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单词 CrossProduct
释义

cross product


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The cross productMathworldPlanetmath (or vector product) of two vectors in 3 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

𝐚×𝐛=det(𝐢𝐣𝐤a1a2a3b1b2b3)
=(a2b3-a3b2)𝐢+(a3b1-a1b3)𝐣+(a1b2-a2b1)𝐤

with 𝐚=a1𝐢+a2𝐣+a3𝐤 and 𝐛=b1𝐢+b2𝐣+b3𝐤, where (𝐢,𝐣,𝐤) is a right-handed orthonormal basis for 3.

If we regard vectors in 3 as quaternions with real part equal to zero, with i=𝐢, j=𝐣 and k=𝐤, 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 “cross product” in 7 which shares many of the properties of the 3 cross product.)

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

𝐚×𝐛=(0-a3a2a30-a1-a2a10)𝐛.

The spectrum of this matrix(and therefore of the map 𝐛𝐚×𝐛)is {0,i|𝐚|,-i|𝐚|}.

Properties of the cross product

In the following, 𝐚, 𝐛 and 𝐜 will be arbitrary vectors in 3, and s and t will be arbitrary real numbers.

  • 𝐚×𝐚=0.

  • 𝐚×(𝐛×𝐜)+𝐛×(𝐜×𝐚)+𝐜×(𝐚×𝐛)=0.

  • The cross product is a bilinear map.This means that (s𝐚)×(t𝐛)=(st)(𝐚×𝐛),and that the cross product is distributivePlanetmathPlanetmath over vector addition,that is, 𝐚×(𝐛+𝐜)=𝐚×𝐛+𝐚×𝐜and(𝐛+𝐜)×𝐚=𝐛×𝐚+𝐜×𝐚.

  • The three properties above mean that the cross product makes 3 into a Lie algebra.

  • 𝐚×𝐛 is orthogonal to both 𝐚 and 𝐛.

  • 𝐚×𝐛=-𝐛×𝐚.

  • The length of 𝐚×𝐛 is the area of the parallelogram spanned by 𝐚 and 𝐛, so |𝐚×𝐛|=|𝐚||𝐛|sinθ, where θ is the angle between 𝐚 and 𝐛. This gives us an expression for the area of a triangle in 3: if the vertices are at 𝐚, 𝐛 and 𝐜, then the area is 12|(𝐚-𝐜)×(𝐛-𝐜)|, which can be written more symmetrically as 12|𝐚×𝐛+𝐛×𝐜+𝐜×𝐚|.

  • 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 sin0=0.

  • One can also see that |𝐚×𝐛|2=|𝐚|2|𝐛|2-|𝐚𝐛|2.

  • 𝐚×(𝐛×𝐜)=(𝐚𝐜)𝐛-(𝐚𝐛)𝐜.This is the vector triple productMathworldPlanetmath.

  • The cross product is rotationally invariant (http://planetmath.org/RotationalInvarianceOfCrossProduct).That is, for any 3×3 rotation matrixMathworldPlanetmath M we haveM(𝐚×𝐛)=(M𝐚)×(M𝐛).

Titlecross product
Canonical nameCrossProduct
Date of creation2013-03-22 11:58:52
Last modified on2013-03-22 11:58:52
Owneryark (2760)
Last modified byyark (2760)
Numerical id46
Authoryark (2760)
Entry typeDefinition
Classificationmsc 15A90
Classificationmsc 15A72
Synonymvector product
Synonymouter product
Related topicVector
Related topicDotProduct
Related topicTripleScalarProduct
Related topicExteriorAlgebra
Related topicDyadProduct
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更新时间:2025/5/3 17:13:34