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3 r 3 1 1 1 z p 1 q 1 r 1 p 2 q 2 r 2 1 1 1 p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3 0 displaystyle lambda x begin vmatrix p_ 2 q_ 2 r_ 2 p_ 3 q_ 3 r_ 3 1 1 1 end vmatrix y begin vmatrix p_ 1 q_ 1 r_ 1 p_ 3 q_ 3 r_ 3 1 1 1 end vmatrix z begin vmatrix p_ 1 q_ 1 r_ 1 p_ 2 q_ 2 r_ 2 1 1 1 end vmatrix begin vmatrix p_ 1 q_ 1 r_ 1 p_ 2 q_ 2 r_ 2 p_ 3 q_ 3 r_ 3 end vmatrix 0 this equation is satisfied by points x α p β q γ r for real values of α β and γ the four components of p q r that define the plane λ are called the grassmann coordinates of the plane because four homogeneous coordinates define both a point and a plane in projective space the geometry of points is dual to the geometry of planes a line as the join of two points in projective space the line λ through two points p and q can be viewed as the intersection of the affine space h w 1 with the plane x α p β q in r 4 the multivector p q provides homogeneous coordinates for the line λ p q p 1 e 1 p 2 e 2 p 3 e 3 e 4 q 1 e 1 q 2 e 2 q 3 e 3 e 4 p 1 q 1 1 1 e 1 e 4 p 2 q 2 1 1 e 2 e 4 p 3 q 3 1 1 e 3 e 4 p 2 q 2 p 3 q 3 e 2 e 3 p 3 q 3 p 1 q 1 e 3 e 1 p 1 q 1 p 2 q 2 e 1 e 2 displaystyle begin aligned lambda mathbf p wedge mathbf q p_ 1 mathbf e _ 1 p_ 2 mathbf e _ 2 p_ 3 mathbf e _ 3 mathbf e _ 4 wedge q_ 1 mathbf e _ 1 q_ 2 mathbf e _ 2 q_ 3 mathbf e _ 3 mathbf e _ 4 begin vmatrix p_ 1 q_ 1 1 1 end vmatrix mathbf e _ 1 wedge mathbf e _ 4 begin vmatrix p_ 2 q_ 2 1 1 end vmatrix mathbf e _ 2 wedge mathbf e _ 4 begin vmatrix p_ 3 q_ 3 1 1 end vmatrix mathbf e _ 3 wedge mathbf e _ 4 begin vmatrix p_ 2 q_ 2 p_ 3 q_ 3 end vmatrix mathbf e _ 2 wedge mathbf e _ 3 begin vmatrix p_ 3 q_ 3 p_ 1 q_ 1 end vmatrix mathbf e _ 3 wedge mathbf e _ 1 begin vmatrix p_ 1 q_ 1 p_ 2 q_ 2 end vmatrix mathbf e _ 1 wedge mathbf e _ 2 end aligned these are known as the plücker coordinates of the line though they are also an example of grassmann coordinates a line as the intersection of two planes a line μ in projective space can also be defined as the set of points x that form the intersection of two planes π and ρ defined by grade three multivectors so the points x are the solutions to the linear equations μ x π 0 x ρ 0 displaystyle mu mathbf x wedge pi 0 mathbf x wedge rho 0 in order to obtain the plucker coordinates of the line μ map the multivectors π and ρ to their dual point coordinates using the right complement denoted by an overline as in 9 e 1 e 2 e 3 e 4 e 2 e 3 e 1 e 4 e 3 e 1 e 2 e 4 e 4 e 1 e 2 e 3 displaystyle mathbf e _ 1 overline mathbf e _ 2 wedge mathbf e _ 3 wedge mathbf e _ 4 quad mathbf e _ 2 overline mathbf e _ 3 wedge mathbf e _ 1 wedge mathbf e _ 4 quad mathbf e _ 3 overline mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 4 quad mathbf e _ 4 overline mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 then π π 1 e 1 π 2 e 2 π 3 e 3 π 4 e 4 ρ ρ 1 e 1 ρ 2 e 2 ρ 3 e 3 ρ 4 e 4 displaystyle overline pi pi _ 1 mathbf e _ 1 pi _ 2 mathbf e _ 2 pi _ 3 mathbf e _ 3 pi _ 4 mathbf e _ 4 quad overline rho rho _ 1 mathbf e _ 1 rho _ 2 mathbf e _ 2 rho _ 3 mathbf e _ 3 rho _ 4 mathbf e _ 4 so the plücker coordinates of the line μ are given by μ π ρ _ π 1 ρ 1 π 4 ρ 4 e 2 e 3 π 2 ρ 2 π 4 ρ 4 e 3 e 1 π 3 ρ 3 π 4 ρ 4 e 1 e 2 π 2 ρ 2 π 3 ρ 3 e 1 e 4 π 3 ρ 3 π 1 ρ 1 e 2 e 4 π 1 ρ 1 π 2 ρ 2 e 3 e 4 displaystyle begin aligned mu underline overline pi wedge overline rho begin vmatrix pi _ 1 rho _ 1 pi _ 4 rho _ 4 end vmatrix mathbf e _ 2 wedge mathbf e _ 3 begin vmatrix pi _ 2 rho _ 2 pi _ 4 rho _ 4 end vmatrix mathbf e _ 3 wedge mathbf e _ 1 begin vmatrix pi _ 3 rho _ 3 pi _ 4 rho _ 4 end vmatrix mathbf e _ 1 wedge mathbf e _ 2 begin vmatrix pi _ 2 rho _ 2 pi _ 3 rho _ 3 end vmatrix mathbf e _ 1 wedge mathbf e _ 4 begin vmatrix pi _ 3 rho _ 3 pi _ 1 rho _ 1 end vmatrix mathbf e _ 2 wedge mathbf e _ 4 begin vmatrix pi _ 1 rho _ 1 pi _ 2 rho _ 2 end vmatrix mathbf e _ 3 wedge mathbf e _ 4 end aligned where the underline denotes the left complement the left complement of the wedge product of right complements is called the antiwedge product denoted by a downward pointing wedge allowing us to write μ π ρ displaystyle mu pi vee rho clifford product edit w k clifford combined multivectors with the inner product defined on the vector space in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and hamilton s quaternions 10 11 the clifford product between two vectors u and v is bilinear and associative like the exterior product and has the additional property that the multivector uv is coupled to the inner product u v by clifford s relation u v v u 2 u v displaystyle mathbf u mathbf v mathbf v mathbf u 2 mathbf u cdot mathbf v clifford s relation retains the anticommuting property for vectors that are perpendicular this can be seen from the mutually orthogonal unit vectors e i i 1 n in r n clifford s relation yields e i e j e j e i 2 e i e j δ i j displaystyle mathbf e _ i mathbf e _ j mathbf e _ j mathbf e _ i 2 mathbf e _ i cdot mathbf e _ j delta _ i j which shows that the basis vectors mutually anticommute e i e j e j e i i j 1 n displaystyle mathbf e _ i mathbf e _ j mathbf e _ j mathbf e _ i quad i neq j 1 ldots n in contrast to the exterior product the clifford product of a vector with itself is not zero to see this compute the product e i e i e i e i 2 e i e i 2 displaystyle mathbf e _ i mathbf e _ i mathbf e _ i mathbf e _ i 2 mathbf e _ i cdot mathbf e _ i 2 which yields e i e i 1 i 1 n displaystyle mathbf e _ i mathbf e _ i 1 quad i 1 ldots n the set of multivectors constructed using clifford s product yields an associative algebra known as a clifford algebra inner products with different properties can be used to construct different clifford algebras 12 13 geometric algebra edit see also blade geometry the term k blade was used in clifford algebra to geometric calculus 1984 14 multivectors play a central role in the mathematical formulation of physics known as geometric algebra according to david hestenes non scalar k vectors are sometimes called k blades or merely blades to emphasize the fact that in contrast to 0 vectors scalars they have directional properties 15 in 2003 the term blade for a multivector that can be written as the exterior product of a scalar and a set of vectors was used by c doran and a lasenby here by the statement any multivector can be expressed as the sum of blades scalars are implicitly defined as 0 blades 16 in geometric algebra a multivector is defined to be the sum of different grade k blades such as the summation of a scalar a vector and a 2 vector 17 a sum of only k grade components is called a k vector 18 or a homogeneous multivector 19 the highest grade element in a space is called a pseudoscalar if a given element is homogeneous of a grade k then it is a k vector but not necessarily a k blade such an element is a k blade when it can be expressed as the exterior product of k vectors a geometric algebra generated by a four dimensional vector space illustrates the point with an example the sum of any two blades with one taken from the xy plane and the other taken from the zw plane will form a 2 vector that is not a 2 blade in a geometric algebra generated by a vector space of dimension 2 or 3 all sums of 2 blades may be written as a single 2 blade examples edit orientation defined by an ordered set of vectors reversed orientation corresponds to negating the exterior product geometric interpretation of grade n elements in a real exterior algebra for n 0 signed point 1 directed line segment or vector 2 oriented plane element 3 oriented volume the exterior product of n vectors can be visualized as any n dimensional shape e g n parallelotope n ellipsoid with magnitude hypervolume and orientation defined by that on its n 1 dimensional boundary and on which side the interior is 20 21 0 vectors are scalars 1 vectors are vectors 2 vectors are bivectors n 1 vectors are pseudovectors n vectors are pseudoscalars in the presence of a volume form such as given an inner product and an orientation pseudovectors and pseudoscalars can be identified with vectors and scalars which is routine in vector calculus but without a volume form this cannot be done without making an arbitrary choice in the algebra of physical space the geometric algebra of euclidean 3 space used as a model of 3 1 spacetime a sum of a scalar and a vector is called a paravector and represents a point in spacetime the vector the space the scalar the time bivectors edit main article bivector a bivector is an element of the antisymmetric tensor product of a tangent space with itself in geometric algebra also a bivector is a grade 2 element a 2 vector resulting from the wedge product of two vectors and so it is geometrically an oriented area in the same way a vector is an oriented line segment if a and b are two vectors the bivector a b has a norm which is its area given by a b a b sin ϕ a b displaystyle left mathbf a wedge mathbf b right left mathbf a right left mathbf b right sin phi _ a b a direction the plane where that area lies on i e the plane determined by a and b as long as they are linearly independent an orientation out of two determined by the order in which the originating vectors are multiplied bivectors are connected to pseudovectors and are used to represent rotations in geometric algebra as bivectors are elements of a vector space λ 2 v where v is a finite dimensional vector space with dim v n it makes sense to define an inner product on this vector space as follows first write any element f λ 2 v in terms of a basis e i e j 1 i j n of λ 2 v as f f a b e a e b 1 a b n displaystyle f f ab mathbf e _ a wedge mathbf e _ b quad 1 leq a b leq n where the einstein summation convention is being used now define a map g λ 2 v λ 2 v r by insisting that g f h g a b c d f a b h c d displaystyle g f h g_ abcd f ab h cd where g a b c d displaystyle g_ abcd are a set of numbers applications edit bivectors play many important roles in physics for example in the classification of electromagnetic fields see also edit blade geometry multivector field paravector references edit john snygg 2012 a new approach to differential geometry using clifford s geometric algebra birkhäuser p 5 2 12 1 2 3 harley flanders 1989 1963 differential forms with applications to the physical sciences 2 1 the space of p vectors pages 5 7 dover books wendell fleming 1977 1965 functions of several variables section 7 5 multivectors page 295 isbn 978 1 4684 9461 7 élie cartan the theory of spinors p 16 considers only homogeneous vectors particularly simple ones referring to them as multivectors collectively or p vectors specifically william m pezzaglia jr 1992 clifford algebra derivation of the characteristic hypersurfaces of maxwell s equations in julian ławrynowicz ed deformations of mathematical structures ii springer p 131 ff isbn 0 7923 2576 1 hence in 3d we associate the alternate terms of pseudovector for bivector and pseudoscalar for the trivector baylis 1994 theoretical methods in the physical sciences an introduction to problem solving using maple v birkhäuser p 234 see footnote isbn 0 8176 3715 x g e shilov linear algebra trans r a silverman dover publications 1977 w v d hodge and d pedoe methods of algebraic geometry vol 1 cambridge univ press 1947 eric lengyel 2024 projective geometric algebra illuminated 2 2 complements pages 44 46 isbn 979 8 9853582 5 4 w k clifford preliminary sketch of bi quaternions proc london math soc vol 4 1873 pp 381 395 w k clifford mathematical papers ed r tucker london macmillan 1882 j m mccarthy an introduction to theoretical kinematics pp 62 5 mit press 1990 o bottema and b roth theoretical kinematics north holland publ co 1979 david hestenes garret sobczyk 1984 clifford algebra to geometric calculus p 4 d reidel isbn 90 277 1673 0 david hestenes 1999 1986 new foundations for classical mechanics page 34 d reidel isbn 90 277 2090 8 c doran and a lasenby 2003 geometric algebra for physicists page 87 cambridge university press isbn 9780511807497 marcos a rodrigues 2000 1 2 geometric algebra an outline invariants for pattern recognition and classification world scientific p 3 ff isbn 981 02 4278 6 r wareham j cameron j lasenby 2005 applications of conformal geometric algebra in computer vision and graphics in hongbo li peter j olver gerald sommer eds computer algebra and geometric algebra with applications springer p 330 isbn 3 540 26296 2 eduardo bayro corrochano 2004 clifford geometric algebra a promising framework for computer vision robotics and learning in alberto sanfeliu josé francisco martínez trinidad jesús ariel carrasco ochoa eds progress in pattern recognition image analysis and applications springer p 25 isbn 3 540 23527 2 r penrose 2007 the road to reality vintage books isbn 978 0 679 77631 4 j a wheeler c misner k s thorne 1973 gravitation w h freeman co p 83 isbn 0 7167 0344 0 v t e linear algebra outline glossary template matrix classes linear equations linear equation system of linear equations determinant minor cauchy binet formula cramer s rule gaussian elimination gauss jordan elimination overcompleteness strassen algorithm matrices matrix matrix addition matrix multiplication basis transformation matrix characteristic polynomial spectrum trace eigenvalue eigenvector and eigenspace cayley hamilton theorem jordan normal form weyr canonical form rank inverse pseudoinverse adjugate transpose dot product symmetric matrix skew symmetric matrix orthogonal matrix unitary matrix hermitian matrix antihermitian matrix positive semi definite pfaffian projection spectral theorem perron frobenius theorem diagonal matrix triangular matrix tridiagonal matrix block matrix sparse matrix hessenberg matrix hessian matrix vandermonde matrix stochastic matrix toeplitz matrix circulant matrix hankel matrix 0 1 matrix list of matrices matrix decompositions cholesky decomposition lu decomposition qr decomposition polar decomposition spectral theorem singular value decomposition higher order singular value decomposition schur decomposition schur complement haynsworth inertia additivity formula reducing subspace relations and computations matrix equivalence matrix congruence matrix similarity matrix consimilarity row equivalence elementary row operations householder transformation least squares linear least squares gram schmidt process woodbury matrix identity vector spaces vector space linear combination linear span linear independence basis hamel basis change of basis dimension theorem for vector spaces hamel dimension examples of vector spaces linear map shear mapping squeeze mapping linear subspace row and column spaces null space rank nullity theorem nullity theorem cyclic subspace dual space linear functional category of vector spaces structures topological vector space normed vector space inner product space euclidean space orthogonality orthogonal complement orthogonal projection orthogonal group pseudo euclidean space null vector indefi...
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