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<!--{{redirect|Scalar product|the abstract scalar product|Inner product space|the product of a vector and a scalar|Scalar multiplication}}
'''Produk skalar''' ({{lang-en|scalar product}} atau ''dot product'' (="produk dot"), 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]] " '''·''' " 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",]] " '''·''' " 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==
TheProduk dotskalar productsering isdidefinisikan oftenmenurut definedsatu indari onedua ofcara: twomenurut ways:aljabar algebraicallyatau ormenurut geometricallygeometri. <!-- 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.
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=== 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>]}} is defineddidefinisikan assebagai:<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>
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>
</math>
=== Definisi menurut geometri ===
===Geometric definition===
InDalam [[ruang Euclidean space]], asuatu [[Euclideanvektor vector(spasial)|vektor Euclidean]] isadalah asebuah geometricalobyek objectgeometri thatyang possessesmemiliki bothbaik abesaran (''magnitude'') anddan a[[arah (geometri)|arah]] (''direction''). ASebuah vectorvektor candapat bedigambarkan picturedseperti assebuah ananak 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''' is defineddidefinisikan bysebagai<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>
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 otherekstrim 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.
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===Scalar projection and first properties===
[[File:Dot Product.svg|thumb|right|Scalar projection]]
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