In what sense does it have direction? And why is it axial rather than a proper vector?

    Angle of polarization is a scalar I'm guessing, as its just a magnitude (angle) rather than direction?
     
  2. jcsd
  3. Jan 20, 2010 #2

    berkeman

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    Staff: Mentor

    (I think you have a small typo in the question. You mean "scalars or vectors", right?)

    What are the definitions of scalars and vectors? How are they fundamentally different?
     
  4. Jan 20, 2010 #3

    bon

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    yes i did make a typo, sorry.

    vectors have direction, scalars don't. both have magnitude. hence my being confused of the fact that my notes say area is a psuedovector.
     
  5. Jan 20, 2010 #4

    berkeman

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    Staff: Mentor

    Sorry, what's the definition of a pseudovector?
     
  6. Jan 20, 2010 #5

    bon

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  7. Jan 20, 2010 #6

    vela

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    Do you recall doing surface integrals, like when applying Gauss's law?
     
  8. Jan 20, 2010 #7

    bon

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    Haven't covered that yet, sorry :(
     
  9. Jan 20, 2010 #8

    berkeman

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    Staff: Mentor

  10. Jan 20, 2010 #9

    jambaugh

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    To say "area is a pseudo-vector" you need to qualify a couple of things.

    First that you're working in 3-space. (otherwise they are at best bi-vectors = rank two tensors)

    Second that you're talking about oriented area elements (which are negative when you reverse orientation) typically defined by two vectors as the parallelogram they define.

    The thing to remember is that "vectors" and "scalars" and such are so defined for arbitrary linear transformations on the space not just for rotations. A pseudo-scalar transforms like a volume element, being unchanged by rotations but scaling by the determinant of a more general transformation matrix. Since a vector defining a line element plus an area element together define a volume element (parallelepiped = oblique box) the area element must be a pseudo-vector (in 3 dim).

    The vector associated with an area element is the normal vector of length numerically equal to the area. Then dotting a vector with this "pseudo-vector" gives the "pseudo-scalar" volume.

    The types and dimensions follow the combinatorics of the the binomial coefficients. In 3 dim you have: 1,3,3,1 corresponding to scalar,vector, bivector=pseudo-vector, pseudo-scalar.
    In four dimensions you would have: 1, 4, 6, 4, 1 and so:
    scalar, vector, bivector(=pseudo-bivector), trivector=pseudo-vector, pseudo-scalar.

    In five dimensions you would have 1,5, 10, 10, 5, 1 and so ....
     
  11. Jan 20, 2010 #10

    bon

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    wow okay thanks for your help. So am i right in thinking angle of polarization is a scalar?
     
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