Cross

Cross[a,b]

gives the vector cross product of a and b.

Details

  • If a and b are lists of length 3, corresponding to vectors in three dimensions, then Cross[a,b] is also a list of length 3.
  • Cross[a,b] can be entered in StandardForm and InputForm as ab, a cross b or a\[Cross]b. Note the difference between \[Cross] and \[Times].
  • Cross is antisymmetric, so that Cross[b,a] is -Cross[a,b]. »
  • Cross[{x,y}] gives the perpendicular vector {-y,x}.
  • In general, Cross[v1,v2,,vn-1] is a totally antisymmetric product which takes vectors of length n and yields a vector of length n that is orthogonal to all of the vi.
  • Cross[v1,v2,] gives the dual (Hodge star) of the wedge product of the vi, viewed as oneforms in n dimensions.

Examples

open allclose all

Basic Examples  (3)

The cross product of two vectors:

The cross product of a single vector:

Enter using cross:

Scope  (3)

Cross product computed with exact arithmetic:

Computed with machine arithmetic:

Computed with arbitrary-precision arithmetic:

Cross of one vector in 2 dimensions:

Cross product of three vectors in 4 dimensions:

Generalizations & Extensions  (3)

Cross in dimension is the contraction of vectors into the Levi-Civita tensor:

Cross of vectors in dimension is ( times the Hodge dual of their tensor product:

The Hodge dual of the TensorWedge of -vectors coincides with the Cross of those vectors:

TensorWedge can treat higher-rank forms:

Applications  (3)

Find the normal to the plane spanned by two vectors:

The equation for the plane:

Find a vector perpendicular to a vector in the plane:

Find a vector orthogonal to n-1 vectors in n dimensions:

Properties & Relations  (5)

If u and v are linearly independent, u×v is nonzero and orthogonal to u and v:

If u and v are linearly dependent, u×v is zero:

Cross is antisymmetric:

For vectors in 3 dimensions, Cross is bilinear:

The (antisymmetric) matrices for the linear operators and :

Introduced in 1996
 (3.0)