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Vector Algebra Relations
The following are important identities in vector algebra. Identities that involve the magnitude of a vector \, \mathbf A\, , or the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension. Identities that use the cross product (vector product) A×B are defined only in three dimensions. Magnitudes The magnitude of a vector A can be expressed using the dot product: :\, \mathbf A \, ^2 = \mathbf In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem: :\, \mathbf A \, ^2 = A_1^2 + A_2^2 +A_3^2 Inequalities *The Cauchy–Schwarz inequality: \mathbf \cdot \mathbf \le \left\, \mathbf A \right\, \left\, \mathbf B \right\, *The triangle inequality: \, \mathbf\, \le \, \mathbf\, + \, \mathbf\, *The reverse triangle inequality: \, \mathbf\, \ge \Bigl, \, \mathbf\, - \, \mathbf\, \Bigr, Angles The vector product and the scalar product of two vectors define the angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Algebra
In mathematics, vector algebra may mean: * Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. * The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean space \R^3 of dot product and especially cross product. In this sense, vector algebra is contrasted with geometric algebra, which provides an alternative generalization to higher dimensions. * An algebra over a field, a vector space equipped with a bilinear product *Original vector algebras of the nineteenth century like quaternions, tessarines, or coquaternions, each of which has its own product (mathematics), product. The vector algebras biquaternions and hyperbolic quaternions enabled the revolution in physics called special relativity by providing mathematical models. Algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operations On Vectors
Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man Publishing's house organ for articles and discussion about its wargaming products * ''The Operation'' (film), a 1973 British television film * ''The Operation'' (1990), a crime, drama, TV movie starring Joe Penny, Lisa Hartman, and Jason Beghe * ''The Operation'' (1992–1998), a reality television series from TLC * The Operation M.D., formerly The Operation, a Canadian garage rock band * "Operation", a song by Relient K from '' The Creepy EP'', 2001 Business * Business operations, the harvesting of value from assets owned by a business * Manufacturing operations, operation of a facility * Operations management, an area of management concerned with designing and controlling the process of production Military and law enforcement ... [...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] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Vectors
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the ''dimension'' of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Quadruple Product
In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors. Scalar quadruple product The scalar quadruple product is defined as the dot product of two cross products: : (\mathbf)\cdot(\mathbf\times \mathbf) \ , where a, b, c, d are vectors in three-dimensional Euclidean space. It can be evaluated using the identity: : (\mathbf)\cdot(\mathbf\times \mathbf) = (\mathbf)(\mathbf) - (\mathbf)(\mathbf) \ . or using the determinant: :(\mathbf)\cdot(\mathbf\times \mathbf) =\begin \mathbf & \mathbf \\ \mathbf & \mathbf \end \ . Proof We first prove that :\begin \mathbf \times (\mathbf \times \mathbf) \cdot \mathbf = (\mathbf \times \mathbf) \cdot (\mathbf \times \mathbf). \end This can be shown by straightforward matrix algebra using the corresp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange's Identity
In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is: \begin \left( \sum_^n a_k^2\right) \left(\sum_^n b_k^2\right) - \left(\sum_^n a_k b_k\right)^2 & = \sum_^ \sum_^n \left(a_i b_j - a_j b_i\right)^2 \\ & \left(= \frac \sum_^n \sum_^n (a_i b_j - a_j b_i)^2\right), \end which applies to any two sets and of real number, real or complex numbers (or more generally, elements of a commutative ring). This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity. In a more compact vector notation, Lagrange's identity is expressed as: \left\, \mathbf a \right\, ^2 \left\, \mathbf b \right\, ^2 - (\mathbf \cdot \mathbf )^2 = \sum_ \left(a_i b_j - a_j b_i \right)^2 \, , where a and b are ''n''-dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner product or Hermitian dot product. Explicitly, for complex numbers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jacob Jacobi. The cross product a\times b and the Lie bracket operation ,b/math> both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket. Definition Let + and \times be two binary operations, and let 0 be the neutral element for +. The is :x \times (y \times z) \ +\ y \times (z \times x) \ +\ z \times (x \times y)\ =\ 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Triple Product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. Scalar triple product The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation Geometrically, the scalar triple product : \mathbf\cdot(\mathbf\times \mathbf) is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined. Properties * The scalar triple product is unchanged under a circular shift of its three operands (a, b, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Triple Product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. Scalar triple product The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation Geometrically, the scalar triple product : \mathbf\cdot(\mathbf\times \mathbf) is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined. Properties * The scalar triple product is unchanged under a circular shift of its three operands (a, b, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
