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Applications of anticrossproducts and cross divisions

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  • Applications of anticrossproducts and cross divisions

    Some subscribers discovered anticrossproducts and cross divisions (de Leva,
    2008) by reading my recent posting about "moment arms in 3-D", and they
    thought they were designed to define moment arms. Not at all.

    Although a cross division can be used to define moment arms, I designed the
    first cross division for a practical purpose. I hate writing scalar
    equations, so I was looking for a clean "coordinate-free" vector-algebraic
    formula to compute the force applied by a "load-shaft" on a force plate for
    its calibration (see fig. 2 in When I found it, I
    realized it was the missing tessera in Gibbs and Heaviside's mosaic of
    vector-algebraic operations (such as vector sum, dot product and cross
    product), i.e. a cross division (by the way, the dot division is just
    impossible!). Then I found other kinds of cross divisions and other
    applications. For instance:


    An "orthogonal cross division" (the simplest kind of cross division; symbol:
    "/o") can be used to define the angular velocity of a point with respect to
    another point O:

    - omega = v /o r

    where v is the velocity of P along any trajectory, even a non-circular,
    non-spherical and non-planar one, and r the directed distance from O to R
    (see fig. 4 in
    Compare this vectorial formula with its well known scalar version, which is
    valid only for circular or spherical motion (v perpendicular to r):

    - |omega| = |v| / |r|

    Can you see the two advantages of the cross division in this case?

    1) It returns not only the magnitude, but also the direction of omega
    2) It is valid even when v is not perpendicular to r (non-constrained 3-D
    motion of P).


    The center of pressure (CoP) is traditionally computed from force-plate
    readings, by solving a system of three scalar equations. A single cross
    division does exactly the same, but it is easier to write. (Similarly, a
    cross product is just a symbol representing three scalar expressions, but it
    is easier to write.)

    You can find other examples in my paper, together with the definition of the
    elementary concept of anticrossproduct, which is the theoretical foundation
    of my study.


    de Leva P., 2008. Anticrossproducts and cross divisions. Journal of
    Biomechanics, 8, 1790-1800

    With my kindest regards,

    Paolo de Leva
    Laboratory of Locomotor Apparatus Bioengineering
    Department of Human Movement and Sport Sciences
    University of Rome, Foro Italico
    Piazza Lauro de Bosis, 6
    00135 Rome - Italy