Algebraically, the dot product is the sum of theproducts of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of theEuclidean magnitudes of the two vectors and thecosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In moderngeometry,Euclidean spaces are often defined by usingvector spaces. In this case, the dot product is used for defining lengths (the length of a vector is thesquare root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is thequotient of their dot product by the product of their lengths).
The name "dot product" is derived from thedot operator " · " that is often used to designate this operation;[1] the alternative name "scalar product" emphasizes that the result is ascalar, rather than a vector (as with thevector product in three-dimensional space).
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 aCartesian coordinate system for Euclidean space.
In modern presentations ofEuclidean geometry, the points of space are defined in terms of theirCartesian coordinates, andEuclidean space itself is commonly identified with thereal coordinate space. In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as thesquare root of the dot product of the vector by itself, and thecosine of the (non oriented) angle between 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.
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:
Illustration showing how to find the angle between vectors using the dot product Calculating bond angles of a symmetricaltetrahedral molecular geometry using a dot product
InEuclidean space, aEuclidean 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. Themagnitude of a vector is denoted by. The dot product of two Euclidean vectors and is defined by[3][4][1]where is theangle between and.
In particular, if the vectors and areorthogonal (i.e., their angle is or), then, which implies thatAt the other extreme, if they arecodirectional, then the angle between them is zero with andThis implies that the dot product of a vector with itself iswhich givesthe formula for theEuclidean length of the vector.
Thescalar projection (or scalar component) of a Euclidean vector in the direction of a Euclidean vector is given bywhere is the angle between and.
In terms of the geometric definition of the dot product, this can be rewritten aswhere is theunit vector in the direction of.
Distributive law for the dot product
The dot product is thus characterized geometrically by[5]The dot product, defined in this manner, ishomogeneous under scaling in each variable, meaning that for any scalar,It also satisfies thedistributive law, meaning that
These properties may be summarized by saying that the dot product is abilinear form. Moreover, this bilinear form ispositive definite, which means that is never negative, and is zero if and only if, the zero vector.
If are thestandard basis vectors in, then we may writeThe vectors are anorthonormal basis, which means that they have unit length and are at right angles to each other. Since these vectors have unit length,and since they form right angles with each other, if,Thus in general, we can say that:where is theKronecker delta.
Vector components in an orthonormal basis
Also, by the geometric definition, for any vector and a vector, we note thatwhere is the component of vector in the direction of. The last step in the equality can be seen from the figure.
Now applying the distributivity of the geometric version of the dot product giveswhich is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.
which follows from the definition ( is the angle between and):[6] The commutative property can also be easily proven with the algebraic definition, and inmore general spaces (where the notion of angle might not be geometrically intuitive but an analogous product can be defined) the angle between two vectors can be defined as
Bilinear (additive, distributive and scalar-multiplicative in both arguments)
Because the dot product is not defined between a scalar and a vector associativity is meaningless.[7] However, bilinearity implies This property is sometimes called the "associative law for scalar and dot product",[8] and one may say that "the dot product is associative with respect to scalar multiplication".[9]
Unlike multiplication of ordinary numbers, where if, then always equals unless is zero, the dot product does not obey thecancellation law: If and, then we can write: by thedistributive law; the result above says this just means that is perpendicular to, which still allows, and therefore allows.
Given two vectors and separated by angle (see the upper image), they form a triangle with a third side. Let, and denote the lengths of,, and, respectively. The dot product of this with itself is:which is thelaw of cosines.
Thescalar triple product of three vectors is defined asIts value is thedeterminant of the matrix whose columns are theCartesian coordinates of the three vectors. It is the signedvolume of theparallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of theexterior product of three vectors.
Thevector triple product is defined by[2][3]This identity, also known asLagrange's formula,may be remembered as "ACB minus ABC", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations inphysics.
Inphysics, the dot product takes two vectors and returns ascalar quantity. It is also known as the "scalar product". 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. Thus, 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.
For vectors withcomplex 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 could be zero without the vector being the zero vector (e.g. this would happen with the vector). 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 dot product, through the alternative definition[12][2]where is thecomplex conjugate of. When vectors are represented bycolumn vectors, the dot product can be expressed as amatrix product involving aconjugate transpose, denoted with the superscript H:
In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product issesquilinear rather than bilinear, as it isconjugate linear and not linear in. The dot product is not symmetric, sinceThe angle between two complex vectors is then given by
The self dot product of a complex vector, involving the conjugate transpose of a row vector, is also known as thenorm squared,, after theEuclidean norm; it is a vector generalization of theabsolute square of a complex scalar (see also:Squared Euclidean distance).
The inner product of two vectors over the field of complex numbers is, in general, a complex number, and issesquilinear instead of bilinear. An inner product space is anormed vector space, and the inner product of a vector with itself is real and positive-definite.
The dot product is defined for vectors that have a finite number ofentries. Thus these vectors can be regarded asdiscrete functions: a length- vector is, then, a function withdomain, and is a notation for the image of by the function/vector.
This notion can be generalized tosquare-integrable 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 somemeasure space:[2]
Inner products can have aweight 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
A double-dot product formatrices is theFrobenius inner product, which is analogous to the dot product on vectors. It is defined as the sum of the products of the corresponding components of two matrices and of the same size:And for real matrices,
The straightforward algorithm for calculating a floating-point dot product of vectors can suffer fromcatastrophic cancellation. To avoid this, approaches such as theKahan summation algorithm are used.
