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Dot Product
Algebraic operation returning a single number from two equal-length sequences
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. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths).
The name "dot product" is derived from the centered dot " · ", that is often used to designate this operation;[1][2] the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector, as is the case for the vector product in three-dimensional space.
Definition
The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.
In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate spaceRn. In such a presentation, the notions of length and angles are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle of two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.
Algebraic definition
The dot product of two vectors and is defined as:[3]
Expressing the above example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with its unique entry:
.
Geometric definition
Illustration showing how to find the angle between vectors using the dot product
In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction to which the arrow points. The magnitude of a vector a is denoted by . The dot product of two Euclidean vectors a and b is defined by[4][5][2]
These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive definite, which means that
is never negative, and is zero if and only if --the zero vector.
The dot product is thus equivalent to multiplying the norm (length) of b by the norm of the projection of a over b.
The vectors ei are an orthonormal basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length
and since they form right angles with each other, if ,
Not associative because the dot product between a scalar (a ? b) and a vector (c) is not defined, which means that the expressions involved in the associative property, (a ? b) ? c or a ? (b ? c) are both ill-defined.[8] Note however that the previously mentioned scalar multiplication property is sometimes called the "associative law for scalar and dot product"[9] or one can say that "the dot product is associative with respect to scalar multiplication" because c (a ? b) = (ca) ? b = a ? (cb).[10]
Unlike multiplication of ordinary numbers, where if , then b always equals c unless a is zero, the dot product does not obey the cancellation law:
If and , then we can write: by the distributive law; the result above says this just means that a is perpendicular to , which still allows , and therefore allows .
This identity, also known as Lagrange's formula, may be remembered as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations in physics.
Physics
In physics, vector magnitude is a scalar in the physical sense (i.e., a physical quantity independent of the coordinate system), expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. For example:[11][12]
For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance, the dot product of a vector with itself would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called isotropic); this in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the scalar product, through the alternative definition[13][3]
where vectors were assumed represented as row vectors.
Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, this scalar product is thus sesquilinear rather than bilinear: it is conjugate linear and not linear in a, and the scalar product is not symmetric, since
The angle between two complex vectors is then given by
This type of scalar product is nevertheless useful, and leads to the notions of Hermitian form and of general inner product spaces.
The self dot product of a complex vector is a generalization of the absolute square of a complex scalar.
The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.
Functions
The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length-n vector u is, then, a function with domain{k ? N | 1 k n}, and ui is a notation for the image of i by the function/vector u.
This notion can be generalized to continuous functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some intervala x b (also denoted [a, b]):[3]
Generalized further to complex functions?(x) and ?(x), by analogy with the complex inner product above, gives[3]
Weight function
Inner products can have a weight function (i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions and with respect to the weight function is
Dyadics and matrices
Matrices have the Frobenius inner product, which is analogous to the vector inner product. It is defined as the sum of the products of the corresponding components of two matrices A and B having the same size:
The inner product between a tensor of order n and a tensor of order m is a tensor of order , see Tensor contraction for details.
Computation
Algorithms
The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. To avoid this, approaches such as the Kahan summation algorithm are used.
Libraries
A dot product function is included in BLAS level 1.