Cross Product

For vectors and in , the cross product in is defined by

			(1)
			(2)

where 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

(3)

where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.

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

			(4)
			(5)
			(6)

The cross product satisfies the general identity

(7)

Note that 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 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 and is

			(8)
			(9)

where is the determinant.

The magnitude of the cross product is given by

			(10)
			(11)

where is the angle between and , given by the dot product

(12)

Identities involving the cross product include

			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)

In tensor notation,

(20)

where is the permutation symbol, Einstein summation has been used to sum over the repeated indices and , and is a free index denoting each component of the vector .