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''of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965001.png" /> by a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965002.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965003.png" />''
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{{TEX|done}}
  
The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965004.png" />, denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965005.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965006.png" />, satisfying the following requirements:
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''of a vector $a$ by a vector $b$ in $\mathbb{R}^3$''
  
1) the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965007.png" /> is equal to the product of the lengths of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v0965009.png" /> by the sine of the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650010.png" /> between them, i.e.
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The [[vector]] $c$, denoted by the symbol $a\times b$ or $[a,b]$, satisfying the following requirements:
  
<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/v/v096/v096500/v09650011.png" /></td> </tr></table>
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# the length of $c$ is equal to the product of the lengths of the vectors $a$ and $b$ by the [[sine]] of the angle $\phi$ between them, i.e. \begin{equation} |c| = |a\times b| = |a|\cdot |b| \sin\phi; \end{equation}
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# $c$ is orthogonal to both $a$ and $b$;
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# the orientation of the vector triple $a,b,c$ is the same as that of the (standard) triple of basis vectors. See [[Vector algebra]].
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650012.png" /> is orthogonal to both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650014.png" />;
 
  
3) the orientation of the vector triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650015.png" /> is the same as that of the (standard) triple of basis vectors. See [[Vector algebra|Vector algebra]].
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====Comments====
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Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ have coordinates with respect to an orthonormal basis in $\mathbb{R}^3$, then the coordinates of $c=a\times b$ are \begin{equation}c=\begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1\end{pmatrix}.\end{equation}
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The vector product is sometimes called cross product <ref name="Matrix Computations" />, also cf. [[cross product]].
  
  
  
====Comments====
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====References====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650016.png" /> have coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650018.png" /> with respect to an orthonormal basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650019.png" />, then the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650020.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096500/v09650021.png" />.
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<references>
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<ref name="Matrix Computations">Gene H. Golub, Charles F. Van Loan, ''Matrix Computations'', Johns Hopkins Studies in the Mathematical Sciences '''3''', JHU Press (2013) {{ISBN|1421407949}}, p. 70.</ref>
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</references>

Latest revision as of 05:53, 15 April 2023


of a vector $a$ by a vector $b$ in $\mathbb{R}^3$

The vector $c$, denoted by the symbol $a\times b$ or $[a,b]$, satisfying the following requirements:

  1. the length of $c$ is equal to the product of the lengths of the vectors $a$ and $b$ by the sine of the angle $\phi$ between them, i.e. \begin{equation} |c| = |a\times b| = |a|\cdot |b| \sin\phi; \end{equation}
  2. $c$ is orthogonal to both $a$ and $b$;
  3. the orientation of the vector triple $a,b,c$ is the same as that of the (standard) triple of basis vectors. See Vector algebra.


Comments

Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ have coordinates with respect to an orthonormal basis in $\mathbb{R}^3$, then the coordinates of $c=a\times b$ are \begin{equation}c=\begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1\end{pmatrix}.\end{equation}

The vector product is sometimes called cross product [1], also cf. cross product.


References

  1. Gene H. Golub, Charles F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences 3, JHU Press (2013) ISBN 1421407949, p. 70.
How to Cite This Entry:
Vector product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_product&oldid=13392
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article