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单词 DistanceOfNonparallelLines
释义

distance of non-parallel lines


As an application of the vector product (http://planetmath.org/CrossProduct) we derive the expression of the d between two non-parallel lines in 3.

Suppose that the position vectors of the points of the two non-parallel lines are expressed in parametric forms

r=a+su

and

r=b+tv,

where s and t are parameters.  A common vector of the lines is the cross product u×v of the direction vectors of the lines, and it may be normed to a unit vectorMathworldPlanetmath

n:=u×v|u×v|

by dividing it by its , which is distinct from 0 because of the non-parallelity.  The vectors a and b are the position vectors of certain points A and B on the lines, and thus their difference a-b is the vector from B to A.  If we project a-b on the unit normal n, the obtained vector

d:=[(a-b)n]n

has the sought  d=|(a-b)n|,  i.e.

d=|(a-b)(u×v)||u×v|.

For illustrating that d is the minimal distance between points of the two lines we underline, that the construction of d guarantees that it connects two points on the lines and is perpendicularMathworldPlanetmathPlanetmathPlanetmath to both lines — thus any displacement of its end pointPlanetmathPlanetmath makes it longer.

Notes.  The numerator is the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a triple scalar product.  If the lines intersect each other, then the connecting vector a-b is at right anglesMathworldPlanetmathPlanetmath to the common normal vector n of their plane and thus the dot productMathworldPlanetmath of these vectors vanishes, i.e. also  d=0.  If the lines do not intersect, they are called agonic linesMathworldPlanetmath or skew lines;  then  d>0.

Titledistance of non-parallel lines
Canonical nameDistanceOfNonparallelLines
Date of creation2013-03-22 15:27:16
Last modified on2013-03-22 15:27:16
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id15
Authorpahio (2872)
Entry typeDerivation
Classificationmsc 15A72
Synonymdistance of lines
Related topicLineInSpace
Related topicDistanceFromPointToALine
Related topicEuclideanDistance
Related topicAngleBetweenTwoLines
Definesagonic lines
Definesskew lines
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更新时间:2025/5/4 14:50:31