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ching constraints as an external product edit the cross product in relation to the exterior product in red are the orthogonal unit vector and the parallel unit bivector the cross product can be defined in terms of the exterior product it can be generalized to an external product in other than three dimensions 19 this generalization allows a natural geometric interpretation of the cross product in exterior algebra the exterior product of two vectors is a bivector a bivector is an oriented plane element in much the same way that a vector is an oriented line element given two vectors a and b one can view the bivector a b as the oriented parallelogram spanned by a and b the cross product is then obtained by taking the hodge star of the bivector a b mapping 2 vectors to vectors a b a b displaystyle a times b star a wedge b this can be thought of as the oriented multi dimensional element perpendicular to the bivector in a d dimensional space hodge star takes a k vector to a d k vector thus only in d 3 dimensions is the result an element of dimension one 3 2 1 i e a vector for example in d 4 dimensions the cross product of two vectors has dimension 4 2 2 giving a bivector thus only in three dimensions does cross product define an algebra structure to multiply vectors generalizations edit there are several ways to generalize the cross product to higher dimensions lie algebra edit main article lie algebra the cross product can be seen as one of the simplest lie products and is thus generalized by lie algebras which are axiomatized as binary products satisfying the axioms of multilinearity skew symmetry and the jacobi identity many lie algebras exist and their study is a major field of mathematics called lie theory for example the heisenberg algebra gives another lie algebra structure on r 3 displaystyle mathbf r 3 in the basis x y z displaystyle x y z the product is x y z x z y z 0 displaystyle x y z x z y z 0 quaternions edit further information quaternions and spatial rotation the cross product can also be described in terms of quaternions in general if a vector a 1 a 2 a 3 is represented as the quaternion a 1 i a 2 j a 3 k the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result the real part will be the negative of the dot product of the two vectors octonions edit see also seven dimensional cross product and octonion a cross product for 7 dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions the nonexistence of nontrivial vector valued cross products of two vectors in other dimensions is related to the result from hurwitz s theorem that the only normed division algebras are the ones with dimension 1 2 4 and 8 exterior product edit main articles exterior algebra and comparison of vector algebra and geometric algebra cross and exterior products in general dimension there is no direct analogue of the binary cross product that yields specifically a vector there is however the exterior product which has similar properties except that the exterior product of two vectors is now a 2 vector instead of an ordinary vector as mentioned above the cross product can be interpreted as the exterior product in three dimensions by using the hodge star operator to map 2 vectors to vectors the hodge dual of the exterior product yields an n 2 vector which is a natural generalization of the cross product in any number of dimensions the exterior product and dot product can be combined through summation to form the geometric product in geometric algebra external product edit as mentioned above the cross product can be interpreted in three dimensions as the hodge dual of the exterior product in any finite n dimensions the hodge dual of the exterior product of n 1 vectors is a vector so instead of a binary operation in arbitrary finite dimensions the cross product is generalized as the hodge dual of the exterior product of some given n 1 vectors this generalization is called external product 20 commutator product edit main articles geometric algebra extensions of the inner and exterior products and cross product lie algebra interpreting the three dimensional vector space of the algebra as the 2 vector not the 1 vector subalgebra of the three dimensional geometric algebra where i e 2 e 3 displaystyle mathbf i mathbf e_ 2 mathbf e_ 3 j e 1 e 3 displaystyle mathbf j mathbf e_ 1 mathbf e_ 3 and k e 1 e 2 displaystyle mathbf k mathbf e_ 1 mathbf e_ 2 the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol displaystyle times the commutator product is defined for 2 vectors a displaystyle a and b displaystyle b in geometric algebra as a b 1 2 a b b a displaystyle a times b tfrac 1 2 ab ba where a b displaystyle ab is the geometric product 21 the commutator product could be generalised to arbitrary multivectors in three dimensions which results in a multivector consisting of only elements of grades 1 1 vectors true vectors and 2 2 vectors pseudovectors while the commutator product of two 1 vectors is indeed the same as the exterior product and yields a 2 vector the commutator of a 1 vector and a 2 vector yields a true vector corresponding instead to the left and right contractions in geometric algebra the commutator product of two 2 vectors has no corresponding equivalent product which is why the commutator product is defined in the first place for 2 vectors furthermore the commutator triple product of three 2 vectors is the same as the vector triple product of the same three pseudovectors in vector algebra however the commutator triple product of three 1 vectors in geometric algebra is instead the negative of the vector triple product of the same three true vectors in vector algebra generalizations to higher dimensions is provided by the same commutator product of 2 vectors in higher dimensional geometric algebras but the 2 vectors are no longer pseudovectors just as the commutator product cross product of 2 vectors in three dimensions correspond to the simplest lie algebra the 2 vector subalgebras of higher dimensional geometric algebra equipped with the commutator product also correspond to the lie algebras 22 also as in three dimensions the commutator product could be further generalised to arbitrary multivectors multilinear algebra edit in the context of multilinear algebra the cross product can be seen as the 1 2 tensor a mixed tensor specifically a bilinear map obtained from the 3 dimensional volume form note 2 a 0 3 tensor by raising an index in detail the 3 dimensional volume form defines a product v v v r displaystyle v times v times v to mathbf r by taking the determinant of the matrix given by these 3 vectors by duality this is equivalent to a function v v v displaystyle v times v to v fixing any two inputs gives a function v r displaystyle v to mathbf r by evaluating on the third input and in the presence of an inner product such as the dot product more generally a non degenerate bilinear form we have an isomorphism v v displaystyle v to v and thus this yields a map v v v displaystyle v times v to v which is the cross product a 0 3 tensor 3 vector inputs scalar output has been transformed into a 1 2 tensor 2 vector inputs 1 vector output by raising an index translating the above algebra into geometry the function volume of the parallelepiped defined by a b displaystyle a b where the first two vectors are fixed and the last is an input which defines a function v r displaystyle v to mathbf r can be represented uniquely as the dot product with a vector this vector is the cross product a b displaystyle a times b from this perspective the cross product is defined by the scalar triple product v o l a b c a b c displaystyle mathrm vol a b c a times b cdot c in the same way in higher dimensions one may define generalized cross products by raising indices of the n dimensional volume form which is a 0 n displaystyle 0 n tensor the most direct generalizations of the cross product are to define either a 1 n 1 displaystyle 1 n 1 tensor which takes as input n 1 displaystyle n 1 vectors and gives as output 1 vector an n 1 displaystyle n 1 ary vector valued product or a n 2 2 displaystyle n 2 2 tensor which takes as input 2 vectors and gives as output skew symmetric tensor of rank n 2 a binary product with rank n 2 tensor values one can also define k n k displaystyle k n k tensors for other k these products are all multilinear and skew symmetric and can be defined in terms of the determinant and parity the n 1 displaystyle n 1 ary product can be described as follows given n 1 displaystyle n 1 vectors v 1 v n 1 displaystyle v_ 1 dots v_ n 1 in r n displaystyle mathbf r n define their generalized cross product v n v 1 v n 1 displaystyle v_ n v_ 1 times cdots times v_ n 1 as perpendicular to the hyperplane defined by the v i displaystyle v_ i magnitude is the volume of the parallelotope defined by the v i displaystyle v_ i which can be computed as the gram determinant of the v i displaystyle v_ i oriented so that v 1 v n displaystyle v_ 1 dots v_ n is positively oriented this is the unique multilinear alternating product which evaluates to e 1 e n 1 e n displaystyle e_ 1 times cdots times e_ n 1 e_ n e 2 e n e 1 displaystyle e_ 2 times cdots times e_ n e_ 1 and so forth for cyclic permutations of indices in coordinates one can give a formula for this n 1 displaystyle n 1 ary analogue of the cross product in r n by i 0 n 1 v i v 1 1 v 1 n v n 1 1 v n 1 n e 1 e n displaystyle bigwedge _ i 0 n 1 mathbf v _ i begin vmatrix v_ 1 1 cdots v_ 1 n vdots ddots vdots v_ n 1 1 cdots v_ n 1 n mathbf e _ 1 cdots mathbf e _ n end vmatrix this formula is identical in structure to the determinant formula for the normal cross product in r 3 except that the row of basis vectors is the last row in the determinant rather than the first the reason for this is to ensure that the ordered vectors v 1 v n 1 λ n 1 i 0 v i have a positive orientation with respect to e 1 e n if n is odd this modification leaves the value unchanged so this convention agrees with the normal definition of the binary product in the case that n is even however the distinction must be kept this n 1 displaystyle n 1 ary form enjoys many of the same properties as the vector cross product it is alternating and linear in its arguments it is perpendicular to each argument and its magnitude gives the hypervolume of the region bounded by the arguments and just like the vector cross product it can be defined in a coordinate independent way as the hodge dual of the wedge product of the arguments moreover the product v 1 v n i 0 n v i displaystyle v_ 1 ldots v_ n bigwedge _ i 0 n v_ i satisfies the filippov identity x 1 x n y 2 y n i 1 n x 1 x i 1 x i y 2 y n x i 1 x n displaystyle x_ 1 ldots x_ n y_ 2 ldots y_ n sum _ i 1 n x_ 1 ldots x_ i 1 x_ i y_ 2 ldots y_ n x_ i 1 ldots x_ n and so it endows r n 1 with a structure of n lie algebra see proposition 1 of 23 history edit in 1773 joseph louis lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions 24 note 3 in 1843 william rowan hamilton introduced the quaternion product and with it the terms vector and scalar given two quaternions 0 u and 0 v where u and v are vectors in r 3 their quaternion product can be summarized as u v u v james clerk maxwell used hamilton s quaternion tools to develop his famous electromagnetism equations and for this and other reasons quaternions for a time were an essential part of physics education in 1844 hermann grassmann published a geometric algebra not tied to dimension two or three grassmann developed several products including a cross product represented then by uv 25 see also exterior algebra in 1853 augustin louis cauchy a contemporary of grassmann published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product 26 27 in 1878 william kingdon clifford known for a precursor to the clifford algebra named in his honor published elements of dynamic in which the term vector product is attested in the book this product of two vectors is defined to have magnitude equal to the area of the parallelogram of which they are two sides and direction perpendicular to their plane 28 in lecture notes from 1881 gibbs represented the cross product by u v displaystyle u times v and called it the skew product 29 30 in 1901 gibb s student edwin bidwell wilson edited and extended these lecture notes into the textbook vector analysis wilson kept the term skew product but observed that the alternative terms cross product note 4 and vector product were more frequent 31 in 1908 cesare burali forti and roberto marcolongo introduced the vector product notation u v 25 this is used in france and other areas until this day as the symbol displaystyle times is already used to denote multiplication and the cartesian product citation needed see also edit cartesian product a product of two sets geometric algebra rotating systems multiple cross products products involving more than three vectors multiplication of vectors quadruple product the symbol notes edit here formal means that this notation has the form of a determinant but does not strictly adhere to the definition it is a mnemonic used to remember the expansion of the cross product by a volume form one means a function that takes in n vectors and gives out a scalar the volume of the parallelotope defined by the vectors v v r displaystyle v times cdots times v to mathbf r this is an n ary multilinear skew symmetric form in the presence of a basis such as on r n displaystyle mathbf r n this is given by the determinant but in an abstract vector space this is added structure in terms of g structures a volume form is an s l displaystyle sl structure in modern notation lagrange defines ξ y z displaystyle mathbf xi mathbf y times mathbf z η z x displaystyle boldsymbol eta mathbf z times mathbf x and ζ x y displaystyle boldsymbol zeta mathbf x times boldsymbol y thereby the modern x displaystyle mathbf x corresponds to the three variables x x x displaystyle x x x in lagrange s notation since a b is read as a cross b references edit 1 2 3 4 5 6 weisstein eric w cross product wolfram mathworld retrieved 2020 09 06 1 2 cross product www mathsisfun com retrieved 2020 09 06 massey william s december 1983 cross products of vectors in higher dimensional euclidean spaces pdf the american mathematical monthly 90 10 697 701 doi 10 2307 2323537 jstor 2323537 s2cid 43318100 archived from the original pdf on 2021 02 26 if one requires only three basic properties of the cross product it turns out that a cross product of vectors exists only in 3 dimensional and 7 dimensional euclidean space arfken george b mathematical methods for physicists 4th ed elsevier jeffreys h jeffreys b s 1999 methods of mathematical physics cambridge...
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