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In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used.
A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot product representation. [1] [2] [3]
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.
The scalar projection is defined as [2] = ‖ ‖ = ^ where the operator ⋅ denotes a dot product, ‖a‖ is the length of a, and θ is the angle between a and b. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b ...
In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not depend on the ...
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication.
where the operator denotes a dot product, ^ is the unit vector in the direction of , ‖ ‖ is the length of , and is the angle between and . [ 1 ] The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates , the components of a vector are the scalar projections in the directions of the coordinate axes .
In this sense, the unit dyadic ij is the function from 3-space to itself sending a 1 i + a 2 j + a 3 k to a 2 i, and jj sends this sum to a 2 j. Now it is revealed in what (precise) sense ii + jj + kk is the identity: it sends a 1 i + a 2 j + a 3 k to itself because its effect is to sum each unit vector in the standard basis scaled by the ...