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Cross Product Matrix

The cross product matrix of a three-dimensional vector u=(u_1,u_2,u_3) is the antisymmetric matrix

S(u)=[0 -u_3 u_2; u_3 0 -u_1; -u_2 u_1 0].

It represents the linear transformation given by taking the cross product with u on the left. Thus, for every three-dimensional column vector v,

S(u)v=uxv.

The order matters: vxu=-S(u)v, since the cross product is anticommutative. The notation [u] is also used for this matrix (Lynch and Park 2017).

See also

Antisymmetric Matrix, Cross Product, Linear Transformation

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References

Lynch, K. M. and Park, F. C. Modern Robotics: Mechanics, Planning, and Control. Cambridge, England: Cambridge University Press, 2017. https://doi.org/10.1017/9781316661239. https://hades.mech.northwestern.edu/index.php/Modern_Robotics.

Cite this as:

Weisstein, Eric W. "Cross Product Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CrossProductMatrix.html

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