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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.
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector.
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.
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.
In vector notation, a plane can be expressed as the set of points for which =where is a normal vector to the plane and is a point on the plane. (The notation denotes the dot product of the vectors and .)
As at Cross product, this presumably relates to language, where we use a form of abbreviation, e.g. "the dot product of two vectors a and b" to mean "the result of applying the dot product to two vectors a and b". Thus, the "dot product" does not mean the result, even though the language use would suggest this on first reading.
The dot products are x i * x j in variant #3, h i * s j in variant 1, and column i ( Kw * H ) * column j ( Qw * S ) in variant 2, and column i ( Kw * X ) * column j ( Qw * X ) in variant 4. Variant 5 uses a fully-connected layer to determine the coefficients. If the variant is QKV, then the dot products are normalized by the √ d where d is ...