In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and then summing those products. The name "dot product" is derived from the centered dot " Failed to parse (Missing texvc executable; please see math/README to configure.): \cdot " that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar (rather than vector) nature of the result.
When two Euclidean vectors are expressed in terms of coordinate vectors on an orthonormal basis, the inner product of the former is equal to the dot product of the latter. For more general vector spaces, while both the inner and the dot product can be defined in different contexts (for instance with complex numbers as scalars) their definitions in these contexts may not coincide.
In three dimensional space, the dot product contrasts with the cross product, which produces a vector as result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions.
The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n
where Σ denotes summation notation and n is the dimension of the vector space.
In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd. Similarly, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf. For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is
- Failed to parse (Missing texvc executable; please see math/README to configure.): [1, 3, -5] \cdot [4, -2, -1] = (1)(4) + (3)(-2) + (-5)(-1) = 4 - 6 + 5 = 3.
Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix. The operation of extracting the coefficient of such a matrix can be written as taking its determinant or its trace (which is the same thing for 1 × 1 matrices); since in general tr(AB) = tr(BA) whenever AB or equivalently BA is a square matrix, one may write
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot \mathbf{b} = \det( \mathbf{a}^{\mathrm{T}}\mathbf{b} ) = \det( \mathbf{b}^{\mathrm{T}}\mathbf{a} ) = \mathrm{tr} ( \mathbf{a}^{\mathrm{T}}\mathbf{b} ) = \mathrm{tr} ( \mathbf{b}^{\mathrm{T}}\mathbf{a} ) = \mathrm{tr} ( \mathbf{a}\mathbf{b}^{\mathrm{T}} ) = \mathrm{tr} ( \mathbf{b}\mathbf{a}^{\mathrm{T}} )
More generally the coefficient (i,j) of a product of matrices is the dot product of the transpose of row i of the first matrix and column j of the second matrix.
Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{A}_B = \left\|\mathbf{A}\right\| \cos\theta
is the scalar projection of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{A}
onto Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}
.
Since Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{A} \cdot \mathbf{B} = \left\|\mathbf{A}\right\| \left\|\mathbf{B}\right\| \cos\theta , then Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{A}_B = \frac{\mathbf{A} \cdot \mathbf{B}}{\left\|\mathbf{B}\right\|}
.
In Euclidean geometry, the dot product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} , the dot product Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}\cdot\mathbf{a}
is the square of the length (magnitude) of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
, denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \left\|\mathbf{a}\right\|
- Failed to parse (Missing texvc executable; please see math/README to configure.): {\mathbf{a} \cdot \mathbf{a}}=\left\|\mathbf{a}\right\|^2
If Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
is another such vector, and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
is the angle between them:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot \mathbf{b}=\left\|\mathbf{a}\right\| \, \left\|\mathbf{b}\right\| \cos \theta \,
This formula can be rearranged to determine the size of the angle between two nonzero vectors:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \theta=\arccos \left( \frac {\bold{a}\cdot\bold{b}} {\left\|\bold{a}\right\|\left\|\bold{b}\right\|}\right).
The Cauchy–Schwarz inequality guarantees that the argument of Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos
is valid.
One can also first convert the vectors to unit vectors by dividing by their magnitude:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol{\hat{a}} = \frac{\bold{a}}{\left\|\bold{a}\right\|}
then the angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
is given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arccos ( \boldsymbol{\hat a}\cdot\boldsymbol{\hat b}).
The terminal points of both unit vectors lie on the unit circle. The unit circle is where the trigonometric values for the six trig functions are found. After substitution, the first vector component is cosine and the second vector component is sine, i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos x, \sin x)
for some angle Failed to parse (Missing texvc executable; please see math/README to configure.): x
. The dot product of the two unit vectors then takes Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos x, \sin x)
and Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos y, \sin y)
for angles Failed to parse (Missing texvc executable; please see math/README to configure.): x
and Failed to parse (Missing texvc executable; please see math/README to configure.): y
and returns Failed to parse (Missing texvc executable; please see math/README to configure.): \cos x \, \cos y + \sin x \, \sin y = \cos(x - y)
where Failed to parse (Missing texvc executable; please see math/README to configure.): x - y = \theta
.
As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero. Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length. This property provides a simple method to test the condition of orthogonality.
Sometimes these properties are also used for "defining" the dot product, especially in 2 and 3 dimensions; this definition is equivalent to the above one. For higher dimensions the formula can be used to define the concept of angle.
The geometric properties rely on the basis being orthonormal, i.e. composed of pairwise perpendicular vectors with unit length.
If both Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
have length one (i.e., they are unit vectors), their dot product simply gives the cosine of the angle between them.
If only Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
is a unit vector, then the dot product Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}\cdot\mathbf{b}
gives Failed to parse (Missing texvc executable; please see math/README to configure.): \left\|\mathbf{a}\right\|\cos\theta
, i.e., the magnitude of the projection of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
in the direction of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
, with a minus sign if the direction is opposite. This is called the scalar projection of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
onto Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
, or scalar component of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
in the direction of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
(see figure). This property of the dot product has several useful applications (for instance, see next section).
If neither Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
nor Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
is a unit vector, then the magnitude of the projection of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
in the direction of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}\cdot\frac{\mathbf{b}}{\left\|\mathbf{b}\right\|}
, as the unit vector in the direction of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{b}
is Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathbf{b}}{\left\|\mathbf{b}\right\|}
.
When an orthonormal basis that the vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
is represented in terms of is rotated, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
's matrix in the new basis is obtained through multiplying Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
by a rotation matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}
. This matrix multiplication is just a compact representation of a sequence of dot products.
For instance, let
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_1 = \{\mathbf{x}, \mathbf{y}, \mathbf{z}\}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_2 = \{\mathbf{u}, \mathbf{v}, \mathbf{w}\}
be two different orthonormal bases of the same space Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^3
, with Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_2
obtained by just rotating Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_1
,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}_1 = (a_x, a_y, a_z)
represent vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
in terms of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_1
,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}_2 = (a_u, a_v, a_w)
represent the same vector in terms of the rotated basis Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_2
,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}_1
, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_1 , Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{w}_1 , be the rotated basis vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u} , Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} , Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{w}
represented in terms of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_1
. Then the rotation from Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_1
to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}_2
is performed as follows:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \bold a_2 = \bold{Ra}_1 = \begin{bmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{bmatrix} \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} = \begin{bmatrix} \bold u_1\cdot\bold a_1 \\ \bold v_1\cdot\bold a_1 \\ \bold w_1\cdot\bold a_1 \end{bmatrix} = \begin{bmatrix} a_u \\ a_v \\ a_w \end{bmatrix} .
Notice that the rotation matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}
is assembled by using the rotated basis vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}_1
, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_1 , Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{w}_1
as its rows, and these vectors are unit vectors. By definition, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Ra}_1
consists of a sequence of dot products between each of the three rows of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}
and vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}_1
. Each of these dot products determines a scalar component of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}
in the direction of a rotated basis vector (see previous section).
If Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}_1
is a row vector, rather than a column vector, then Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}
must contain the rotated basis vectors in its columns, and must post-multiply Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}_1
- Failed to parse (Missing texvc executable; please see math/README to configure.): \bold a_2 = \bold a_1 \bold R = \begin{bmatrix} a_x & a_y & a_z \end{bmatrix} \begin{bmatrix} u_x & v_x & w_x \\ u_y & v_y & w_y \\ u_z & v_z & w_z \end{bmatrix} = \begin{bmatrix} \bold u_1\cdot\bold a_1 & \bold v_1\cdot\bold a_1 & \bold w_1\cdot\bold a_1 \end{bmatrix} = \begin{bmatrix} a_u & a_v & a_w \end{bmatrix} .
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. Example:
The following properties hold if a, b, and c are real vectors and r is a scalar.
The dot product is commutative:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.
The dot product is distributive over vector addition:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.
The dot product is bilinear:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot (r\mathbf{b} + \mathbf{c}) = r(\mathbf{a} \cdot \mathbf{b}) +(\mathbf{a} \cdot \mathbf{c}).
When multiplied by a scalar value, dot product satisfies:
- Failed to parse (Missing texvc executable; please see math/README to configure.): (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = (c_1c_2) (\mathbf{a} \cdot \mathbf{b})
(these last two properties follow from the first two).
Two non-zero vectors a and b are orthogonal if and only if a • b = 0.
Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law:
- If a • b = a • c and a ≠ 0, then we can write: a • (b − c) = 0 by the distributive law; the result above says this just means that a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠ c.
Provided that the basis is orthonormal, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed. The above mentioned geometric interpretation relies on this property. In other words, for an orthonormal space with any number of dimensions, the dot product is invariant under a coordinate transformation based on an orthogonal matrix. This corresponds to the following two conditions:
- The new basis is again orthonormal (i.e., it is orthonormal expressed in the old one).
- The new base vectors have the same length as the old ones (i.e., unit length in terms of the old basis).
If a and b are functions, then the derivative of a • b is a' • b + a • b'
Main article:
Triple productThis is a very useful identity (also known as Lagrange's formula) involving the dot- and cross-products. It is written as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})
which is easier to remember as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula is commonly used to simplify vector calculations in physics.
Consider the element of Rn
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} = v_1 \mathbf{\hat{e}}_1 + v_2 \mathbf{\hat{e}}_2 + ... + v_n \mathbf{\hat{e}}_n. \,
Repeated application of the Pythagorean theorem yields for its length |v|
- Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathbf{v}|^2 = v_1^2 + v_2^2 + ... + v_n^2. \,
But this is the same as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 + ... + v_n^2, \,
so we conclude that taking the dot product of a vector v with itself yields the squared length of the vector.
- Lemma 1
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} \cdot \mathbf{v} = |\mathbf{v}|^2. \,
Now consider two vectors a and b extending from the origin, separated by an angle θ. A third vector c may be defined as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{c} \ \stackrel{\mathrm{def}}{=}\ \mathbf{a} - \mathbf{b}. \,
creating a triangle with sides a, b, and c. According to the law of cosines, we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathbf{c}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2 |\mathbf{a}||\mathbf{b}| \cos \theta. \,
Substituting dot products for the squared lengths according to Lemma 1, we get
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{c} \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} - 2 |\mathbf{a}||\mathbf{b}| \cos\theta. \,
(1)
But as c ≡ a − b, we also have
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{c} \cdot \mathbf{c} = (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \,
, which, according to the distributive law, expands to
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{c} \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} -2(\mathbf{a} \cdot \mathbf{b}). \,
(2)
Merging the two c • c equations, (1) and (2), we obtain
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} -2(\mathbf{a} \cdot \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} - 2 |\mathbf{a}||\mathbf{b}| \cos\theta. \,
Subtracting a • a + b • b from both sides and dividing by −2 leaves
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos\theta. \,
Q.E.D.
The inner product generalizes the dot product to abstract vector spaces over the real numbers and is usually denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\mathbf{a}\, , \mathbf{b}\rangle . More precisely, if Failed to parse (Missing texvc executable; please see math/README to configure.): V
is a vector space over Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}
, the inner product is a function Failed to parse (Missing texvc executable; please see math/README to configure.): V\times V \rightarrow \mathbb{R} . Owing to the geometric interpretation of the dot product, the norm ||a|| of a vector a in such an inner product space is defined as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \|\mathbf{a}\| = \sqrt{\langle\mathbf{a}\, , \mathbf{a}\rangle}
such that it generalizes length, and the angle θ between two vectors a and b by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos{\theta} = \frac{\langle\mathbf{a}\, , \mathbf{b}\rangle}{\|\mathbf{a}\| \, \|\mathbf{b}\|}.
In particular, two vectors are considered orthogonal if their inner product is zero
- Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\mathbf{a}\, , \mathbf{b}\rangle = 0.
For vectors with complex entries, using the given definition of the dot product would lead to quite different geometric properties. For instance the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in turn would have severe consequences for notions like length and angle. Many geometric properties can be salvaged, at the cost of giving up the symmetric and bilinear properties of the scalar product, by alternatively defining
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a}\cdot \mathbf{b} = \sum{a_i \overline{b_i}}
where bi is the complex conjugate of bi. 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 not linear in b (but rather conjugate linear), and the scalar product is not symmetric either, since
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}}.
The angle between two complex vectors is then given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos\theta = \frac{\operatorname{Re}(\mathbf{a}\cdot\mathbf{b})}{\|\mathbf{a}\|\,\|\mathbf{b}\|}.
This type of scalar product is nevertheless quite useful, and leads to the notions of Hermitian form and of general inner product spaces.
The Frobenius inner product generalizes the dot product to matrices. It is defined as the sum of the products of the corresponding components of two matrices having the same size.
The dot product between a tensor of order n and a tensor of order m is a tensor of order n+m-2. The dot product is calculated by multiplying and summing across a single index in both tensors. If Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{A}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}
are two tensors with element representation Failed to parse (Missing texvc executable; please see math/README to configure.): A_{ij\dots}^{k\ell\dots}
and Failed to parse (Missing texvc executable; please see math/README to configure.): B_{mn\dots}^{p{\dots}i}
the elements of the dot product Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{A} \cdot \mathbf{B}
are given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): A_{ij\dots}^{k\ell\dots}\cdot B_{mn\dots}^{p{\dots}i} = \sum_{i=1}^n A_{ij\dots}^{k\ell\dots}B_{mn\dots}^{p{\dots}i}.
This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices[citation needed].
Occasionally, a double dot product is used to represent multiplying and summing across two indices. The double dot product between two 2nd order tensors is a scalar quantity.
