In
mathematics, the dot product or scalar product
[The term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other ]symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinea ...
s, for example in a pseudo-Euclidean space. is an
algebraic operation
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
that takes two equal-length sequences of numbers (usually
coordinate vectors), and returns a single number. In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, 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
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
,
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s are often defined by using
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. 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 between two vectors is the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
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;
the alternative name "scalar product" emphasizes that the result is a
scalar, rather than a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, as is the case for the
vector product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, the points of space are defined in terms of their
Cartesian coordinates, and
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
itself is commonly identified with the
real coordinate space R
''n''. 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 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.
Coordinate definition
The dot product of two vectors and specified with respect to an
orthonormal basis, is defined as:
:
where Σ denotes
summation and ''n'' is the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. For instance, in
three-dimensional space, the dot product of vectors and is:
:
Likewise, the dot product of the vector with itself is:
:
If vectors are identified with
row matrices, the dot product can also be written as a
matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
:
where
denotes the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of
.
Expressing the above example in this way, a 1 × 3 matrix (
row vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
) is multiplied by a 3 × 1 matrix (
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
) to get a 1 × 1 matrix that is identified with its unique entry:
:
.
Geometric definition
In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, 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
:
where is the
angle between and .
In particular, if the vectors and are
orthogonal (i.e., their angle is or 90°), then
, which implies that
:
At the other extreme, if they are codirectional, then the angle between them is zero with
and
:
This implies that the dot product of a vector a with itself is
:
which gives
:
the formula for the
Euclidean length of the vector.
Scalar projection and first properties
The
scalar projection
In mathematics, the scalar projection of a vector \mathbf on (or onto) a vector \mathbf, also known as the scalar resolute of \mathbf in the direction of \mathbf, is given by:
:s = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf,
wher ...
(or scalar component) of a Euclidean vector a in the direction of a Euclidean vector b is given by
:
where is the angle between a and b.
In terms of the geometric definition of the dot product, this can be rewritten
:
where
is the
unit vector in the direction of b.
The dot product is thus characterized geometrically by
:
The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar ''α'',
:
It also satisfies a
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
, meaning that
:
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.
Equivalence of the definitions
If e
1, ..., e
''n'' are the
standard basis vectors in R
''n'', then we may write
:
The vectors e
''i'' 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 ,
:
Thus in general, we can say that:
:
Where δ
ij is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
.
Also, by the geometric definition, for any vector e
''i'' and a vector a, we note
:
where ''a''
''i'' is the component of vector a in the direction of e
''i''. The last step in the equality can be seen from the figure.
Now applying the distributivity of the geometric version of the dot product gives
:
which is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.
Properties
The dot product fulfills the following properties if a, b, and c are real
vectors and ''r'' is a
scalar.
#
Commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
:
#:
#: which follows from the definition (''θ'' is the angle between a and b):
#:
#
Distributive over vector addition:
#:
#
Bilinear:
#:
#
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
:
#:
# 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. Note however that the previously mentioned scalar multiplication property is sometimes called the "associative law for scalar and dot product"
or one can say that "the dot product is associative with respect to scalar multiplication" because ''c'' (a ⋅ b) = (''c'' a) ⋅ b = a ⋅ (''c'' b).
#
Orthogonal:
#: Two non-zero vectors a and b are ''orthogonal''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
.
# No
cancellation:
#: 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
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
; the result above says this just means that a is perpendicular to , which still allows , and therefore allows .
#
Product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
:
#: If a and b are (vector-valued)
differentiable functions, then the derivative (
denoted by a prime ) of is given by the rule .
Application to the law of cosines
Given two vectors a and b separated by angle ''θ'' (see image right), they form a triangle with a third side . The dot product of this with itself is:
:
which is the
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
.
Triple product
There are two
ternary operation
In mathematics, a ternary operation is an ''n''-ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''.
In computer science, a ternary operator i ...
s involving dot product and
cross product.
The scalar triple product of three vectors is defined as
:
Its value is the
determinant of the matrix whose columns are the
Cartesian coordinates of the three vectors. It is the signed
volume of the
parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
of three vectors.
The vector triple product is defined by
:
This identity, also known as ''Lagrange'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 in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
.
Physics
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, vector magnitude is a
scalar in the physical sense (i.e., a
physical quantity independent of the coordinate system), expressed as the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of a
numerical value
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
and a
physical unit
A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
, 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:
*
Mechanical work
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
is the dot product of
force and
displacement vectors,
*
Power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
is the dot product of
force and
velocity.
Generalizations
Complex vectors
For vectors with
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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 a =
i. 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
:
where
is the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of
. When vectors are represented by
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
s, the dot product can be expressed as a
matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
involving a
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
, 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 is
sesquilinear rather than bilinear, as it is
conjugate linear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y \ ...
and not linear in a. The dot product is not symmetric, since
:
The angle between two complex vectors is then given by
:
The complex dot product leads to the notions of
Hermitian forms and general
inner product spaces, which are widely used in mathematics and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
.
The self dot product of a complex vector
, involving the conjugate transpose of a row vector, is also known as the norm squared,
, after the
Euclidean norm; it is a vector generalization of the ''
absolute square'' of a complex scalar (see also:
squared Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
).
Inner product
The inner product generalizes the dot product to
abstract vector spaces over a
field of
scalars, being either the field of
real numbers
or the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s
. It is usually denoted using
angular brackets by
.
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
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, 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- vector is, then, a function with
domain , and is a notation for the image of by the function/vector .
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
interval (also denoted ):
:
Generalized further to
complex functions and , by analogy with the complex inner product above, gives
:
Weight function
Inner products can have a
weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
(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
A double-dot product for
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is the
Frobenius 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 A and B of the same size:
:
:
(For real matrices)
Writing a matrix as a
dyadic, we can define a different double-dot product (see ,) however it is not an inner product.
Tensors
The inner product between a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
of order ''n'' and a tensor of order ''m'' is a tensor of order , see
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tens ...
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 real SDOT, DDOT; complex CDOTU, ZDOTU = X^T * Y, CDOTC ZDOTC = X^H * Y
*
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...
as
*
Matlab as or or
* GNU Octave as
* Intel oneAPI Math Kernel Library real p?dot dot = sub(x)'*sub(y); complex p?dotc dotc = conjg(sub(x)')*sub(y)
See also
*
Cauchy–Schwarz inequality
*
Cross product
*
Dot product representation of a graph
*
Euclidean norm, the square-root of the self dot product
*
Matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
*
Metric tensor
*
Multiplication of vectors
*
Outer product
Notes
References
External links
*
Explanation of dot product including with complex vectors"Dot Product"by Bruce Torrence,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007.
{{tensors
Articles containing proofs
Bilinear forms
Operations on vectors
Analytic geometry
Tensors
Scalars