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Homework Help: Oblique Cross Product

  1. Nov 20, 2014 #1
    1. The problem statement, all variables and given/known data

    Find vector product of [itex]C = A \times B [/itex] of two vectors in oblique coord. system. Give explicit expressions of components of C in covariant and contravariant components (constructing reciprocal basis from direct basis will be useful).

    2. Relevant equations

    I am basically just crossing two vectors, one product is that of two arbitrary contravariant vectors, and one is a product of two arbitrary covariant vectors. I understand this part, I just am always confused by the word "explicit."

    3. The attempt at a solution

    I take this determinant (contravariant)
    [itex] \begin{array}{ccc} a_1 & a_2 & a_3 \\ A^1 & A^2 & A^3 \\ B^1 & B^2 & B^3 \end{array} [/itex]

    and this one as well (covariant)
    [itex] \begin{array}{ccc} a^1 & a^2 & a^3 \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{array} [/itex]

    and for my covariant components I can set [itex] a_i = \frac{e_i}{sin(\alpha)} [/itex]

    I am not sure what else is being asked, if anything at all.

    I have this relation for components:

    [itex] x_1 = x^1 + x^2cos(\alpha) [/itex]

    but not sure if I should apply it to get vector terms as [itex]a_1A_1 +...[/itex]:
  2. jcsd
  3. Nov 23, 2014 #2


    Staff: Mentor

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