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 .