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Two straight lines in space that do not lie in a plane. The angle between two skew lines is defined as either of the angles between any two lines parallel to them and passing through a point of space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085690/s0856901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085690/s0856902.png" /> are the direction vectors of two skew lines, then the cosine of the angle between them is given by
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<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/s/s085/s085690/s0856903.png" /></td> </tr></table>
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The common perpendicular of two skew lines is the line intersecting both of them at right angles. Any two skew lines have a unique common perpendicular. The equation (as the line of intersection of two planes) of this common perpendicular to the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085690/s0856904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085690/s0856905.png" /> has the form
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Two straight lines in space that do not lie in a plane. The angle between two skew lines is defined as either of the angles between any two lines parallel to them and passing through a point of space. If  $  \mathbf a $
 +
and  $  \mathbf b $
 +
are the direction vectors of two skew lines, then the cosine of the angle between them 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/s/s085/s085690/s0856906.png" /></td> </tr></table>
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$$
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\cos  \phi _ {12}  = \pm 
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\frac{( \mathbf a , \mathbf b ) }{|
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\mathbf a | \cdot | \mathbf b | }
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.
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$$
  
<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/s/s085/s085690/s0856907.png" /></td> </tr></table>
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The common perpendicular of two skew lines is the line intersecting both of them at right angles. Any two skew lines have a unique common perpendicular. The equation (as the line of intersection of two planes) of this common perpendicular to the lines  $  \mathbf r = \mathbf r _ {1} + \mathbf a t _ {1} $
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and  $  \mathbf r = \mathbf r _ {2} + \mathbf b t _ {2} $
 +
has the form
  
The distance between two skew lines is the length of the segment of their common perpendicular whose end points lie on the lines (or, the distance between the two parallel planes containing the two lines). The distance between two skew lines is given by
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$$
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(( \mathbf r - \mathbf r _ {1} ) , \mathbf a , [ \mathbf a , \mathbf b ]) =  0,
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$$
  
<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/s/s085/s085690/s0856908.png" /></td> </tr></table>
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$$
 +
(( \mathbf r - \mathbf r _ {2} ) , \mathbf b , [ \mathbf a , \mathbf b ])  = 0 .
 +
$$
  
 +
The distance between two skew lines is the length of the segment of their common perpendicular whose end points lie on the lines (or, the distance between the two parallel planes containing the two lines). The distance between two skew lines is given by
  
 +
$$
 +
d  = 
 +
\frac{| (( \mathbf r _ {1} - \mathbf r _ {2} ) , \mathbf a ,\
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\mathbf b ) | }{| [ \mathbf a , \mathbf b ] | }
 +
.
 +
$$
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Pedoe,  "Geometry. A comprehensive introduction" , Dover, reprint  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Pedoe,  "Geometry. A comprehensive introduction" , Dover, reprint  (1988)</TD></TR></table>

Revision as of 08:14, 6 June 2020


Two straight lines in space that do not lie in a plane. The angle between two skew lines is defined as either of the angles between any two lines parallel to them and passing through a point of space. If $ \mathbf a $ and $ \mathbf b $ are the direction vectors of two skew lines, then the cosine of the angle between them is given by

$$ \cos \phi _ {12} = \pm \frac{( \mathbf a , \mathbf b ) }{| \mathbf a | \cdot | \mathbf b | } . $$

The common perpendicular of two skew lines is the line intersecting both of them at right angles. Any two skew lines have a unique common perpendicular. The equation (as the line of intersection of two planes) of this common perpendicular to the lines $ \mathbf r = \mathbf r _ {1} + \mathbf a t _ {1} $ and $ \mathbf r = \mathbf r _ {2} + \mathbf b t _ {2} $ has the form

$$ (( \mathbf r - \mathbf r _ {1} ) , \mathbf a , [ \mathbf a , \mathbf b ]) = 0, $$

$$ (( \mathbf r - \mathbf r _ {2} ) , \mathbf b , [ \mathbf a , \mathbf b ]) = 0 . $$

The distance between two skew lines is the length of the segment of their common perpendicular whose end points lie on the lines (or, the distance between the two parallel planes containing the two lines). The distance between two skew lines is given by

$$ d = \frac{| (( \mathbf r _ {1} - \mathbf r _ {2} ) , \mathbf a ,\ \mathbf b ) | }{| [ \mathbf a , \mathbf b ] | } . $$

Comments

References

[a1] D. Pedoe, "Geometry. A comprehensive introduction" , Dover, reprint (1988)
How to Cite This Entry:
Skew lines. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew_lines&oldid=48727
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article