Cross product

From CGAFaq

The cross product or vector product of two 3-dimensional vectors produces a 3-dimensional vector result. The following properties define it uniquely:

1. u×v is bilinear (linear in u and linear in v)
2. u×u = 0 for all u (it is an alternating function)
3. x×y = z, y×z = x, z×x = y

Historically, the cross product was defined (separately) by Josiah Willard Gibbs and Oliver Heaviside as a portion of the quaternion product, useful for physical applications (such as Maxwell's theory of electromagnetism). The other portion was the dot product, u·v. This partition was highly controversial at the time, but all three products have proved useful. In terms of the two partial products, the quaternion product of two vectors [u, 0] and [v, 0] is [u×v, −u·v].

Although, properly speaking, the cross product exists only in 3D, an occasional abuse of terminology is to refer to the “cross product” of planar vectors, a scalar result that can be interpreted as the z component of a vector in the xy plane.

A definition of the cross product in terms of components can be remembered using the “magic word” xyzzy:

x = y₁z₂ − z₁y₂

y = z₁x₂ − x₁z₂

z = x₁y₂ − y₁x₂

Though perhaps not obvious from its definition, the cross product has useful geometric properties. For example, the cross product of two vectors is the zero vector if they are collinear, or perpendicular to their common plane if they are independent. In either case its length is twice the area of the triangle with those two sides, ‖u‖ ‖v‖ sin θ, where θ is the angle between u and v. Because the cross product is an alternating function, the result vector is negated if u and v are swapped: u×v = −v×u.

Wedge product

Following Hermann Grassmann, we can define a wedge product, “∧”, much like the cross product, for vectors in any dimension; however the result is not a vector, but a bivector or dyad. It is an accident of the dimension 3 that we can associate a unique vector with each bivector, and thus define the cross product as producing a vector result. The wedge product is heavily used in the exterior calculus of differential forms, where dx dy is taken as shorthand for dx∧dy. Some authors use the wedge symbol for the cross product; the context should make the meaning clear.

Generalized cross product

The perpendicularity production of cross product can be generalized to ℝⁿ. Given n−1 linearly independent vectors, we can find a vector that is perpendicular to each of the given vectors. It is given by the following (notational) determinant:

where

contains basis vectors of ℝⁿ. Note we have abused notation.

For example, in ℝ² we find a perpendicular to vector u by:

In ℝ³ we find a perpendicular to vectors u and v by:

Properties of generalized cross product:

Antisymmetry 

Multilinearity 

Perpendicularity 

, for all i in [0, n-1]

A variation

Sometimes one might see the generalized cross product given as:

It is related to our definition by:

There is some preference to using the form given in this page. For assume we are in an even dimension and calculate the generalized cross product of the first vectors of the natural basis. Our definition produces (0, 0, ..., 0, 1). The latter definition produces (0, 0, ..., 0, -1).