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The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
The operations of vector addition and scalar multiplication of a vector space; The algebraic operations in vector calculus (vector analysis) – including the dot and cross products of 3-dimensional Euclidean space; Algebra over a field – a vector space equipped with a bilinear product; Any of the original vector algebras of the nineteenth ...
Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly due to historical reasons. Vector quaternion, a quaternion with a zero real part; Multivector or p-vector, an element of the exterior algebra of a vector space.
The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds: The following relationship holds: a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b ...
Geometric algebra (GA) is an extension or completion of vector algebra (VA). [1] The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in the GA of 3D space (nor is this article intended to be mathematically rigorous). In GA, vectors are not normally ...