cross product


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The cross productMathworldPlanetmath (or vector product) of two vectors in ℝ3 is a vector http://planetmath.org/node/1285orthogonalMathworldPlanetmathPlanetmathPlanetmath to the plane of the two vectors being crossed, whose magnitude is equal to the area of the parallelogramMathworldPlanetmath defined by the two vectors. Notice there can be two such vectors. The cross product produces the vector that would be in a right-handed coordinate system with the plane.

We write the cross product of the vectors 𝐚 and 𝐛 as

πšΓ—π› = det⁑(𝐒𝐣𝐀a1a2a3b1b2b3)
= (a2⁒b3-a3⁒b2)⁒𝐒+(a3⁒b1-a1⁒b3)⁒𝐣+(a1⁒b2-a2⁒b1)⁒𝐀

with 𝐚=a1⁒𝐒+a2⁒𝐣+a3⁒𝐀 and 𝐛=b1⁒𝐒+b2⁒𝐣+b3⁒𝐀, where (𝐒,𝐣,𝐀) is a right-handed orthonormal basis for ℝ3.

If we regard vectors in ℝ3 as quaternions with real part equal to zero, with i=𝐒, j=𝐣 and k=𝐀, then the cross product of two vectors can be obtained by zeroing the real part of the product of the two quaternions. (A similarMathworldPlanetmathPlanetmath construction using octonions instead of quaternions gives a β€œcross product” in ℝ7 which shares many of the properties of the ℝ3 cross product.)

If we write vectors in the form 𝐚=(a1a2a3), then we can express the cross product as

πšΓ—π›=(0-a3a2a30-a1-a2a10)⁒𝐛.

The spectrum of this matrix (and therefore of the map π›β†¦πšΓ—π›) is {0,i⁒|𝐚|,-i⁒|𝐚|}.

Properties of the cross product

In the following, 𝐚, 𝐛 and 𝐜 will be arbitrary vectors in ℝ3, and s and t will be arbitrary real numbers.

  • β€’

    πšΓ—πš=0.

  • β€’

    πšΓ—(π›Γ—πœ)+𝐛×(πœΓ—πš)+πœΓ—(πšΓ—π›)=0.

  • β€’

    The cross product is a bilinear map. This means that (s⁒𝐚)Γ—(t⁒𝐛)=(s⁒t)⁒(πšΓ—π›), and that the cross product is distributivePlanetmathPlanetmath over vector addition, that is, πšΓ—(𝐛+𝐜)=πšΓ—π›+πšΓ—πœ and (𝐛+𝐜)Γ—πš=π›Γ—πš+πœΓ—πš.

  • β€’

    The three properties above mean that the cross product makes ℝ3 into a Lie algebra.

  • β€’

    πšΓ—π› is orthogonal to both 𝐚 and 𝐛.

  • β€’

    πšΓ—π›=-π›Γ—πš.

  • β€’

    The length of πšΓ—π› is the area of the parallelogram spanned by 𝐚 and 𝐛, so |πšΓ—π›|=|𝐚|⁒|𝐛|⁒sin⁑θ, where ΞΈ is the angle between 𝐚 and 𝐛. This gives us an expression for the area of a triangle in ℝ3: if the vertices are at 𝐚, 𝐛 and 𝐜, then the area is 12⁒|(𝐚-𝐜)Γ—(𝐛-𝐜)|, which can be written more symmetrically as 12⁒|πšΓ—π›+π›Γ—πœ+πœΓ—πš|.

  • β€’

    From the above, you can see that the cross product of any vector with 𝟎 is 𝟎. More generally, the cross product of two parallel vectors is 𝟎, since sin⁑0=0.

  • β€’

    One can also see that |πšΓ—π›|2=|𝐚|2⁒|𝐛|2-|πšβ‹…π›|2.

  • β€’

    πšΓ—(π›Γ—πœ)=(πšβ‹…πœ)⁒𝐛-(πšβ‹…π›)⁒𝐜. This is the vector triple productMathworldPlanetmath.

  • β€’

    The cross product is rotationally invariant (http://planetmath.org/RotationalInvarianceOfCrossProduct). That is, for any 3Γ—3 rotation matrixMathworldPlanetmath M we have M⁒(πšΓ—π›)=(M⁒𝐚)Γ—(M⁒𝐛).

Title cross product
Canonical name CrossProduct
Date of creation 2013-03-22 11:58:52
Last modified on 2013-03-22 11:58:52
Owner yark (2760)
Last modified by yark (2760)
Numerical id 46
Author yark (2760)
Entry type Definition
Classification msc 15A90
Classification msc 15A72
Synonym vector product
Synonym outer product
Related topic Vector
Related topic DotProduct
Related topic TripleScalarProduct
Related topic ExteriorAlgebra
Related topic DyadProduct