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

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Basic Examples  (3)

The cross product of two vectors:

The cross product of a single vector:

Enter using cross:

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:

Create vectors a and b:

Cross is antisymmetric:

Cross of one vector in two dimensions:

The result is perpendicular to the original vector:

Cross product of three vectors in four 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:

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:

Find the area of the parallelogram defined by two vectors:

Compare with a direct computation using Area:

Visualize the parallelogram:

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 :

Wolfram Research (1996), Cross, Wolfram Language function, https://reference.wolfram.com/language/ref/Cross.html.

Text

Wolfram Research (1996), Cross, Wolfram Language function, https://reference.wolfram.com/language/ref/Cross.html.

BibTeX

@misc{reference.wolfram_2020_cross, author="Wolfram Research", title="{Cross}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Cross.html}", note=[Accessed: 12-May-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_cross, organization={Wolfram Research}, title={Cross}, year={1996}, url={https://reference.wolfram.com/language/ref/Cross.html}, note=[Accessed: 12-May-2021 ]}

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