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Cross Product


CrossProduct

For vectors u=(u_x,u_y,u_z) and v=(v_x,v_y,v_z) in R^3, the cross product in is defined by

uxv=x^^(u_yv_z-u_zv_y)-y^^(u_xv_z-u_zv_x)+z^^(u_xv_y-u_yv_x)
(1)
=x^^(u_yv_z-u_zv_y)+y^^(u_zv_x-u_xv_z)+z^^(u_xv_y-u_yv_x),
(2)

where (x^^,y^^,z^^) is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant

 uxv=|x^^ y^^ z^^; u_x u_y u_z; v_x v_y v_z|,
(3)

where x^^, y^^, and z^^ are unit vectors. Here, uxv is always perpendicular to both u and v, with the orientation determined by the right-hand rule.

Special cases involving the unit vectors in three-dimensional Cartesian coordinates are given by

x^^xy^^=z^^
(4)
y^^xz^^=x^^
(5)
z^^xx^^=y^^.
(6)

The cross product satisfies the general identity

 AxB=-BxA.
(7)

Note that uxv is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation u ^ v to denote the cross product.

The cross product is implemented in the Wolfram Language as Cross[a, b].

A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing: you can't cross a scaler with a vector," a reference to the fact the cross product can be applied only to two vectors and not a scalar and a vector (or two scalars, for that matter). Another joke presented on the television sitcom Head of the Class asks, "What do you get when you cross an elephant and a grape?" The answer is "Elephant grape sine-of-theta."

In two dimensions, the analog of the cross product for u=(u_x,u_y) and v=(v_x,v_y) is

uxv=det(uv)
(8)
=u_xv_y-u_yv_x,
(9)

where det(A) is the determinant.

The magnitude of the cross product is given by

|uxv|=|u||v|sintheta
(10)
=|u||v|sqrt(1-(u^^·v^^)^2),
(11)

where theta is the angle between u and v, given by the dot product

 costheta=u^^·v^^.
(12)

Identities involving the cross product include

d/(dt)[r_1(t)xr_2(t)]=r_1(t)x(dr_2)/(dt)+(dr_1)/(dt)xr_2(t)
(13)
AxB=-BxA
(14)
Ax(B+C)=AxB+AxC
(15)
(tA)xB=t(AxB)
(16)
A·(BxC)=det(ABC)
(17)
Ax(BxC)=B(A·C)-C(A·B)
(18)
(AxB)x(CxD)=det(ABD)C-det(ABC)D.
(19)

In tensor notation,

 AxB=epsilon_(ijk)A^jB^k,
(20)

where epsilon_(ijk) is the permutation symbol, Einstein summation has been used to sum over the repeated indices j and k, and i is a free index denoting each component of the vector (AxB)_i.


See also

Cartesian Product, Determinant, Dot Product, Permutation symbol, Right-Hand Rule, Scalar Triple Product, Vector, Vector Direct Product, Vector Multiplication Explore this topic in the MathWorld classroom

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References

Arfken, G. "Vector or Cross Product." §1.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 18-26, 1985.Jeffreys, H. and Jeffreys, B. S. "Vector Product." §2.07 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 67-73, 1988.

Referenced on Wolfram|Alpha

Cross Product

Cite this as:

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

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