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The term vector was coined by W. R. Hamilton around 1843, as he revealed quaternions, a system which uses vectors and scalars to span a four-dimensional space. For a quaternion q = a + b i + c j + d k, Hamilton used two projections: S q = a , for the scalar part of q , and V q = b i + c j + d k, the vector part.
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
A vector database, vector store or vector search engine is a database that can store vectors (fixed-length lists of numbers) along with other data items. Vector databases typically implement one or more Approximate Nearest Neighbor algorithms, [1] [2] [3] so that one can search the database with a query vector to retrieve the closest matching database records.
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.
Cases of 0 ≤ n ≤ 1 do not offer anything new: R 1 is the real line, whereas R 0 (the space containing the empty column vector) is a singleton, understood as a zero vector space. However, it is useful to include these as trivial cases of theories that describe different n .
Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.
A vector pointing from A to B. In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector [1] or spatial vector [2]) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space.
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j and k. In mathematics , the standard basis (also called natural basis or canonical basis ) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of whose ...