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atrix left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right w_ 3 begin vmatrix u_ 1 v_ 1 u_ 2 v_ 2 end vmatrix left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right w_ 1 begin vmatrix u_ 2 v_ 2 u_ 3 v_ 3 end vmatrix left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right w_ 2 begin vmatrix u_ 1 v_ 1 u_ 3 v_ 3 end vmatrix left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right w_ 3 begin vmatrix u_ 1 v_ 1 u_ 2 v_ 2 end vmatrix left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right left w_ 1 begin vmatrix u_ 2 v_ 2 u_ 3 v_ 3 end vmatrix w_ 2 begin vmatrix u_ 1 v_ 1 u_ 3 v_ 3 end vmatrix w_ 3 begin vmatrix u_ 1 v_ 1 u_ 2 v_ 2 end vmatrix right left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right begin vmatrix u_ 1 v_ 1 w_ 1 u_ 2 v_ 2 w_ 2 u_ 3 v_ 3 w_ 3 end vmatrix left mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 right end aligned this shows that the magnitude of the three vector u v w is the volume of the parallelepiped spanned by the three vectors u v and w in higher dimensional spaces the component three vectors are projections of the volume of a parallelepiped onto the coordinate three spaces and the magnitude of the three vector is the volume of the parallelepiped as it sits in the higher dimensional space grassmann coordinates edit in this section we consider multivectors on a projective space p n which provide a convenient set of coordinates for lines planes and hyperplanes that have properties similar to the homogeneous coordinates of points called grassmann coordinates 8 points in a real projective space p n are defined to be lines through the origin of the vector space r n 1 for example the projective plane p 2 is the set of lines through the origin of r 3 thus multivectors defined on r n 1 can be viewed as multivectors on p n a convenient way to view a multivector on p n is to examine it in an affine component of p n which is the intersection of the lines through the origin of r n 1 with a selected hyperplane such as h x n 1 1 lines through the origin of r 3 intersect the plane e z 1 to define an affine version of the projective plane that only lacks the points for which z 0 called the points at infinity multivectors on the projective plane p 2 edit points in the affine component e z 1 of the projective plane p 2 have coordinates x x y 1 a linear combination of two points p p 1 p 2 1 and q q 1 q 2 1 defines a plane in r 3 that intersects e in the line joining p and q the multivector p q defines a parallelogram in r 3 given by p q p 2 q 2 e 2 e 3 p 1 q 1 e 1 e 3 p 1 q 2 q 1 p 2 e 1 e 2 displaystyle mathbf p wedge mathbf q p_ 2 q_ 2 mathbf e _ 2 wedge mathbf e _ 3 p_ 1 q_ 1 mathbf e _ 1 wedge mathbf e _ 3 p_ 1 q_ 2 q_ 1 p_ 2 mathbf e _ 1 wedge mathbf e _ 2 notice that substitution of α p β q for p multiplies this multivector by a constant therefore the components of p q are homogeneous coordinates for the plane through the origin of r 3 the set of points x x y 1 on the line through p and q is the intersection of the plane defined by p q with the plane e z 1 these points satisfy x p q 0 that is x p q x e 1 y e 2 e 3 p 2 q 2 e 2 e 3 p 1 q 1 e 1 e 3 p 1 q 2 q 1 p 2 e 1 e 2 0 displaystyle mathbf x wedge mathbf p wedge mathbf q x mathbf e _ 1 y mathbf e _ 2 mathbf e _ 3 wedge big p_ 2 q_ 2 mathbf e _ 2 wedge mathbf e _ 3 p_ 1 q_ 1 mathbf e _ 1 wedge mathbf e _ 3 p_ 1 q_ 2 q_ 1 p_ 2 mathbf e _ 1 wedge mathbf e _ 2 big 0 which simplifies to the equation of a line λ x p 2 q 2 y p 1 q 1 p 1 q 2 q 1 p 2 0 displaystyle lambda x p_ 2 q_ 2 y p_ 1 q_ 1 p_ 1 q_ 2 q_ 1 p_ 2 0 this equation is satisfied by points x α p β q for real values of α and β the three components of p q that define the line λ are called the grassmann coordinates of the line because three homogeneous coordinates define both a point and a line the geometry of points is said to be dual to the geometry of lines in the projective plane this is called the principle of duality multivectors on projective 3 space p 3 edit three dimensional projective space p 3 consists of all lines through the origin of r 4 let the three dimensional hyperplane h w 1 be the affine component of projective space defined by the points x x y z 1 the multivector p q r defines a parallelepiped in r 4 given by p q r p 2 q 2 r 2 p 3 q 3 r 3 1 1 1 e 2 e 3 e 4 p 1 q 1 r 1 p 3 q 3 r 3 1 1 1 e 1 e 3 e 4 p 1 q 1 r 1 p 2 q 2 r 2 1 1 1 e 1 e 2 e 4 p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3 e 1 e 2 e 3 displaystyle mathbf p wedge mathbf q wedge mathbf r begin vmatrix p_ 2 q_ 2 r_ 2 p_ 3 q_ 3 r_ 3 1 1 1 end vmatrix mathbf e _ 2 wedge mathbf e _ 3 wedge mathbf e _ 4 begin vmatrix p_ 1 q_ 1 r_ 1 p_ 3 q_ 3 r_ 3 1 1 1 end vmatrix mathbf e _ 1 wedge mathbf e _ 3 wedge mathbf e _ 4 begin vmatrix p_ 1 q_ 1 r_ 1 p_ 2 q_ 2 r_ 2 1 1 1 end vmatrix mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 4 begin vmatrix p_ 1 q_ 1 r_ 1 p_ 2 q_ 2 r_ 2 p_ 3 q_ 3 r_ 3 end vmatrix mathbf e _ 1 wedge mathbf e _ 2 wedge mathbf e _ 3 notice that substitution of α p β q γ r for p multiplies this multivector by a constant therefore the components of p q r are homogeneous coordinates for the 3 space through the origin of r 4 a plane in the affine component h w 1 is the set of points x x y z 1 in the intersection of h with the 3 space defined by p q r these points satisfy x p q r 0 that is x p q r x e 1 y e 2 z e 3 e 4 p q r 0 displaystyle mathbf x wedge mathbf p wedge mathbf q wedge mathbf r x mathbf e _ 1 y mathbf e _ 2 z mathbf e _ 3 mathbf e _ 4 wedge mathbf p wedge mathbf q wedge mathbf r 0 which simplifies to the equation of a plane λ x p 2 q 2 r 2 p 3 q 3 r 3 1 1 1 y p 1 q 1 r 1 p 3 q 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 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