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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.
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 generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule
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]
Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic.
The cross product occurs frequently in the study of rotation, where it is used to calculate torque and angular momentum. It can also be used to calculate the Lorentz force exerted on a charged particle moving in a magnetic field. The dot product is used to determine the work done by a constant force.
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 ...
The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors. In 1881, Josiah Willard Gibbs, [10] and independently Oliver Heaviside, introduced the notation for both the dot product and the cross product using a period (a ⋅ b) and an "×" (a × b), respectively, to denote ...