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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.
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.
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.
For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB. [1]
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]
In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: Dot product – also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the ...
The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products: [1] (()) (()) = [] This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product.
If one views any real m × n matrix as a vector of length mn (an operation called vectorization) then the above operation on A and B coincides with the standard dot product. According to the above expression, tr( A ⊤ A ) is a sum of squares and hence is nonnegative, equal to zero if and only if A is zero.