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end bmatrix times begin bmatrix c_ y c_ z end bmatrix when doing this for a y displaystyle a_ y the next two elements down should wrap around the matrix so that after the z component comes the x component for clarity when performing this operation for a y displaystyle a_ y the next two components should be z and x in that order while for a z displaystyle a_ z the next two components should be taken as x and y a y b z b x c z c x a z b x b y c x c y displaystyle a_ y begin bmatrix b_ z b_ x end bmatrix times begin bmatrix c_ z c_ x end bmatrix a_ z begin bmatrix b_ x b_ y end bmatrix times begin bmatrix c_ x c_ y end bmatrix for a x displaystyle a_ x then if we visualize the cross operator as pointing from an element on the left to an element on the right we can take the first element on the left and simply multiply by the element that the cross points to in the right hand matrix we then subtract the next element down on the left multiplied by the element that the cross points to here as well this results in our a x displaystyle a_ x formula a x b y c z b z c y displaystyle a_ x b_ y c_ z b_ z c_ y we can do this in the same way for a y displaystyle a_ y and a z displaystyle a_ z to construct their associated formulas applications edit the cross product has applications in various contexts for example it is used in computational geometry physics and engineering a non exhaustive list of examples follows computational geometry edit the cross product appears in the calculation of the distance of two skew lines lines not in the same plane from each other in three dimensional space the cross product can be used to calculate the normal for a triangle or polygon an operation frequently performed in computer graphics for example the winding of a polygon clockwise or anticlockwise about a point within the polygon can be calculated by triangulating the polygon like spoking a wheel and summing the angles between the spokes using the cross product to keep track of the sign of each angle in computational geometry of the plane the cross product is used to determine the sign of the acute angle defined by three points p 1 x 1 y 1 p 2 x 2 y 2 displaystyle p_ 1 x_ 1 y_ 1 p_ 2 x_ 2 y_ 2 and p 3 x 3 y 3 displaystyle p_ 3 x_ 3 y_ 3 it corresponds to the direction upward or downward of the cross product of the two coplanar vectors defined by the two pairs of points p 1 p 2 displaystyle p_ 1 p_ 2 and p 1 p 3 displaystyle p_ 1 p_ 3 the sign of the acute angle is the sign of the expression p x 2 x 1 y 3 y 1 y 2 y 1 x 3 x 1 displaystyle p x_ 2 x_ 1 y_ 3 y_ 1 y_ 2 y_ 1 x_ 3 x_ 1 which is the signed length of the cross product of the two vectors to use the cross product simply extend the 2d vectors p 1 p 2 p 3 displaystyle p_ 1 p_ 2 p_ 3 to co planar 3d vectors by setting z k 0 displaystyle z_ k 0 for each of them in the right handed coordinate system if the result is 0 the points are collinear if it is positive the three points constitute a positive angle of rotation around p 1 displaystyle p_ 1 from p 2 displaystyle p_ 2 to p 3 displaystyle p_ 3 otherwise a negative angle from another point of view the sign of p displaystyle p tells whether p 3 displaystyle p_ 3 lies to the left or to the right of line p 1 p 2 displaystyle p_ 1 p_ 2 the cross product is used in calculating the volume of a polyhedron such as a tetrahedron or parallelepiped angular momentum and torque edit the angular momentum l of a particle about a given origin is defined as l r p displaystyle mathbf l mathbf r times mathbf p where r is the position vector of the particle relative to the origin p is the linear momentum of the particle in the same way the moment m of a force f b applied at point b around point a is given as m a r a b f b displaystyle mathbf m _ mathrm a mathbf r _ mathrm ab times mathbf f _ mathrm b in mechanics the moment of a force is also called torque and written as τ displaystyle mathbf tau since position r linear momentum p and force f are all true vectors both the angular momentum l and the moment of a force m are pseudovectors or axial vectors rigid body edit the cross product frequently appears in the description of rigid motions two points p and q on a rigid body can be related by v p v q ω r p r q displaystyle mathbf v _ p mathbf v _ q boldsymbol omega times left mathbf r _ p mathbf r _ q right where r displaystyle mathbf r is the point s position v displaystyle mathbf v is its velocity and ω displaystyle boldsymbol omega is the body s angular velocity since position r displaystyle mathbf r and velocity v displaystyle mathbf v are true vectors the angular velocity ω displaystyle boldsymbol omega is a pseudovector or axial vector lorentz force edit see also lorentz force the cross product is used to describe the lorentz force experienced by a moving electric charge q e f q e e v b displaystyle mathbf f q_ e left mathbf e mathbf v times mathbf b right since velocity v force f and electric field e are all true vectors the magnetic field b is a pseudovector other edit in vector calculus the cross product is used to define the formula for the vector operator curl the trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi view geometry in particular when deriving matching 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 math...
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