Dot product

The dot product of two vectors a = ${\displaystyle {\color {red}[a_{1},a_{2},\cdots ,a_{n}]}}$ and b = ${\displaystyle {\color {blue}[b_{1},b_{2},\cdots ,b_{n}]}}$ is defined as:[2]

${\displaystyle \mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} =\sum _{i=1}^{n}{\color {red}a}_{i}{\color {blue}b}_{i}={\color {red}a}_{1}{\color {blue}b}_{1}+{\color {red}a}_{2}{\color {blue}b}_{2}+\cdots +{\color {red}a}_{n}{\color {blue}b}_{n}}$

where Σ denotes summation and n is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors [1, 3, −5] and [4, −2, −1] is:

${\displaystyle {\begin{aligned}\ [{\color {red}1,3,-5}]\cdot [{\color {blue}4,-2,-1}]&=({\color {red}1}\times {\color {blue}4})+({\color {red}3}\times {\color {blue}-2})+({\color {red}-5}\times {\color {blue}-1})\\&=4-6+5\\&=3\end{aligned}}}$

Likewise, the dot product of the vector [1, 3, −5] with itself is:

${\displaystyle {\begin{aligned}\ [{\color {red}1,3,-5}]\cdot [{\color {red}1,3,-5}]&=({\color {red}1}\times {\color {red}1})+({\color {red}3}\times {\color {red}3})+({\color {red}-5}\times {\color {red}-5})\\&=1+9+25\\&=35\end{aligned}}}$

If vectors are identified with row matrices, the dot product can also be written as a matrix product

${\displaystyle \mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} =\mathbf {\color {red}a} \mathbf {\color {blue}b} ^{\mathsf {T}},}$

where ${\displaystyle \mathbf {\color {blue}b} ^{\mathsf {T}}}$ denotes the transpose of ${\displaystyle \mathbf {\color {blue}b} }$.

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:

${\displaystyle {\begin{bmatrix}\color {red}1&\color {red}3&\color {red}-5\end{bmatrix}}{\begin{bmatrix}\color {blue}4\\\color {blue}-2\\\color {blue}-1\end{bmatrix}}=\color {purple}3}$.

Illustration showing how to find the angle between vectors using the dot product

Calculating bond angles of a symmetrical tetrahedral molecular geometry using a dot product

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 a is denoted by ${\displaystyle \left\|\mathbf {a} \right\|}$. The dot product of two Euclidean vectors a and b is defined by[3][4][1]

${\displaystyle \mathbf {a} \cdot \mathbf {b} =\|\mathbf {a} \|\ \|\mathbf {b} \|\cos \theta ,}$

where θ is the angle between a and b.

In particular, if the vectors a and b are orthogonal (i.e., their angle is π / 2 or 90°), 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 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

The scalar projection (or scalar component) of a Euclidean vector a in the direction of a Euclidean vector b is given by

${\displaystyle a_{b}=\left\|\mathbf {a} \right\|\cos \theta ,}$

where θ is the angle between a and b.

In terms of the geometric definition of the dot product, this can be rewritten

${\displaystyle a_{b}=\mathbf {a} \cdot {\widehat {\mathbf {b} }},}$

where ${\displaystyle {\widehat {\mathbf {b} }}=\mathbf {b} /\left\|\mathbf {b} \right\|}$ is the unit vector in the direction of b.

Distributive law for the dot product

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 α,

${\displaystyle (\alpha \mathbf {a} )\cdot \mathbf {b} =\alpha (\mathbf {a} \cdot \mathbf {b} )=\mathbf {a} \cdot (\alpha \mathbf {b} ).}$

It also satisfies a 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.

The dot product is thus equivalent to multiplying the norm (length) of b by the norm of the projection of a over b.

If e₁, ..., e_n are the standard basis vectors in Rⁿ, 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 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

${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{i}=1}$

and since they form right angles with each other, if i ≠ 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 δ _ij is the Kronecker delta.

Vector components in an orthonormal basis

Also, by the geometric definition, for any vector e_i and a vector a, we note

${\displaystyle \mathbf {a} \cdot \mathbf {e} _{i}=\left\|\mathbf {a} \right\|\,\left\|\mathbf {e} _{i}\right\|\cos \theta _{i}=\left\|\mathbf {a} \right\|\cos \theta _{i}=a_{i},}$

where 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

${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \sum _{i}b_{i}\mathbf {e} _{i}=\sum _{i}b_{i}(\mathbf {a} \cdot \mathbf {e} _{i})=\sum _{i}b_{i}a_{i}=\sum _{i}a_{i}b_{i},}$

which is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.

Triangle with vector edges a and b, separated by angle θ.

Given two vectors a and b separated by angle θ (see image right), they form a triangle with a third side c = a − b. The dot product of this with itself is:

${\displaystyle {\begin{aligned}\mathbf {\color {orange}c} \cdot \mathbf {\color {orange}c} &=(\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\cdot (\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\\&=\mathbf {\color {red}a} \cdot \mathbf {\color {red}a} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {blue}b} \cdot \mathbf {\color {red}a} +\mathbf {\color {blue}b} \cdot \mathbf {\color {blue}b} \\&=\mathbf {\color {red}a} ^{2}-\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +\mathbf {\color {blue}b} ^{2}\\&=\mathbf {\color {red}a} ^{2}-2\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +\mathbf {\color {blue}b} ^{2}\\\mathbf {\color {orange}c} ^{2}&=\mathbf {\color {red}a} ^{2}+\mathbf {\color {blue}b} ^{2}-2\mathbf {\color {red}a} \mathbf {\color {blue}b} \cos \mathbf {\color {purple}\theta } \\\end{aligned}}}$

which is the law of cosines.

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 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 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).

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.

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-n vector u is, then, a function with domain {k ∈ N ∣ 1 ≤ k ≤ n}, and u_i is a notation for the image of i by the function/vector u.

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 a ≤ x ≤ b (also denoted [a, b]):[2]

${\displaystyle \left\langle u,v\right\rangle =\int _{a}^{b}u(x)v(x)dx}$

Generalized further to complex functions ψ(x) and χ(x), by analogy with the complex inner product above, gives[2]

${\displaystyle \left\langle \psi ,\chi \right\rangle =\int _{a}^{b}\psi (x){\overline {\chi (x)}}dx.}$

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

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

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 A and 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}}).}

${\displaystyle \mathbf {A}$ :\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}}).} (For real matrices)

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.

The inner product between a tensor of order n and a tensor of order m is a tensor of order n + m − 2, see Tensor contraction for details.

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.

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 as  A' * B 
• Matlab as  A' * B  or  conj(transpose(A)) * B  or  sum(conj(A) .* B) 
• GNU Octave as  sum(conj(X) .* Y, dim) 
• Intel oneAPI Math Kernel Library real p?dot dot = sub(x)'*sub(y); complex p?dotc dotc = conjg(sub(x)')*sub(y)