Dot product explained

In mathematics, the dot product or scalar product^[1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, 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 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, Euclidean spaces are often defined by using vector spaces. 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 of their dot product by the product of their lengths).

The name "dot product" is derived from the dot operator " · " that is often used to designate this operation;^[2] the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product 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.

Rⁿ

. In such a presentation, the notions of length and angle 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

a=[a_1,a_2, … ,a_n]

and specified with respect to an orthonormal basis, is defined as:^[3]

$\mathbf a \cdot \mathbf b = \sum_^n a_i b_i = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n$

where

\Sigma

denotes summation and

is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors and is:

$\begin\ [1, 3, -5] \cdot [4, -2, -1] &= (1 \times 4) + (3\times-2) + (-5\times-1) \\&= 4 - 6 + 5 \\&= 3\end$

Likewise, the dot product of the vector with itself is:

$\begin\ [1, 3, -5] \cdot [1, 3, -5] &= (1 \times 1) + (3\times 3) + (-5\times -5) \\&= 1 + 9 + 25 \\&= 35\end$

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

$\mathbf a \cdot \mathbf b = \mathbf a^ \mathbf b,$ where

a{^T}

denotes the transpose of

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: $\begin 1 & 3 & -5\end\begin 4 \\ -2 \\ -1\end = 3 \, .$

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

is denoted by

\left\|a\right\|

. The dot product of two Euclidean vectors

and

is defined by^[4] ^[5]

\mathbf\cdot\mathbf= \left\|\mathbf\right\| \left\|\mathbf\right\|\cos\theta,

where

\theta

is the angle between

and

In particular, if the vectors

and

are orthogonal (i.e., their angle is

	\pi
	2

90^\circ

), then

\cos

	\pi
	2

, which implies that

$\mathbf a \cdot \mathbf b = 0 .$ At the other extreme, if they are codirectional, then the angle between them is zero with

\cos0=1

and

\mathbf a \cdot \mathbf b = \left\| \mathbf a \right\| \, \left\| \mathbf b \right\|

This implies that the dot product of a vector

with itself is

\mathbf a \cdot \mathbf a = \left\| \mathbf a \right\| ^2,

which gives

\left\| \mathbf a \right\| = \sqrt,

the formula for the Euclidean length of the vector.

Scalar projection and first properties

The scalar projection (or scalar component) of a Euclidean vector

in the direction of a Euclidean vector

is given by

a_b = \left\| \mathbf a \right\| \cos \theta,

where

\theta

is the angle between

and

In terms of the geometric definition of the dot product, this can be rewritten as $a_b = \mathbf a \cdot \widehat,$ where

\widehat{b}=b/\left\|b\right\|

is the unit vector in the direction of

The dot product is thus characterized geometrically by^[6] $\mathbf a \cdot \mathbf b = a_b \left\| \mathbf \right\| = b_a \left\| \mathbf \right\| .$ The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar

\alpha

(\alpha \mathbf) \cdot \mathbf b = \alpha (\mathbf a \cdot \mathbf b) = \mathbf a \cdot (\alpha \mathbf b) .

It also satisfies the distributive law, meaning that

\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

a ⋅ a

is never negative, and is zero if and only if

a=0

, the zero vector.

Equivalence of the definitions

e_{1, … ,e}_n

are the standard basis vectors in

Rⁿ

, then we may write

\begin\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

The vectors

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,

\mathbf e_i \cdot \mathbf e_i = 1

and since they form right angles with each other, if

i ≠ j

\mathbf e_i \cdot \mathbf e_j = 0 .

Thus in general, we can say that:

\mathbf e_i \cdot \mathbf e_j = \delta_,

where

\delta_ij

is the Kronecker delta.

Also, by the geometric definition, for any vector

e_i

and a vector

, we note that

\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

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 $\mathbf a \cdot \mathbf b = \mathbf a \cdot \sum_i b_i \mathbf e_i = \sum_i b_i (\mathbf a \cdot \mathbf e_i) = \sum_i b_i a_i= \sum_i a_i b_i,$ which is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.

Properties

The dot product fulfills the following properties if

, and

are real vectors and

c₁

and

c₂

are scalars.

Commutative

\mathbf \cdot \mathbf = \mathbf \cdot \mathbf,

which follows from the definition (

\theta

is the angle between

and

):^[7]

\mathbf \cdot \mathbf = \left\| \mathbf \right\| \left\| \mathbf \right\| \cos \theta = \left\| \mathbf \right\| \left\| \mathbf \right\| \cos \theta = \mathbf \cdot \mathbf .

Distributive over vector addition : $\mathbf \cdot (\mathbf + \mathbf) = \mathbf \cdot \mathbf + \mathbf \cdot \mathbf .$

a ⋅ b

and a vector

is not defined, which means that the expressions involved in the associative property,

(a ⋅ b) ⋅ c

a ⋅ (b ⋅ c)

are both ill-defined.^[8] Note however that the previously mentioned scalar multiplication property is sometimes called the "associative law for scalar and dot product"^[9] or one can say that "the dot product is associative with respect to scalar multiplication" because

c(a ⋅ b)=(ca) ⋅ b=a ⋅ (cb)

.^[10]

Orthogonal

Two non-zero vectors

and

are orthogonal if and only if

a ⋅ b=0

No cancellation

Unlike multiplication of ordinary numbers, where if

ab=ac

, then

always equals

unless

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

is perpendicular to

(b-c)

, which still allows

(b-c) ≠ 0

, and therefore allows

b ≠ c

Product rule

and

are vector-valued differentiable functions, then the derivative (denoted by a prime

{}'

) of

a ⋅ b

is given by the rule

(\mathbf\cdot\mathbf)' = \mathbf'\cdot\mathbf + \mathbf\cdot\mathbf'.

Application to the law of cosines

See main article: Law of cosines.

Given two vectors

{\color{red}a

} and

{\color{blue}b

} separated by angle

\theta

(see the upper image), they form a triangle with a third side

{\color{orange}c

} = - . Let

and

denote the lengths of

{\color{red}a

{\color{blue}b

}, and

{\color{orange}c

}, respectively. The dot product of this with itself is:

$\begin\mathbf \cdot \mathbf & = (\mathbf - \mathbf) \cdot (\mathbf - \mathbf) \\ & = \mathbf \cdot \mathbf - \mathbf \cdot \mathbf - \mathbf \cdot \mathbf + \mathbf \cdot \mathbf \\ & = ^2 - \mathbf \cdot \mathbf - \mathbf \cdot \mathbf + ^2 \\ & = ^2 - 2 \mathbf \cdot \mathbf + ^2 \\^2 & = ^2 + ^2 - 2 \cos \mathbf \\\end$

which is the law of cosines.

Triple product

See main article: Triple product.

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

The scalar triple product of three vectors is defined as $\mathbf \cdot (\mathbf \times \mathbf) = \mathbf \cdot (\mathbf \times \mathbf)=\mathbf \cdot (\mathbf \times \mathbf).$ 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 $\mathbf \times (\mathbf \times \mathbf) = (\mathbf \cdot \mathbf)\, \mathbf - (\mathbf \cdot \mathbf)\, \mathbf .$ 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, $\mathbf \cdot \mathbf = |\mathbf| \, |\mathbf| \cos \theta$ 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:^[11] ^[12]

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 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^[13] $\mathbf \cdot \mathbf = \sum_i,$ where

\overline{b_i}

is the complex conjugate of

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:

$\mathbf \cdot \mathbf = \mathbf^\mathsf \mathbf .$

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

. The dot product is not symmetric, since

\mathbf \cdot \mathbf = \overline .

The angle between two complex vectors is then given by

\cos \theta = \frac .

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

a ⋅ a=a^Ha

, involving the conjugate transpose of a row vector, is also known as the norm squared,

\mathbf \cdot \mathbf = \|\mathbf\|^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

See main article: Inner product space. 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 numbers

\Complex

. It is usually denoted using angular brackets by

\left\langlea,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-

vector

is, then, a function with domain

\{k\inN:1\leqk\leqn\}

, and

u_i

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 :

$\left\langle u, v \right\rangle = \int_a^b u(x) v(x) \,dx.$

Generalized further to complex functions

\psi(x)

and

\chi(x)

, by analogy with the complex inner product above, gives

$\left\langle \psi, \chi \right\rangle = \int_a^b \psi(x) \overline \,dx.$

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

u(x)

and

v(x)

with respect to the weight function

r(x)>0

$\left\langle u, v \right\rangle_r = \int_a^b r(x) u(x) v(x) \, d x.$

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

and

of the same size:

$\mathbf : \mathbf = \sum_i \sum_j A_ \overline = \operatorname (\mathbf^\mathsf \mathbf) = \operatorname (\mathbf \mathbf^\mathsf) .$ And for real matrices, $\mathbf : \mathbf = \sum_i \sum_j A_ B_ = \operatorname (\mathbf^\mathsf \mathbf) = \operatorname (\mathbf \mathbf^\mathsf) = \operatorname (\mathbf^\mathsf \mathbf) = \operatorname (\mathbf \mathbf^\mathsf) .$

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 of order

and a tensor of order

is a tensor of order

n+m-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, ; complex,,,
Fortran as or
Julia as or standard library LinearAlgebra as
R (programming language) as for vectors or, more generally for matrices, as
Matlab as or or or
Python (package NumPy) as or or
GNU Octave as , and similar code as Matlab
Intel oneAPI Math Kernel Library real p?dot ; complex p?dotc

External links

- Explanation of dot product including with complex vectors
"Dot Product" by Bruce Torrence, Wolfram Demonstrations Project, 2007.

Notes and References

The term scalar product means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication.
Web site: Dot Product. 2020-09-06. www.mathsisfun.com.
Book: S. Lipschutz . M. Lipson . Linear Algebra (Schaum's Outlines) . 4th . 2009. McGraw Hill. 978-0-07-154352-1.
Book: M.R. Spiegel . S. Lipschutz . D. Spellman . Vector Analysis (Schaum's Outlines). 2nd . 2009. McGraw Hill. 978-0-07-161545-7.
Book: A I Borisenko. I E Taparov. Vector and tensor analysis with applications . Dover . Richard Silverman . 1968 . 14.
Book: Arfken . G. B. . Weber . H. J. . Mathematical Methods for Physicists . . Boston, MA . 5th . 978-0-12-059825-0 . 2000 . 14–15 .
Web site: Nykamp. Duane. The dot product. September 6, 2020. Math Insight.
Weisstein, Eric W. "Dot Product." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DotProduct.html
Book: T. Banchoff. J. Wermer . Linear Algebra Through Geometry. 1983. Springer Science & Business Media. 978-1-4684-0161-5 . 12.
Book: A. Bedford. Wallace L. Fowler. Engineering Mechanics: Statics. 2008. Prentice Hall . 978-0-13-612915-8 . 5th . 60.
Book: K.F. Riley . M.P. Hobson . S.J. Bence . Mathematical methods for physics and engineering. registration . 3rd. 2010. Cambridge University Press . 978-0-521-86153-3.
Book: M. Mansfield . C. O'Sullivan . Understanding Physics . 4th . 2011. John Wiley & Sons. 978-0-47-0746370.
Book: Berberian, Sterling K. . 287 . Linear Algebra . 2014 . 1992 . Dover . 978-0-486-78055-9.

Dot product explained

Definition

Coordinate definition

Geometric definition

Scalar projection and first properties

Equivalence of the definitions

Properties

Application to the law of cosines

Triple product

Physics

Generalizations

Complex vectors

Inner product

Functions

Weight function

Dyadics and matrices

Tensors

Computation

Algorithms

Libraries

See also

External links

Notes and References