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 one‐forms in n dimensions.
Examples
open allclose allBasic Examples (3)
Scope (7)
Find the cross product of machine-precision vectors:
Cross product of complex vectors:
Cross product of exact vectors:
Cross product of symbolic vectors:
Cross is antisymmetric:
Cross of one vector in two 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 (4)
Find the normal to the plane spanned by two vectors:
Find a vector perpendicular to a vector in the plane:
Find a vector orthogonal to n-1 vectors in n dimensions:
Find the area of the parallelogram defined by two vectors:
Compare with a direct computation using Area:
Text
Wolfram Research (1996), Cross, Wolfram Language function, https://reference.wolfram.com/language/ref/Cross.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1996. "Cross." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Cross.html.
APA
Wolfram Language. (1996). Cross. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Cross.html