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''scalar product, dot product, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512401.png" /> of two non-zero vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512403.png" />''
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''scalar product, dot product, $\def\vect#1{\mathbf{#1}}\def\modulus#1{\left|#1\right|}(\vect a,\vect b)$ of two non-zero vectors $\vect a$ and $\vect b$''
  
The product of their lengths and the cosine of the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512404.png" /> between them:
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The product of their lengths and the cosine of the angle $\phi$ between them:
 
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\[(\vect a,\vect b)=\modulus{\vect a}\modulus{\vect b}\cos\phi.\]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512405.png" /></td> </tr></table>
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$\phi$ is taken to be that angle between the vectors not exceeding $\pi$. When $\vect a$ or $\vect b$ is zero, the inner product is taken to be zero. The inner product $(\vect a,\vect a)=\vect a^2=\modulus{\vect a}^2$ is called the scalar square of the vector $\vect a$ (see [[Vector algebra]]).
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512406.png" /> is taken to be that angle between the vectors not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512407.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512408.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i0512409.png" /> is zero, the inner product is taken to be zero. The inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124010.png" /> is called the scalar square of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124011.png" /> (see [[Vector algebra|Vector algebra]]).
 
 
 
The inner product of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124012.png" />-dimensional vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124014.png" /> over the real numbers is given by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124015.png" /></td> </tr></table>
 
  
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The inner product of two $n$-dimensional vectors $\vect a=(a_1,\dotsc,a_n)$ and $\vect b=(b_1,\dotsc,b_n)$ over the real numbers is given by
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\[(\vect a,\vect b)=a_1b_1+\dotsb+a_nb_n.\]
 
In the complex case it is given by
 
In the complex case it is given by
 
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\[(\vect a,\vect b)=a_1\bar b_1+\dotsb+a_n\bar b_n\]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124016.png" /></td> </tr></table>
 
 
 
 
An infinite-dimensional [[Vector space|vector space]] admitting an inner product and complete with respect to it is called a [[Hilbert space|Hilbert space]].
 
An infinite-dimensional [[Vector space|vector space]] admitting an inner product and complete with respect to it is called a [[Hilbert space|Hilbert space]].
  
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====Comments====
 
====Comments====
More generally, an inner product on a real vector space is a symmetric [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124017.png" /> which is positive definite, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124019.png" />. A (unitary) inner product on a complex vector space is likewise defined as a Hermitian (i.e., with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124020.png" />) [[Sesquilinear form|sesquilinear form]], with complex conjugation as automorphism, which is positive definite. In finite-dimensional spaces one can always find an orthonormal basis in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124021.png" /> takes the standard form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124022.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051240/i05124023.png" />.
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More generally, an inner product on a real vector space is a symmetric [[Bilinear form|bilinear form]] $f$ which is positive definite, i.e., $f(x,x)>0$ for all $x\neq 0$. A (unitary) inner product on a complex vector space is likewise defined as a Hermitian (i.e., with $f(y,x)=\overline{f(x,y)}$) [[Sesquilinear form|sesquilinear form]], with complex conjugation as automorphism, which is positive definite. In finite-dimensional spaces one can always find an orthonormal basis in which $f$ takes the standard form $f(x,y)=\sum_{i=1}^n x_iy_i$, respectively $\sum_{i=1}^n x_i\bar y_i$.
  
 
Besides the inner product (which can be defined in arbitrary dimensions), in three-dimensional space one also has the [[Vector product|vector product]].
 
Besides the inner product (which can be defined in arbitrary dimensions), in three-dimensional space one also has the [[Vector product|vector product]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrăţescu,  "Inner product structures" , Reidel  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrăţescu,  "Inner product structures" , Reidel  (1987)</TD></TR></table>
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Latest revision as of 04:43, 22 March 2013

scalar product, dot product, $\def\vect#1{\mathbf{#1}}\def\modulus#1{\left|#1\right|}(\vect a,\vect b)$ of two non-zero vectors $\vect a$ and $\vect b$

The product of their lengths and the cosine of the angle $\phi$ between them: \[(\vect a,\vect b)=\modulus{\vect a}\modulus{\vect b}\cos\phi.\] $\phi$ is taken to be that angle between the vectors not exceeding $\pi$. When $\vect a$ or $\vect b$ is zero, the inner product is taken to be zero. The inner product $(\vect a,\vect a)=\vect a^2=\modulus{\vect a}^2$ is called the scalar square of the vector $\vect a$ (see Vector algebra).

The inner product of two $n$-dimensional vectors $\vect a=(a_1,\dotsc,a_n)$ and $\vect b=(b_1,\dotsc,b_n)$ over the real numbers is given by \[(\vect a,\vect b)=a_1b_1+\dotsb+a_nb_n.\] In the complex case it is given by \[(\vect a,\vect b)=a_1\bar b_1+\dotsb+a_n\bar b_n\] An infinite-dimensional vector space admitting an inner product and complete with respect to it is called a Hilbert space.


Comments

More generally, an inner product on a real vector space is a symmetric bilinear form $f$ which is positive definite, i.e., $f(x,x)>0$ for all $x\neq 0$. A (unitary) inner product on a complex vector space is likewise defined as a Hermitian (i.e., with $f(y,x)=\overline{f(x,y)}$) sesquilinear form, with complex conjugation as automorphism, which is positive definite. In finite-dimensional spaces one can always find an orthonormal basis in which $f$ takes the standard form $f(x,y)=\sum_{i=1}^n x_iy_i$, respectively $\sum_{i=1}^n x_i\bar y_i$.

Besides the inner product (which can be defined in arbitrary dimensions), in three-dimensional space one also has the vector product.

References

[a1] V.I. Istrăţescu, "Inner product structures" , Reidel (1987)
How to Cite This Entry:
Inner product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inner_product&oldid=15855
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article