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Math Help - cross product

  1. #1
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    cross product

    In case you’re not familiar with it, the cross product is very handy in 3d geometry:

    aXb is a vector perpendicular to a and b in direction given by right hand rule (rotate (screw) a into b thru the acute angle θ between them), and magnitude absinθ.

    aXb = “det” |i, j, k ; a1, a2, a3 ; b1, b2, b3| , = |row; row; row|, (not obvious, proof in textbooks)

    Ex (0,0,1) X (a1,a2,a3) = |i,j,k ; 0,0,1 ; a1,a2,a3 | = -a2i + a1j = (-a2,a1,0)

    It follows that a.bXc = det |a;b;c|

    i,j,k is orthonormal basis (also called e1,e2,e3). "det" not really determinant- write out as determinant and evaluate as determinant.

    It would be visually clearer if I could print determinants, sorry.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: cross product

    Is you question to show that  a(b \times c) = Det(a;b;c) or are you telling us an interesting fact about cross products?
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  3. #3
    Super Member ILikeSerena's Avatar
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    Re: cross product

    Quote Originally Posted by Hartlw View Post
    It would be visually clearer if I could print determinants, sorry.
    You mean like this?

    \mathbf a \times \mathbf b = \det \begin{pmatrix}\mathbf i & \mathbf j & \mathbf k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix}

    or like this?

    \mathbf a \times \mathbf b = \begin{vmatrix}\mathbf i & \mathbf j & \mathbf k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}
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    Re: cross product

    ILikeSerena:

    In either case it’s the determinant, written with any convention you like.

    aXb = i(a2b3-a1b2) –j(a1b3-b1a3) +k(a1b2-a1b2).

    It’s just short hand for (a1i + a2j + a3k) X (b1i + b2j + b3k) and noting iXi =0, iXj = k, etc. Sketch a little right hand triad of i,j,k to instantly get all the possibilities.

    Geometrically a.bXc is volume of the triad a,b,c, and is usually evaluated as det |a; b; c|

    This isn’t a question, just a point of information, since I get the impression the cross product is not generally handy.
    For a complete lucid explanation of cross product google “cross product” wiki.
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