# Difference between revisions of "Dot product"

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− | {{redirect|Scalar product|the abstract scalar product|Inner product space | + | {{redirect|Scalar product|the abstract scalar product|Inner product space|the product of a vector and a scalar|Scalar multiplication}} |

− | In [[mathematics]], the '''dot product''', or '''scalar product''' (or sometimes '''inner product''' in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually [[coordinate vector]]s) and returns a single number | + | In [[mathematics]], the '''dot product''', or '''scalar product''' (or sometimes '''inner product''' in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually [[coordinate vector]]s) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the [[Euclidean vector#Length|magnitudes]] of the two vectors and the [[cosine]] of the angle between them. The name "dot product" is derived from the [[Dot operator|centered dot]] " '''·''' " that is often used to designate this operation; the alternative name "scalar product" emphasizes the [[scalar (mathematics)|scalar]] (rather than [[Euclidean vector|vectorial]]) nature of the result. |

− | In | + | In three-dimensional space, the dot product contrasts with the [[cross product]] of two vectors, which produces a [[pseudovector]] as the result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions. |

− | + | ==Definition== | |

+ | The dot product is often defined in one of two ways: 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. | ||

− | + | In modern presentations of [[Euclidean geometry]], the points of space are defined in terms of their Cartesian coordinates, and [[Euclidean space]] itself is commonly identified with the [[real coordinate space]] '''R'''<sup>''n''</sup>. In such a presentation, the notions of length and angles are not primitive. They 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 of 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. | |

− | ==='' | + | ===Algebraic definition=== |

+ | The dot product of two vectors {{nowrap|1='''a''' = [''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>]}} and {{nowrap|1='''b''' = [''b''<sub>1</sub>, ''b''<sub>2</sub>, ..., ''b''<sub>''n''</sub>]}} is defined as:<ref name="Lipschutz2009">{{cite book |author= S. Lipschutz, M. Lipson |first1= |title= Linear Algebra (Schaum’s Outlines)|edition= 4th |year= 2009|publisher= McGraw Hill|isbn=978-0-07-154352-1}}</ref> | ||

− | + | :<math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n</math> | |

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

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− | where Σ denotes [[Summation|summation notation]] and ''n'' is the dimension of the vector space. | ||

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:<math> | :<math> | ||

− | [1, 3, -5] \cdot [4, -2, -1] | + | \begin{align} |

− | = (1)(4) + (3)(-2) + (-5)(-1) | + | \ [1, 3, -5] \cdot [4, -2, -1] &= (1)(4) + (3)(-2) + (-5)(-1) \\ |

− | = 4 - 6 + 5 | + | &= 4 - 6 + 5 \\ |

− | = 3. | + | &= 3. |

+ | \end{align} | ||

</math> | </math> | ||

− | + | ===Geometric definition=== | |

− | + | In [[Euclidean space]], a [[Euclidean vector]] is a geometrical 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 the arrow points. The magnitude of a vector '''A''' is denoted by <math>\|\mathbf{A}\|</math>. The dot product of two Euclidean vectors '''A''' and '''B''' is defined by<ref name="Spiegel2009">{{cite book |author= M.R. Spiegel, S. Lipschutz, D. Spellman|first1= |title= Vector Analysis (Schaum’s Outlines)|edition= 2nd |year= 2009|publisher= McGraw Hill|isbn=978-0-07-161545-7}}</ref> | |

− | + | :<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,</math> | |

− | + | where θ is the [[angle]] between '''A''' and '''B'''. | |

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− | + | In particular, if '''A''' and '''B''' are [[orthogonal]], then the angle between them is 90° and | |

− | + | :<math>\mathbf A\cdot\mathbf B=0.</math> | |

+ | At the other extreme, if they are codirectional, then the angle between them is 0° and | ||

+ | :<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|</math> | ||

+ | This implies that the dot product of a vector '''A''' by itself is | ||

+ | :<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2,</math> | ||

+ | which gives | ||

+ | : <math> \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A},</math> | ||

+ | the formula for the [[Euclidean length]] of the vector. | ||

− | + | ===Scalar projection and first properties=== | |

+ | [[File:Dot Product.svg|thumb|right|Scalar projection]] | ||

+ | The [[scalar projection]] (or scalar component) of a Euclidean vector '''A''' in the direction of a Euclidean vector '''B''' is given by | ||

+ | :<math>A_B=\|\mathbf A\|\cos\theta</math> | ||

+ | where θ is the angle between '''A''' and '''B'''. | ||

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

+ | :<math>A_B = \mathbf A\cdot\widehat{\mathbf B}</math> | ||

+ | where <math>\widehat{\mathbf B} = \mathbf B/\|\mathbf B\|</math> is the [[unit vector]] in the direction of '''B'''. | ||

− | === | + | [[File:Dot product distributive law.svg|thumb|right|Distributive law for the dot product]] |

− | + | The dot product is thus characterized geometrically by<ref>{{Cite book | last1=Arfken | first1=G. B. | last2=Weber | first2=H. J. | title=Mathematical Methods for Physicists | publisher=[[Academic Press]] | location=Boston, MA | edition=5th | isbn=978-0-12-059825-0 | year=2000 | pages=14–15 }}.</ref> | |

+ | :<math>\mathbf A\cdot\mathbf B = A_B\|\mathbf{B}\|=B_A\|\mathbf{A}\|.</math> | ||

+ | The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α, | ||

+ | :<math>(\alpha\mathbf{A})\cdot\mathbf B=\alpha(\mathbf A\cdot\mathbf B)=\mathbf A\cdot(\alpha\mathbf B).</math> | ||

+ | It also satisfies a [[distributive law]], meaning that | ||

+ | :<math>\mathbf A\cdot(\mathbf B+\mathbf C) = \mathbf A\cdot\mathbf B+\mathbf A\cdot\mathbf C.</math> | ||

− | + | These properties may be summarized by saying that the dot product is a [[bilinear form]]. Moreover, this bilinear form is [[positive definite bilinear form|positive definite]], which means that | |

− | + | <math>\mathbf A\cdot \mathbf A</math> | |

− | + | is never negative and is zero if and only if <math>\mathbf A = \mathbf 0.</math> | |

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− | :<math> \ | + | ===Equivalence of the definitions=== |

− | \ | + | If '''e'''<sub>1</sub>,...,'''e'''<sub>''n''</sub> are the [[standard basis|standard basis vectors]] in '''R'''<sup>''n''</sup>, then we may write |

− | \ | + | :<math>\begin{align} |

− | + | \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{align} | ||

</math> | </math> | ||

+ | The vectors '''e'''<sub>''i''</sub> 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 | ||

+ | :<math>\mathbf e_i\cdot\mathbf e_i=1</math> | ||

+ | and since they form right angles with each other, if ''i'' ≠ ''j'', | ||

+ | :<math>\mathbf e_i\cdot\mathbf e_j = 0.</math> | ||

− | + | Now applying the distributivity of the geometric version of the dot product gives | |

+ | :<math>\mathbf A\cdot\mathbf B = \sum_i B_i(\mathbf A\cdot\mathbf e_i) = \sum_i B_iA_i</math> | ||

+ | which is precisely the algebraic definition of the dot product. So the (geometric) dot product equals the (algebraic) dot product. | ||

− | + | ==Properties== | |

+ | The dot product fulfils the following properties if '''a''', '''b''', and '''c''' are real [[vector (geometry)|vectors]] and ''r'' is a [[scalar (mathematics)|scalar]].<ref name="Lipschutz2009" /><ref name="Spiegel2009" /> | ||

− | :<math> \ | + | # '''[[Commutative]]:''' |

− | \ | + | #: <math> \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.</math> |

− | \ | + | #: which follows from the definition (''θ'' is the angle between '''a''' and '''b'''): |

− | \ | + | #: <math>\mathbf{a}\cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta = \|\mathbf{b}\|\|\mathbf{a}\|\cos\theta = \mathbf{b}\cdot\mathbf{a} </math> |

+ | # '''[[Distributive]] over vector addition:''' | ||

+ | #: <math> \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}. </math> | ||

+ | # '''[[bilinear form|Bilinear]]''': | ||

+ | #: <math> \mathbf{a} \cdot (r\mathbf{b} + \mathbf{c}) | ||

+ | = r(\mathbf{a} \cdot \mathbf{b}) + (\mathbf{a} \cdot \mathbf{c}). | ||

</math> | </math> | ||

+ | # '''[[Scalar multiplication]]:''' | ||

+ | #: <math> (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = c_1 c_2 (\mathbf{a} \cdot \mathbf{b}) </math> | ||

+ | # '''[[Orthogonal]]:''' | ||

+ | #: Two non-zero vectors '''a''' and '''b''' are ''orthogonal'' [[if and only if]] {{nowrap|1='''a''' ⋅ '''b''' = 0}}. | ||

+ | # '''No [[cancellation law|cancellation]]:''' | ||

+ | #: Unlike multiplication of ordinary numbers, where if {{nowrap|1=''ab'' = ''ac''}}, then ''b'' always equals ''c'' unless ''a'' is zero, the dot product does not obey the [[cancellation law]]: | ||

+ | #: If {{nowrap|1='''a''' ⋅ '''b''' = '''a''' ⋅ '''c'''}} and {{nowrap|'''a''' ≠ '''0'''}}, then we can write: {{nowrap|1='''a''' ⋅ ('''b''' − '''c''') = 0}} by the [[distributive law]]; the result above says this just means that '''a''' is perpendicular to {{nowrap|('''b''' − '''c''')}}, which still allows {{nowrap|('''b''' − '''c''') ≠ '''0'''}}, and therefore {{nowrap|'''b''' ≠ '''c'''}}. | ||

+ | # '''[[Derivative]]:''' If '''a''' and '''b''' are [[function (mathematics)|functions]], then the derivative ([[Notation for differentiation#Lagrange's notation|denoted by a prime]] ′) of {{nowrap|'''a''' ⋅ '''b'''}} is {{nowrap|'''a'''′ ⋅ '''b''' + '''a''' ⋅ '''b'''′}}. | ||

− | == | + | ===Application to the cosine law=== |

− | + | [[File:Dot product cosine rule.svg|100px|thumb|Triangle with vector edges '''a''' and '''b''', separated by angle ''θ''.]] | |

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− | + | {{main|law of cosines}} | |

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− | + | Given two vectors '''a''' and '''b''' separated by angle ''θ'' (see image right), they form a triangle with a third side {{nowrap|1='''c''' = '''a''' − '''b'''}}. The dot product of this with itself is: | |

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− | + | :<math> | |

− | :<math> \mathbf{a} \cdot ( | + | \begin{align} |

− | + | \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{align} | ||

</math> | </math> | ||

− | + | which is the [[law of cosines]]. | |

− | + | {{clear}} | |

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==Triple product expansion== | ==Triple product expansion== | ||

{{Main|Triple product}} | {{Main|Triple product}} | ||

− | This is a very useful identity (also known as '''Lagrange's formula''') involving the dot- and | + | This is a very useful identity (also known as '''Lagrange's formula''') involving the dot- and [[Cross product|cross-products]]. It is written as:<ref name="Lipschutz2009" /><ref name="Spiegel2009" /> |

:<math>\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})</math> | :<math>\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})</math> | ||

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which is [[mnemonic|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]]. | which is [[mnemonic|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]]. | ||

− | == | + | ==Physics== |

− | + | In [[physics]], vector magnitude is a [[scalar (physics)|scalar]] in the physical sense, i.e. a [[physical quantity]] independent of the coordinate system, expressed as the [[product (mathematics)|product]] of a [[number|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. | |

− | + | Examples include:<ref name="Riley2010">{{cite book |author= K.F. Riley, M.P. Hobson, S.J. Bence |title= Mathematical methods for physics and engineering|edition= 3rd|year= 2010|publisher= Cambridge University Press|isbn=978-0-521-86153-3}}</ref><ref>{{cite book |author= M. Mansfield, C. O’Sullivan|title= Understanding Physics|edition= 4th |year= 2011|publisher= John Wiley & Sons|isbn=978-0-47-0746370}}</ref> | |

− | + | * [[Mechanical work]] is the dot product of [[force]] and [[Displacement (vector)|displacement]] vectors. | |

− | + | * [[Magnetic flux]] is the dot product of the [[magnetic field]] and the [[Area vector|area]] vectors. | |

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− | [[ | ||

==Generalizations== | ==Generalizations== | ||

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===Complex vectors=== | ===Complex vectors=== | ||

− | + | For vectors with [[complex number|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 would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called [[Isotropic quadratic form|isotropic]]); 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 scalar product, through the alternative definition<ref name="Lipschutz2009" /> | |

− | For vectors with [[complex number|complex]] entries, using the given definition of the dot product would lead to quite different | ||

:<math>\mathbf{a}\cdot \mathbf{b} = \sum{a_i \overline{b_i}} </math> | :<math>\mathbf{a}\cdot \mathbf{b} = \sum{a_i \overline{b_i}} </math> | ||

− | where <span style="text-decoration: overline">''b<sub>i</sub>''</span> is the [[complex conjugate]] of ''b<sub>i</sub>''. 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''' | + | where <span style="text-decoration: overline">''b<sub>i</sub>''</span> is the [[complex conjugate]] of ''b<sub>i</sub>''. 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 thus [[sesquilinear]] rather than bilinear: it is [[conjugate linear]] and not linear in '''b''', and the scalar product is not symmetric, since |

:<math> \mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}}. </math> | :<math> \mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}}. </math> | ||

The angle between two complex vectors is then given by | The angle between two complex vectors is then given by | ||

:<math>\cos\theta = \frac{\operatorname{Re}(\mathbf{a}\cdot\mathbf{b})}{\|\mathbf{a}\|\,\|\mathbf{b}\|}.</math> | :<math>\cos\theta = \frac{\operatorname{Re}(\mathbf{a}\cdot\mathbf{b})}{\|\mathbf{a}\|\,\|\mathbf{b}\|}.</math> | ||

− | This type of scalar product is nevertheless | + | This type of scalar product is nevertheless useful, and leads to the notions of [[Hermitian form]] and of general [[inner product space]]s. |

− | === | + | ===Inner product=== |

+ | {{main|Inner product space}} | ||

+ | The inner product generalizes the dot product to [[vector space|abstract vector spaces]] over a [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]], being either the field of [[real number]]s <math>\mathbb{R}</math> or the field of [[complex number]]s <math>\mathbb{C}</math>. It is usually denoted by <math>\langle\mathbf{a}\, , \mathbf{b}\rangle</math>. | ||

− | + | The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is [[Sesquilinear form|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 [[coordinate vector|entries]]. Thus these vectors can be regarded as [[discrete function]]s: a length-{{mvar|n}} vector {{mvar|u}} is, then, a function with [[domain of a function|domain]] {{math|{''k'' ∈ ℕ ∣ 1 ≤ ''k'' ≤ ''n''}}}, and {{math|''u''<sub>''i''</sub>}} is a notation for the image of {{math|''i''}} by the function/vector {{math|''u''}}. | ||

− | + | This notion can be generalized to [[continuous function]]s: 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 (mathematics)|interval]] {{math|''a'' ≤ ''x'' ≤ ''b''}} (also denoted {{math|[''a'', ''b'']}}):<ref name="Lipschutz2009" /> | |

− | :<math> | + | :<math>\langle u , v \rangle = \int_a^b u(x)v(x)dx </math> |

− | + | Generalized further to [[complex function]]s {{math|''ψ''(''x'')}} and {{math|''χ''(''x'')}}, by analogy with the complex inner product above, gives<ref name="Lipschutz2009" /> | |

− | :<math> | + | :<math>\langle \psi , \chi \rangle = \int_a^b \psi(x)\overline{\chi(x)}dx.</math> |

===Weight function=== | ===Weight function=== | ||

− | + | Inner products can have a [[weight function]], i.e. a function which weight each term of the inner product with a value. | |

− | Inner products can have a [[weight function]], i.e. a function which weight each term of the inner product with a value. | ||

===Dyadics and matrices=== | ===Dyadics and matrices=== | ||

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[[Matrix (mathematics)|Matrices]] have the [[Frobenius inner product]], which is analogous to the vector inner product. It is defined as the sum of the products of the corresponding components of two matrices '''A''' and '''B''' having the same size: | [[Matrix (mathematics)|Matrices]] have the [[Frobenius inner product]], which is analogous to the vector inner product. It is defined as the sum of the products of the corresponding components of two matrices '''A''' and '''B''' having the same size: | ||

− | :<math>\bold{A}:\bold{B} = \sum_i\sum_j A_{ij}B_{ij}</math> | + | :<math>\bold{A}:\bold{B} = \sum_i\sum_j A_{ij}\overline{B_{ij}} = \mathrm{tr}(\mathbf{A}^* \mathbf{B}) = \mathrm{tr}(\mathbf{A} \mathbf{B}^*).</math> |

+ | :<math>\bold{A}:\bold{B} = \sum_i\sum_j A_{ij}B_{ij} = \mathrm{tr}(\mathbf{A}^\mathrm{T} \mathbf{B}) = \mathrm{tr}(\mathbf{A} \mathbf{B}^\mathrm{T}).</math> (For real matrices) | ||

− | [[Dyadics]] have a dot product and "double" dot product defined on them, see [[Dyadics# | + | [[Dyadics]] have a dot product and "double" dot product defined on them, see [[Dyadics#Product of dyadic and dyadic|Dyadics (Product of dyadic and dyadic)]] for their definitions. |

===Tensors=== | ===Tensors=== | ||

− | + | The inner product between a [[tensor]] of order ''n'' and a tensor of order ''m'' is a tensor of order {{nowrap|''n'' + ''m'' − 2}}, see [[tensor contraction]] for details. | |

− | 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. | ||

==See also== | ==See also== | ||

* [[Cauchy–Schwarz inequality]] | * [[Cauchy–Schwarz inequality]] | ||

+ | * [[Cross product]] | ||

* [[Matrix multiplication]] | * [[Matrix multiplication]] | ||

==References== | ==References== | ||

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{{reflist}} | {{reflist}} | ||

==External links== | ==External links== | ||

+ | * {{springer|title=Inner product|id=p/i051240}} | ||

* {{mathworld|urlname=DotProduct|title=Dot product}} | * {{mathworld|urlname=DotProduct|title=Dot product}} | ||

* [http://www.mathreference.com/la,dot.html Explanation of dot product including with complex vectors] | * [http://www.mathreference.com/la,dot.html Explanation of dot product including with complex vectors] | ||

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[[Category:Vectors]] | [[Category:Vectors]] | ||

[[Category:Analytic geometry]] | [[Category:Analytic geometry]] | ||

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## Revision as of 18:49, 1 February 2014

{{#invoke:Hatnote|hatnote}}Template:Main other

In mathematics, the **dot product**, or **scalar product** (or sometimes **inner product** in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the magnitudes of the two vectors and the cosine of the angle between them. The name "dot product" is derived from the centered dot " **·** " that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar (rather than vectorial) nature of the result.

In three-dimensional space, the dot product contrasts with the cross product of two vectors, which produces a pseudovector as the result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions.

## Definition

The dot product is often defined in one of two ways: 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.

In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space **R**^{n}. In such a presentation, the notions of length and angles are not primitive. They 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 of 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.

### Algebraic definition

The dot product of two vectors **a** = [*a*_{1}, *a*_{2}, ..., *a*_{n}] and **b** = [*b*_{1}, *b*_{2}, ..., *b*_{n}] is defined as:^{[1]}

where Σ denotes summation notation 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:

### Geometric definition

In Euclidean space, a Euclidean vector is a geometrical 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 the arrow points. The magnitude of a vector **A** is denoted by . The dot product of two Euclidean vectors **A** and **B** is defined by^{[2]}

where θ is the angle between **A** and **B**.

In particular, if **A** and **B** are orthogonal, then the angle between them is 90° and

At the other extreme, if they are codirectional, then the angle between them is 0° and

This implies that the dot product of a vector **A** by itself is

which gives

the formula for the Euclidean length of the vector.

### Scalar projection and first properties

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

where θ is the angle between **A** and **B**.

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

where is the unit vector in the direction of **B**.

The dot product is thus characterized geometrically by^{[3]}

The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α,

It also satisfies a distributive law, meaning that

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 is never negative and is zero if and only if

### Equivalence of the definitions

If **e**_{1},...,**e**_{n} are the standard basis vectors in **R**^{n}, then we may write

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

and since they form right angles with each other, if *i* ≠ *j*,

Now applying the distributivity of the geometric version of the dot product gives

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

## Properties

The dot product fulfils the following properties if **a**, **b**, and **c** are real vectors and *r* is a scalar.^{[1]}^{[2]}

**Commutative:****Distributive over vector addition:****Bilinear**:**Scalar multiplication:****Orthogonal:**- Two non-zero vectors
**a**and**b**are*orthogonal*if and only if**a**⋅**b**= 0.

- Two non-zero vectors
**No cancellation:**- 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**.

- Unlike multiplication of ordinary numbers, where if
**Derivative:**If**a**and**b**are functions, then the derivative (denoted by a prime ′) of**a**⋅**b**is**a**′ ⋅**b**+**a**⋅**b**′.

### Application to the cosine law

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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:

which is the law of cosines.

## Triple product expansion

{{#invoke:main|main}}

This is a very useful identity (also known as **Lagrange's formula**) involving the dot- and cross-products. It is written as:^{[1]}^{[2]}

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.

## Physics

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.
Examples include:^{[4]}^{[5]}

- Mechanical work is the dot product of force and displacement vectors.
- Magnetic flux is the dot product of the magnetic field and the area vectors.

## 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 would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called isotropic); 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 scalar product, through the alternative definition^{[1]}

where *b _{i}* is the complex conjugate of

*b*. 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 thus sesquilinear rather than bilinear: it is conjugate linear and not linear in

_{i}**b**, and the scalar product is not symmetric, since

The angle between two complex vectors is then given by

This type of scalar product is nevertheless useful, and leads to the notions of Hermitian form and of general inner product spaces.

### Inner product

{{#invoke:main|main}} 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 . It is usually denoted by .

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-Template:Mvar vector Template:Mvar 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*]):^{[1]}

Generalized further to complex functions *ψ*(*x*) and *χ*(*x*), by analogy with the complex inner product above, gives^{[1]}

### Weight function

Inner products can have a weight function, i.e. a function which weight each term of the inner product with a value.

### Dyadics and matrices

Matrices have the Frobenius inner product, which is analogous to the vector inner product. It is defined as the sum of the products of the corresponding components of two matrices **A** and **B** having the same size:

Dyadics have a dot product and "double" dot product defined on them, see Dyadics (Product of dyadic and dyadic) for their definitions.

### Tensors

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.

## See also

## References

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## External links

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

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