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Antivector
An antivector is an element of grade in an ''n''-dimensional exterior algebra. An antivector is always a blade, and it gets its name from the fact that its components each involve a combination of all except one basis vector, thus being the opposite of a vector whose components each involve exactly one basis vector. Like a vector, an antivector has ''n'' components in ''n''-dimensional space, and this sometimes leads to an inadequate distinction being made between the two types of entities. However, antivectors transform differently with a change of basis than vectors do, which shows that they are different kinds of quantities. In physics, the names ''pseudovector'' and ''axial vector'' are used to describe vectors that transform in the same way that an antivector transforms. These typically arise as the result of cross products between two vectors. See also * Exterior algebra *Geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its opposite if the orientation of the space is changed, or an improper rigid transformation such as a reflection is applied to the whole figure. Geometrically, the direction of a reflected pseudovector is opposite to its mirror image, but with equal magnitude. In contrast, the reflection of a ''true'' (or polar) vector is exactly the same as its mirror image. In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors. One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blade (geometry)
In the study of geometric algebras, a -blade or a simple -vector is a generalization of the concept of scalars and vectors to include ''simple'' bivectors, trivectors, etc. Specifically, a -blade is a -vector that can be expressed as the exterior product (informally ''wedge product'') of 1-vectors, and is of '' grade'' . In detail: *A 0-blade is a scalar. *A 1-blade is a vector. Every vector is simple. *A 2-blade is a ''simple'' bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and : *:a \wedge b . *A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors , , and : *:a \wedge b \wedge c. *In a vector space of dimension , a blade of grade is called a ''pseudovector'' or an ''antivector''. *The highest grade element in a space is called a '' pseudoscalar'', and in a space of dimension is an -blade. *In a vector space of dimension , there are di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
