Dot product

Properties

The dot product fulfills the following properties if ${\displaystyle \mathbf{a}}$, ${\displaystyle \mathbf{b}}$, ${\displaystyle \mathbf{c}}$ and ${\displaystyle \mathbf{d}}$ are real vectors and ${\displaystyle \alpha }$, ${\displaystyle \beta }$, ${\displaystyle \gamma }$ and ${\displaystyle \delta }$ are scalars.

Commutative

$${\displaystyle \mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a},}$$ which follows from the definition (${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf{a}}$ and ${\displaystyle \mathbf{b}}$): $${\displaystyle \mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta =\Vert \mathbf{b}\Vert \Vert \mathbf{a}\Vert \cos \theta =\mathbf{b}\cdot \mathbf{a}.}$$ The commutative property can also be easily proven with the algebraic definition, and in more 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

$${\displaystyle \theta =\arccos \left(\frac{\mathbf{a}\cdot \mathbf{b}}{\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert }\right).}$$

Bilinear (additive, distributive and scalar-multiplicative in both arguments)

$${\displaystyle \begin{array}{cc}(\alpha \mathbf{a}+\beta \mathbf{b})& \cdot (\gamma \mathbf{c}+\delta \mathbf{d})\\ & =\alpha \gamma (\mathbf{a}\cdot \mathbf{c})+\alpha \delta (\mathbf{a}\cdot \mathbf{d})+\beta \gamma (\mathbf{b}\cdot \mathbf{c})+\beta \delta (\mathbf{b}\cdot \mathbf{d}).\end{array}}$$

Not associative

Because the dot product is not defined between a scalar ${\displaystyle \mathbf{a}\cdot \mathbf{b}}$ and a vector ${\displaystyle \mathbf{c},}$ associativity is meaningless. However, bilinearity implies $${\displaystyle c(\mathbf{a}\cdot \mathbf{b})=(c\mathbf{a})\cdot \mathbf{b}=\mathbf{a}\cdot (c\mathbf{b}).}$$ This property is sometimes called the "associative law for scalar and dot product", and one may say that "the dot product is associative with respect to scalar multiplication".

Orthogonal

Two non-zero vectors ${\displaystyle \mathbf{a}}$ and ${\displaystyle \mathbf{b}}$ are orthogonal if and only if ${\displaystyle \mathbf{a}\cdot \mathbf{b}=0}$.

No cancellation

Unlike multiplication of ordinary numbers, where if ${\displaystyle ab=ac}$, then ${\displaystyle b}$ always equals ${\displaystyle c}$ unless ${\displaystyle a}$ is zero, the dot product does not obey the cancellation law: If ${\displaystyle \mathbf{a}\cdot \mathbf{b}=\mathbf{a}\cdot \mathbf{c}}$ and ${\displaystyle \mathbf{a}\ne \mathbf{0}}$, then we can write: ${\displaystyle \mathbf{a}\cdot (\mathbf{b}-\mathbf{c})=0}$ by the distributive law; the result above says this just means that ${\displaystyle \mathbf{a}}$ is perpendicular to ${\displaystyle (\mathbf{b}-\mathbf{c})}$, which still allows ${\displaystyle (\mathbf{b}-\mathbf{c})\ne \mathbf{0}}$, and therefore allows ${\displaystyle \mathbf{b}\ne \mathbf{c}}$.

Product rule

If ${\displaystyle \mathbf{a}}$ and ${\displaystyle \mathbf{b}}$ are vector-valued differentiable functions, then the derivative (denoted by a prime ${\displaystyle {}^{\prime }}$) of ${\displaystyle \mathbf{a}\cdot \mathbf{b}}$ is given by the rule $${\displaystyle (\mathbf{a}\cdot \mathbf{b}{)}^{\prime }={\mathbf{a}}^{\prime }\cdot \mathbf{b}+\mathbf{a}\cdot {\mathbf{b}}^{\prime }.}$$

Application to the law of cosines

Given two vectors ${\displaystyle \mathbf{a}}$ and ${\displaystyle \mathbf{b}}$ separated by angle ${\displaystyle \theta }$ (see the upper image), they form a triangle with a third side ${\displaystyle \mathbf{c}=\mathbf{a}-\mathbf{b}}$. Let ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ denote the lengths of ${\displaystyle \mathbf{a}}$, ${\displaystyle \mathbf{b}}$, and ${\displaystyle \mathbf{c}}$, respectively. The dot product of ${\displaystyle \mathbf{c}}$ with itself is: $${\displaystyle \begin{array}{cc}\mathbf{c}\cdot \mathbf{c}& =(\mathbf{a}-\mathbf{b})\cdot (\mathbf{a}-\mathbf{b})\\ & =\mathbf{a}\cdot \mathbf{a}-\mathbf{a}\cdot \mathbf{b}-\mathbf{b}\cdot \mathbf{a}+\mathbf{b}\cdot \mathbf{b}\\ & =a^2-\mathbf{a}\cdot \mathbf{b}-\mathbf{a}\cdot \mathbf{b}+b^2\\ & =a^2-2\mathbf{a}\cdot \mathbf{b}+b^2\\ c^2& =a^2+b^2-2ab\cos \theta \end{array}}$$ which is the law of cosines.

Triple product

There are two ternary operations involving dot product and cross product.

The scalar triple product of three vectors is defined as $${\displaystyle \mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\mathbf{b}\cdot (\mathbf{c}\times \mathbf{a})=\mathbf{c}\cdot (\mathbf{a}\times \mathbf{b}).}$$ 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 of three vectors.

The vector triple product is defined by $${\displaystyle \mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\cdot \mathbf{c})\mathbf{b}-(\mathbf{a}\cdot \mathbf{b})\mathbf{c}.}$$ 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

In physics, the dot product takes two vectors and returns a scalar 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, $${\displaystyle \mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta }$$ 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 example:

• Mechanical work is the dot product of force and displacement vectors,
• Power is the dot product of force and velocity.

Generalizations

Complex vectors

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 could be zero without the vector being the zero vector (e.g. this would happen with the vector ${\displaystyle \mathbf{a}=[1i]}$). 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 $${\displaystyle \mathbf{a}\cdot \mathbf{b}=\sum _i{a}_i\overline{b_i},}$$ where ${\displaystyle \overline{b_i}}$ is the complex conjugate of ${\displaystyle b_i}$. When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H: $${\displaystyle \mathbf{a}\cdot \mathbf{b}={\mathbf{b}}^{\mathsf{H}}\mathbf{a}.}$$

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 and not linear in ${\displaystyle \mathbf{a}}$. The dot product is not symmetric, since $${\displaystyle \mathbf{a}\cdot \mathbf{b}=\overline{\mathbf{b}\cdot \mathbf{a}}.}$$ The angle between two complex vectors is then given by $${\displaystyle \cos \theta =\frac{\mathrm{Re}(\mathbf{a}\cdot \mathbf{b})}{\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert }.}$$

The complex dot product leads to the notions of Hermitian forms and general inner product spaces, which are widely used in mathematics and physics.

The self dot product of a complex vector ${\displaystyle \mathbf{a}\cdot \mathbf{a}={\mathbf{a}}^{\mathsf{H}}\mathbf{a}}$, involving the conjugate transpose of a row vector, is also known as the norm squared, $\mathbf{a}\cdot \mathbf{a}=\Vert \mathbf{a}{\Vert }^2$, after the Euclidean norm; it is a vector generalization of the absolute square of a complex scalar (see also: Squared Euclidean distance).

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 ${\displaystyle \mathbb{R}}$ or the field of complex numbers ${\displaystyle \mathbb{C}}$. It is usually denoted using angular brackets by ${\displaystyle ⟨\mathbf{a},\mathbf{b}⟩}$.

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-${\displaystyle n}$ vector ${\displaystyle u}$ is, then, a function with domain ${\displaystyle \{k\in \mathbb{N}:1\le k\le n\}}$, and ${\displaystyle u_i}$ is a notation for the image of ${\displaystyle i}$ by the function/vector ${\displaystyle u}$.

This notion can be generalized to square-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 some measure space ${\displaystyle (X,\mathcal{A},\mu )}$: $${\displaystyle ⟨u,v⟩={\int }_{X}uv\text{d}\mu .}$$

For example, if ${\displaystyle f}$ and ${\displaystyle g}$ are continuous functions over a compact subset ${\displaystyle K}$ of ${\displaystyle {\mathbb{R}}^n}$ with the standard Lebesgue measure, the above definition becomes: $${\displaystyle ⟨f,g⟩={\int }_{K}f(\mathbf{x})g(\mathbf{x}){\mathrm{d}}^n\mathbf{x}.}$$

Generalized further to complex continuous functions ${\displaystyle \psi }$ and ${\displaystyle \chi }$, by analogy with the complex inner product above, gives: $${\displaystyle ⟨\psi ,\chi ⟩={\int }_K\psi (z)\overline{\chi (z)}\text{d}z.}$$

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 ${\displaystyle u(x)}$ and ${\displaystyle v(x)}$ with respect to the weight function ${\displaystyle r(x)>0}$ is $${\displaystyle {⟨u,v⟩}_r={\int }_a^{b}r(x)u(x)v(x)dx.}$$

Dyadics and matrices

A double-dot product for matrices 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 ${\displaystyle \mathbf{A}}$ and ${\displaystyle \mathbf{B}}$ of the same size: $${\displaystyle \mathbf{A}:\mathbf{B}=\sum _i\sum _j{A}_{ij}\overline{B_{ij}}=\mathrm{tr}({\mathbf{B}}^{\mathsf{H}}\mathbf{A})=\mathrm{tr}(\mathbf{A}{\mathbf{B}}^{\mathsf{H}}).}$$ And for real matrices, $${\displaystyle \mathbf{A}:\mathbf{B}=\sum _i\sum _j{A}_{ij}{B}_{ij}=\mathrm{tr}({\mathbf{B}}^{\mathsf{T}}\mathbf{A})=\mathrm{tr}(\mathbf{A}{\mathbf{B}}^{\mathsf{T}})=\mathrm{tr}({\mathbf{A}}^{\mathsf{T}}\mathbf{B})=\mathrm{tr}(\mathbf{B}{\mathbf{A}}^{\mathsf{T}}).}$$

Writing a matrix as a dyadic, we can define a different double-dot product (see Dyadics § Product of dyadic and dyadic) however it is not an inner product.

Tensors

The (single-) dot product between a tensor of order ${\displaystyle n}$ and a tensor of order ${\displaystyle m}$ is a tensor of order ${\displaystyle n+m-2}$ (more generally, each dot reduces the order by 2), 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 real SDOT, DDOT; complex CDOTU, ZDOTU = X^T * Y, CDOTC, ZDOTC = X^H * Y
• Fortran as dot_product(A,B) or sum(conjg(A) * B)
• Julia as A' * B or standard library LinearAlgebra as dot(A, B)
• R (programming language) as sum(A * B) for vectors or, more generally for matrices, as A %*% B
• Matlab as A' * B or conj(transpose(A)) * B or sum(conj(A) .* B) or dot(A, B)
• Python (package NumPy) as np.dot(A, B) or np.inner(A, B)
• GNU Octave as sum(conj(X) .* Y, dim), and similar code as Matlab
• Intel oneAPI Math Kernel Library real p?dot dot = sub(x)'*sub(y); complex p?dotc dotc = conjg(sub(x)')*sub(y)