BUILT-IN MATHEMATICA SYMBOL

Cross

gives the vector cross product of a and b.

If a and b are lists of length 3, corresponding to vectors in three dimensions, then Cross is also a list of length 3.

Cross can be entered in StandardForm and InputForm as , a Esc cross Esc b or . Note the difference between \[Cross] and \[Times].

In general, Cross 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 .

Cross gives the dual (Hodge star) of the wedge product of the , viewed as one-forms in n dimensions.

The cross product of two vectors:

The cross product of a single vector:

Enter using Esc cross Esc:

The cross product of two vectors:

Out[1]=

The cross product of a single vector:

Out[1]=

Enter using Esc cross Esc:

Out[1]=

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:

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:

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

: