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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 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 ...
The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors. The product of a scalar k with a vector v can be represented in any of the following fashions:
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 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 ...
From January 2008 to December 2012, if you bought shares in companies when E. William Barnett joined the board, and sold them when he left, you would have a 57.6 percent return on your investment, compared to a -2.8 percent return from the S&P 500.
The dot product operator involving vectors is a good example of a covector. To illustrate, assume we have a covector defined as v ⋅ {\displaystyle \mathbf {v} \ \cdot } , where v {\displaystyle \mathbf {v} } is a vector.