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Difference between revisions of "Normal equation"

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''normalized equation''
 
''normalized equation''
  
 
The equation of a line in a plane of the form
 
The equation of a line in a plane of the form
  
<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/n/n067/n067490/n0674901.png" /></td> </tr></table>
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$$x\cos\alpha+y\sin\alpha-p=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674903.png" /> are rectangular Cartesian coordinates in the plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674905.png" /> are the coordinates of the unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674906.png" /> perpendicular to the line, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674907.png" /> is the distance of the coordinate origin from the line. An equation of a line of the form
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where $x$ and $y$ are rectangular Cartesian coordinates in the plane, $\cos\alpha$ and $\sin\alpha$ are the coordinates of the unit vector $\{\cos\alpha,\sin\alpha\}$ perpendicular to the line, and $p\geq0$ is the distance of the coordinate origin from the line. An equation of a line of the form
  
<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/n/n067/n067490/n0674908.png" /></td> </tr></table>
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$$Ax+By+C=0$$
  
reduces to normal form after multiplication by the normalizing factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n0674909.png" /> of absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749010.png" /> and of sign opposite to that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749011.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749012.png" />, the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749013.png" /> is arbitrary).
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reduces to normal form after multiplication by the normalizing factor $\lambda$ of absolute value $(A^2+B^2)^{-1/2}$ and of sign opposite to that of $C$ (if $C=0$, the sign of $\lambda$ is arbitrary).
  
 
Similarly, the equation of a plane
 
Similarly, the equation of a plane
  
<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/n/n067/n067490/n06749014.png" /></td> </tr></table>
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$$Ax+By+Cz+D=0$$
  
 
reduces to the normal form
 
reduces to the normal form
  
<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/n/n067/n067490/n06749015.png" /></td> </tr></table>
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$$x\cos\alpha+y\cos\beta+z\cos\gamma-p=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749018.png" /> are the direction cosines of a vector perpendicular to the plane, after multiplication by the normalizing factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749019.png" /> of absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749020.png" /> and of sign opposite to that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067490/n06749021.png" />.
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where $\cos\alpha$, $\cos\beta$ and $\cos\gamma$ are the direction cosines of a vector perpendicular to the plane, after multiplication by the normalizing factor $\lambda$ of absolute value $(A^2+B^2+C^2)^{-1/2}$ and of sign opposite to that of $D$.

Latest revision as of 07:50, 15 July 2014

normalized equation

The equation of a line in a plane of the form

$$x\cos\alpha+y\sin\alpha-p=0,$$

where $x$ and $y$ are rectangular Cartesian coordinates in the plane, $\cos\alpha$ and $\sin\alpha$ are the coordinates of the unit vector $\{\cos\alpha,\sin\alpha\}$ perpendicular to the line, and $p\geq0$ is the distance of the coordinate origin from the line. An equation of a line of the form

$$Ax+By+C=0$$

reduces to normal form after multiplication by the normalizing factor $\lambda$ of absolute value $(A^2+B^2)^{-1/2}$ and of sign opposite to that of $C$ (if $C=0$, the sign of $\lambda$ is arbitrary).

Similarly, the equation of a plane

$$Ax+By+Cz+D=0$$

reduces to the normal form

$$x\cos\alpha+y\cos\beta+z\cos\gamma-p=0,$$

where $\cos\alpha$, $\cos\beta$ and $\cos\gamma$ are the direction cosines of a vector perpendicular to the plane, after multiplication by the normalizing factor $\lambda$ of absolute value $(A^2+B^2+C^2)^{-1/2}$ and of sign opposite to that of $D$.

How to Cite This Entry:
Normal equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_equation&oldid=17334
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article