Dot product

Coordinate definition

The dot product of two vectors ${\displaystyle \mathbf {a} =[a_{1},a_{2},\cdots ,a_{n}]}$ and ${\displaystyle \mathbf {b} =[b_{1},b_{2},\cdots ,b_{n}]}$, specified with respect to an orthonormal basis, is defined as:^[2] $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}}$$ where ${\displaystyle \Sigma }$ denotes summation and ${\displaystyle n}$ is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors ${\displaystyle [1,3,-5]}$ and ${\displaystyle [4,-2,-1]}$ is: $${\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [4,-2,-1]&=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}}$$

Likewise, the dot product of the vector ${\displaystyle [1,3,-5]}$ with itself is: $${\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [1,3,-5]&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}}$$

If vectors are identified with column vectors, the dot product can also be written as a matrix product $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathsf {T}}\mathbf {b} ,}$$ where ${\displaystyle \mathbf {a} {^{\mathsf {T}}}}$ denotes the transpose of ${\displaystyle \mathbf {a} }$.

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: $${\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}{\begin{bmatrix}4\\-2\\-1\end{bmatrix}}=3\,.}$$

Geometric definition

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 ${\displaystyle \mathbf {a} }$ is denoted by ${\displaystyle \left\|\mathbf {a} \right\|}$. The dot product of two Euclidean vectors ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$ is defined by^[3][4][1] $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,}$$ where ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$.

In particular, if the vectors ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$ are orthogonal (i.e., their angle is ${\displaystyle {\frac {\pi }{2}}}$ or ${\displaystyle 90^{\circ }}$), then ${\displaystyle \cos {\frac {\pi }{2}}=0}$, which implies that $${\displaystyle \mathbf {a} \cdot \mathbf {b} =0.}$$ At the other extreme, if they are codirectional, then the angle between them is zero with ${\displaystyle \cos 0=1}$ and $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}$$ This implies that the dot product of a vector ${\displaystyle \mathbf {a} }$ with itself is $${\displaystyle \mathbf {a} \cdot \mathbf {a} =\left\|\mathbf {a} \right\|^{2},}$$ which gives $${\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }},}$$ the formula for the Euclidean length of the vector.

Scalar projection and first properties

The scalar projection (or scalar component) of a Euclidean vector ${\displaystyle \mathbf {a} }$ in the direction of a Euclidean vector ${\displaystyle \mathbf {b} }$ is given by $${\displaystyle a_{b}=\left\|\mathbf {a} \right\|\cos \theta ,}$$ where ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$.

In terms of the geometric definition of the dot product, this can be rewritten as $${\displaystyle a_{b}=\mathbf {a} \cdot {\widehat {\mathbf {b} }},}$$ where ${\displaystyle {\widehat {\mathbf {b} }}=\mathbf {b} /\left\|\mathbf {b} \right\|}$ is the unit vector in the direction of ${\displaystyle \mathbf {b} }$.

The dot product is thus characterized geometrically by^[5] $${\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{b}\left\|\mathbf {b} \right\|=b_{a}\left\|\mathbf {a} \right\|.}$$ The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar ${\displaystyle \alpha }$, $${\displaystyle (\alpha \mathbf {a} )\cdot \mathbf {b} =\alpha (\mathbf {a} \cdot \mathbf {b} )=\mathbf {a} \cdot (\alpha \mathbf {b} ).}$$ It also satisfies the distributive law, meaning that $${\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .}$$

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 ${\displaystyle \mathbf {a} \cdot \mathbf {a} }$ is never negative, and is zero if and only if ${\displaystyle \mathbf {a} =\mathbf {0} }$, the zero vector.

Equivalence of the definitions

If ${\displaystyle \mathbf {e} _{1},\cdots ,\mathbf {e} _{n}}$ are the standard basis vectors in ${\displaystyle \mathbf {R} ^{n}}$, then we may write $${\displaystyle {\begin{aligned}\mathbf {a} &=[a_{1},\dots ,a_{n}]=\sum _{i}a_{i}\mathbf {e} _{i}\\\mathbf {b} &=[b_{1},\dots ,b_{n}]=\sum _{i}b_{i}\mathbf {e} _{i}.\end{aligned}}}$$ The vectors ${\displaystyle \mathbf {e} _{i}}$ are an orthonormal basis, which means that they have unit length and are at right angles to each other. Since these vectors have unit length, $${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{i}=1}$$ and since they form right angles with each other, if ${\displaystyle i\neq j}$, $${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=0.}$$ Thus in general, we can say that: $${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij},}$$ where ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

Also, by the geometric definition, for any vector ${\displaystyle \mathbf {e} _{i}}$ and a vector ${\displaystyle \mathbf {a} }$, we note that $${\displaystyle \mathbf {a} \cdot \mathbf {e} _{i}=\left\|\mathbf {a} \right\|\left\|\mathbf {e} _{i}\right\|\cos \theta _{i}=\left\|\mathbf {a} \right\|\cos \theta _{i}=a_{i},}$$ where ${\displaystyle a_{i}}$ is the component of vector ${\displaystyle \mathbf {a} }$ in the direction of ${\displaystyle \mathbf {e} _{i}}$. 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 $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \sum _{i}b_{i}\mathbf {e} _{i}=\sum _{i}b_{i}(\mathbf {a} \cdot \mathbf {e} _{i})=\sum _{i}b_{i}a_{i}=\sum _{i}a_{i}b_{i},}$$ which is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.

Application to the law of cosines

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

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 ${\displaystyle \mathbf {a} =[1\ i]}$). 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^[12][2] $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i}{{a_{i}}\,{\overline {b_{i}}}},}$$ 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 $${\displaystyle \cos \theta ={\frac {\operatorname {Re} (\mathbf {a} \cdot \mathbf {b} )}{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}.}$$

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, ${\textstyle \mathbf {a} \cdot \mathbf {a} =\|\mathbf {a} \|^{2}}$, 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 ${\displaystyle \left\langle \mathbf {a} \,,\mathbf {b} \right\rangle }$.

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 ${\displaystyle \{k\in \mathbb {N} :1\leq k\leq n\}}$, 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 ${\displaystyle (X,{\mathcal {A}},\mu )}$:^[2] $${\displaystyle \left\langle u,v\right\rangle =\int _{X}uv\,{\text{d}}\mu .}$$

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: $${\displaystyle \left\langle f,g\right\rangle =\int _{K}f(\mathbf {x} )g(\mathbf {x} )\,\operatorname {d} ^{n}\mathbf {x} .}$$

Generalized further to complex continuous functions ${\displaystyle \psi }$ and ${\displaystyle \chi }$, by analogy with the complex inner product above, gives: $${\displaystyle \left\langle \psi ,\chi \right\rangle =\int _{K}\psi (z){\overline {\chi (z)}}\,{\text{d}}z.}$$

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}$0}"> is $${\displaystyle \left\langle u,v\right\rangle _{r}=\int _{a}^{b}r(x)u(x)v(x)\,dx.}$$

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}}}=\operatorname {tr} (\mathbf {B} ^{\mathsf {H}}\mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {B} ^{\mathsf {H}}).} And for real matrices, $${\displaystyle \mathbf {A}$$ :\mathbf {B} =\sum _{i}\sum _{j}A_{ij}B_{ij}=\operatorname {tr} (\mathbf {B} ^{\mathsf {T}}\mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {B} ^{\mathsf {T}})=\operatorname {tr} (\mathbf {A} ^{\mathsf {T}}\mathbf {B} )=\operatorname {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 inner product between a tensor of order ${\displaystyle n}$ and a tensor of order ${\displaystyle m}$ is a tensor of order ${\displaystyle n+m-2}$, see Tensor contraction for details.

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.matmul(A, B)  or  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)