Lihat sumber Perkalian titik

{{redirect|Hasil kali skalar | hasil kali skalar abstrak | Hasil kali dalam | hasil kali vektor dan skalar | Perkalian skalar}}

'''Perkalian titik''' <ref>https://jagostat.com/aljabar-linear/perkalian-titik-dan-silang</ref> ({{Lang-en|dot product}}, {{lang-en|scalar product}}) (juga disebut '''darab skalar''', '''produk dot''', '''darab bintik''', '''produk skalar''', atau "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 ini dapat didefinisikan menurut aljabar maupun geometri.
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]]).

Dalam ruang tiga dimensi, produk skalar dikontraskan dengan [[perkalian silang]] (en: ''cross product'') dua vektor, yang menghasilkan suatu [[pseudovector]]. Produk skalar berkaitan langsung dengan kosinus sudut yang dibentuk oleh dua vektor dalam ruang Euclidean dari seberapapun banyaknya dimensi.

== Definisi ==
Perkalian titik 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.

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.

=== Definisi menurut aljabar ===
Produk skalar dua vektor {{nowrap|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>, ..., ''A''<sub>''n''</sub>]}} dan {{nowrap|1='''B''' = [''B''<sub>1</sub>, ''B''<sub>2</sub>, ..., ''B''<sub>''n''</sub>]}} didefinisikan sebagai:<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>

di mana Σ melambangkan [[Summation|summation notation]] dan ''n'' adalah dimensi [[ruang vektor]]. Misalnya, dalam [[ruang tiga dimensi]], produk skalar vektor-vektor {{nowrap|[1, 3, −5]}} dan {{nowrap|[4, −2, −1]}} adalah:

:<math>
\begin{align}
\ [1, 3, -5] \cdot [4, -2, -1] &= (1)(4) + (3)(-2) + (-5)(-1) \\
&= 4 - 6 + 5 \\
&= 3.
\end{align}
</math>

=== Definisi menurut geometri ===
Dalam [[Ruang Euklides|ruang Euclidean]], suatu [[vektor (spasial)|vektor Euclidean]] adalah sebuah objek geometri yang memiliki baik besaran (''magnitude'') dan [[arah (geometri)|arah]] (''direction''). Sebuah vektor dapat digambarkan seperti sebuah anak panah. Besarannya adalah panjangnya, sedangkan arahnya adalah yang ditunjuk oleh ujung panah. Besaran vektor '''A''' dilambangkan dengan <math>\|\mathbf{A}\|</math>. Produk skalar dua vektor Euclidean '''A''' dan '''B''' didefinisikan sebagai<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>
di mana θ adalah [[sudut]] di antara '''A''' dan '''B'''.

Secara khusus, jika '''A''' dan '''B''' adalah [[ortogonal]], maka sudut di antara keduanya adalah 90° dan
:<math>\mathbf A\cdot\mathbf B=0.</math>
Pada keadaan ekstrem lain, jika kedua vektor itu mempunyai arah yang sama (''codirectional''), maka sudut di antara keduanya adalah 0° dan
:<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|</math>
Ini menyiratkan bahwa produk skalar suatu vektor '''A''' dengan dirinya sendiri adalah
:<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2,</math>
yang menghasilkan
: <math> \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A},</math>
rumus untuk [[panjang Euclidean]] vektor itu.
<!--
===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>

===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>
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''&nbsp;≠&nbsp;''j'',
:<math>\mathbf e_i\cdot\mathbf e_j = 0.</math>

Also, by the geometric definition, for any vector '''e'''<sub>''i''</sub> and a vector '''A''', we note
:<math>\mathbf A\cdot\mathbf e_i = \|\mathbf A\|\,\|\mathbf e_i\|\cos\theta = \|\mathbf A\|\cos\theta = A_i,</math>
where ''A''<sub>''i''</sub> is the component of vector '''A''' in the direction of '''e'''<sub>''i''</sub>.

Now applying the distributivity of the geometric version of the dot product gives
:<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.
-->

== Sifat ==
Produk skalar memenuhi sifat-sifat berikut jika '''a''', '''b''', dan '''c''' adalah [[vektor (spasial)|vektor]] [[bilangan real|real]] dan ''r'' adalah suatu [[skalar (matematika)|bilangan skalar]].<ref name="Lipschutz2009" /><ref name="Spiegel2009" />

# '''[[Komutatif]]:'''
#: <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>
# '''[[Distributif property|Distributif]] 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>
# '''[[Perkalian skalar]]:'''
#: <math> (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = c_1 c_2 (\mathbf{a} \cdot \mathbf{b}) </math>
# '''[[Ortogonal]]:'''
#: Dua vektor bukan-nol '''a''' dan '''b''' adalah ''[[ortogonal]]'' [[jika dan hanya jika]] {{nowrap|1='''a''' ⋅ '''b''' = 0}}.
# '''Tidak ada [[:en:cancellation law|cancellation]]:'''
#: Berbeda dengan perkalian angka biasa, di mana jika {{nowrap|1=''ab'' = ''ac''}}, maka ''b'' selalu sama dengan ''c'' kecuali ''a'' sama dengan [[nol]], produk skalar tidak menuruti [[cancellation law]]:
#: Jika {{nowrap|1='''a''' ⋅ '''b''' = '''a''' ⋅ '''c'''}} dan {{nowrap|'''a''' ≠ '''0'''}}, maka dapat ditulis: {{nowrap|1='''a''' ⋅ ('''b''' − '''c''') = 0}} dengan [[hukum distributif]]; hasil di atas mengatakan bahwa ini hanya berarti '''a''' tegak lurus dengan {{nowrap|('''b''' − '''c''')}}, di mana masih mengizinkan {{nowrap|('''b''' − '''c''') ≠ '''0'''}}, sehingga {{nowrap|'''b''' ≠ '''c'''}}.
# '''[[Product Rule]]:''' Jika '''a''' dan '''b''' adalah suatu [[fungsi (matematika)|fungsi]], maka [[turunan]] ([[Notation for differentiation#Lagrange's notation|dilambangkan oleh tanda ''prime'']] ′) dari {{nowrap|'''a''' ⋅ '''b'''}} adalah {{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''.]]

{{main|law of cosines}}

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:

:<math>
\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>

which is the [[law of cosines]].
{{clear}}

==Triple product expansion==
{{Main|Triple product}}

This is an 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>

which [[mnemonic|may be remembered]] as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula finds application in simplifying vector calculations in [[physics]].

== Fisika ==
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.
-->
== Generalisasi ==
<!--
===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" />

:<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>

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

[[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 ==
* [[Pertidaksamaan Cauchy–Schwarz]]
* [[Perkalian matriks]]
* [[Perkalian silang]]
* [[Perkalian skalar]]
* [[Perkalian vektor]]

== Referensi ==
{{reflist}}

== Pranala luar ==
* {{springer|title=Inner product|id=p/i051240}}
* {{mathworld|urlname=DotProduct|title=Dot product}}
* [http://www.mathreference.com/la,dot.html Explanation of dot product including with complex vectors]
* [http://demonstrations.wolfram.com/DotProduct/ "Dot Product"] by Bruce Torrence, [[Wolfram Demonstrations Project]], 2007.

{{linear algebra}}

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