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{{redirect|Hasil kali skalar | hasil kali skalar abstrak | Hasil kali dalam | hasil kali vektor dan skalar | Perkalian skalar}}
<!--{{redirect|Scalar product|the abstract scalar product|Inner product space|the product of a vector and a scalar|Scalar multiplication}}
 
'''Produk skalardot''', juga disebut '''darab bintik''' ({{langLang-en|scalarDot product}}) atau ''dot'produk productskalar''', juga disebut '''darab skalar''' ({{lang-en|scalar product}}), juga disebut ''inner product'' (="produk dalam") dalam konteks ruang Euclid) dalam [[matematika]] adalah suatu operasi aljabar yang memasukkan dua [[urutan]] bilangan dengan panjang yang sama (biasanya [[vektor koordinat]]) dan menghasilkan suatu bilangan tunggal.<!-- Operasi Thisini operationdapat candidefinisikan bemenurut 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|Euclidean magnitude]]s of the two vectors and the [[cosine]] of the angle between them. The name "dot product" is derived from the [[Dot operator|centered dot]] "&nbsp;'''·'''&nbsp;" that is often used to designate this operation; the alternative name "scalar product" emphasizes the [[scalar (mathematics)|scalar]] (rather than [[Euclidean vector|vectorial]]) nature ofaljabar themaupun resultgeometri.
Menurut aljabar, produk skalar merupakan jumlah dari produk-produk masukan yang bersangkutan dari bilangan-bilangan pada dua urutan tersebut. Menurut geometri, produk skalar adalah produk dari [[Vektor (spasial)#Panjang|"besaran Euclidean" atau "panjang vektor"]] dua vektor dan [[kosinus]] sudut di antara keduanya. Nama "''produk dot''" diambil dari tanda [[Dot operator|''dot'', yaitu "tanda titik di tengah",]] "&nbsp;'''·'''&nbsp;" yang sering digunakan untuk melambangkan operasi ini; nama "produk skalar" menekankan sifat [[skalar (matematika)|skalar]] hasilnya (bukan [[Vektor (spasial)|vektorial]]).
 
InDalam three-dimensionalruang spacetiga dimensi, theproduk dotskalar productdikontraskan contrastsdengan with[[produk silang|produk thesilang [[(''cross product'')]] ofdua two vectorsvektor, whichyang producesmenghasilkan asuatu [[pseudovector]] as the result. Produk Theskalar dotberkaitan productlangsung isdengan directlykosinus relatedsudut toyang thedibentuk cosineoleh ofdua thevektor angledalam between two vectors inruang Euclidean space of anydari numberseberapapun ofbanyaknya dimensionsdimensi.
 
==DefinitionDefinisi==
Produk skalar sering didefinisikan menurut satu dari dua cara: menurut aljabar atau menurut geometri. Definisi geometris didasarkan pada pengertian sudut dan jarak (besaran vektor). Persamaan dua definisi ini bergantung pada memiliki [[sistem koordinat Kartesius]] untuk ruang Euklides.
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.
 
Dalam presentasi modern [[geometri Euclidean]], titik-titik ruang ditentukan berdasarkan koordinat Cartesiannya, dan [[ruang Euclidean]] itu sendiri umumnya diidentifikasikan dengan [[ruang kordinat nyata]] '''R'''<sup>''n''</sup>. Dalam presentasi seperti itu, pengertian panjang dan sudut tidaklah primitif. Mereka ditentukan melalui perkalian titik: panjang vektor didefinisikan sebagai akar kuadrat dari hasil kali titik vektor itu sendiri, dan [[kosinus]] dari (tidak berorientasi) sudut dua vektor dengan panjang satu didefinisikan sebagai perkalian titik mereka. Jadi kesetaraan dari dua definisi hasil perkalian titik adalah bagian dari kesetaraan klasik dan formulasi modern geometri Euklides.
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.
 
=== Definisi menurut aljabar ===
===Algebraic definition===
TheProduk dotskalar productdua of two vectorsvektor {{nowrap|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>, ..., ''A''<sub>''n''</sub>]}} anddan {{nowrap|1='''B''' = [''B''<sub>1</sub>, ''B''<sub>2</sub>, ..., ''B''<sub>''n''</sub>]}} isdidefinisikan defined assebagai:<ref name="Lipschutz2009">{{cite book |author= S. Lipschutz, M. Lipson |first1= |title= Linear Algebra (Schaum’s Outlines)|url= https://archive.org/details/linearalgebra0000lips_a2h3|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>
 
wheredi mana Σ denotesmelambangkan [[Summation|summation notation]] anddan ''n'' isadalah thedimensi dimensionruang of the vector spacevektor. For instanceMisalnya, indalam [[three-dimensionalruang tiga spacedimensi]], theproduk dotskalar product of vectorsvektor-vektor {{nowrap|[1, 3, −5]}} anddan {{nowrap|[4, −2, −1]}} isadalah:
 
:<math>
Baris 25 ⟶ 26:
</math>
 
=== Definisi menurut geometri ===
===Geometric definition===
InDalam [[EuclideanRuang spaceEuklides|ruang Euclidean]], asuatu [[Euclideanvektor vector(spasial)|vektor Euclidean]] isadalah asebuah geometricalobjek objectgeometri thatyang possessesmemiliki bothbaik abesaran (''magnitude'') anddan a[[arah (geometri)|arah]] (''direction''). Sebuah Avektor vectordapat candigambarkan beseperti picturedsebuah asanak an arrowpanah. Besarannya Its magnitude is itsadalah lengthpanjangnya, andsedangkan itsarahnya directionadalah isyang theditunjuk directionoleh theujung arrow pointspanah. Besaran The magnitude of a vectorvektor '''A''' isdilambangkan denoted bydengan <math>\|\mathbf{A}\|</math>. Produk Theskalar dotdua product of twovektor Euclidean vectors '''A''' anddan '''B''' isdidefinisikan defined bysebagai<ref name="Spiegel2009">{{cite book |author= M.R. Spiegel, S. Lipschutz, D. Spellman|first1= |title= Vector Analysis (Schaum’s Outlines)|url= https://archive.org/details/vectoranalysisan0000lips|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>
wheredi mana θ is theadalah [[anglesudut]] betweendi antara '''A''' anddan '''B'''.
 
InSecara particularkhusus, ifjika '''A''' anddan '''B''' areadalah [[orthogonalortogonal]], thenmaka thesudut angledi betweenantara themkeduanya isadalah 90° anddan
:<math>\mathbf A\cdot\mathbf B=0.</math>
AtPada thekeadaan otherekstrem extremelain, ifjika theykedua arevektor itu mempunyai arah yang sama (''codirectional''), thenmaka thesudut angledi betweenantara themkeduanya isadalah 0° anddan
:<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|</math>
Ini menyiratkan bahwa produk skalar suatu vektor '''A''' dengan dirinya sendiri adalah
This implies that the dot product of a vector '''A''' by itself is
:<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2,</math>
yang menghasilkan
which gives
: <math> \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A},</math>
therumus formula for theuntuk [[panjang Euclidean length]] of thevektor vectoritu.
<!--
 
===Scalar projection and first properties===
[[File:Dot Product.svg|thumb|right|Scalar projection]]
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:<math>\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_iA_i</math>
which is precisely the algebraic definition of the dot product. So the (geometric) dot product equals the (algebraic) dot product.
-->
 
==Properties Sifat ==
TheProduk dotskalar productmemenuhi fulfillssifat-sifat theberikut following properties ifjika '''a''', '''b''', anddan '''c''' are realadalah [[vectorvektor (geometryspasial)|vectorsvektor]] and[[bilangan real|real]] dan ''r'' isadalah asuatu [[scalarskalar (mathematicsmatematika)|scalarbilangan skalar]].<ref name="Lipschutz2009" /><ref name="Spiegel2009" />
 
# '''[[CommutativeKomutatif]]:'''
#: <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>
# '''[[DistributiveDistributif property|DistributiveDistributif]] 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>
# '''[[ScalarPerkalian multiplicationskalar]]:'''
#: <math> (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = c_1 c_2 (\mathbf{a} \cdot \mathbf{b}) </math>
# '''[[OrthogonalOrtogonal]]:'''
#: TwoDua non-zerovektor vectorsbukan-nol '''a''' anddan '''b''' areadalah ''orthogonal[[ortogonal]]'' [[ifjika anddan onlyhanya ifjika]] {{nowrap|1='''a''' ⋅ '''b''' = 0}}.
# '''NoTidak ada [[:en:cancellation law|cancellation]]:'''
#: UnlikeBerbeda multiplicationdengan ofperkalian ordinaryangka numbersbiasa, wheredi ifmana jika {{nowrap|1=''ab'' = ''ac''}}, thenmaka ''b'' alwaysselalu equalssama dengan ''c'' unlesskecuali ''a'' issama zerodengan [[nol]], the dot product doesproduk notskalar obeytidak themenuruti [[cancellation law]]:
#: IfJika {{nowrap|1='''a''' ⋅ '''b''' = '''a''' ⋅ '''c'''}} anddan {{nowrap|'''a''' ≠ '''0'''}}, thenmaka wedapat can writeditulis: {{nowrap|1='''a''' ⋅ ('''b''' − '''c''') = 0}} by thedengan [[distributivehukum lawdistributif]]; thehasil resultdi aboveatas saysmengatakan thisbahwa justini meanshanya thatberarti '''a''' istegak perpendicularlurus todengan {{nowrap|('''b''' − '''c''')}}, whichdi stillmana allowsmasih mengizinkan {{nowrap|('''b''' − '''c''') ≠ '''0'''}}, and thereforesehingga {{nowrap|'''b''' ≠ '''c'''}}.
# '''[[Product Rule]]:''' IfJika '''a''' anddan '''b''' areadalah suatu [[functionfungsi (mathematicsmatematika)|functionsfungsi]], thenmaka the derivative[[turunan]] ([[Notation for differentiation#Lagrange's notation|denoteddilambangkan byoleh atanda ''prime'']] ′) ofdari {{nowrap|'''a''' ⋅ '''b'''}} isadalah {{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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* [[Magnetic flux]] is the dot product of the [[magnetic field]] and the [[Area vector|area]] vectors.
-->
== Generalisasi ==
<!--
===Complex vectors===
Baris 146 ⟶ 148:
:<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 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>
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>
Baris 154 ⟶ 156:
===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.
Baris 163 ⟶ 165:
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>\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>\langle \psi , \chi \rangle = \int_a^b \psi(x)\overline{\chi(x)}dx.</math>
 
===Weight function===
Baris 180 ⟶ 182:
[[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.
-->
=== Tensor ===
Produk skalar antar suatu [[tensor]] pada ordo ''n'' dan suatu tensor pada ordo ''m'' adalah tensor pada ordo {{nowrap|''n'' + ''m'' − 2}}<!--, lihat [[tensor contraction]] for details.-->
 
== Lihat pula ==
* [[Cauchy–SchwarzPertidaksamaan inequalityCauchy–Schwarz]]
* [[CrossPerkalian productmatriks]]
* [[MatrixPerkalian multiplicationsilang]]
* [[Perkalian skalar]]
* [[Perkalian vektor]]
 
== Referensi ==
{{reflist}}
 
== Pranala luar ==
* {{springer|title=Inner product|id=p/i051240}}
* {{mathworld|urlname=DotProduct|title=Dot product}}
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{{linear algebra}}
 
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