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




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


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


has the sought   d=|(a-b)n|,  i.e.


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.

Title distance of non-parallel lines
Canonical name DistanceOfNonparallelLines
Date of creation 2013-03-22 15:27:16
Last modified on 2013-03-22 15:27:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Derivation
Classification msc 15A72
Synonym distance of lines
Related topic LineInSpace
Related topic DistanceFromPointToALine
Related topic EuclideanDistance
Related topic AngleBetweenTwoLines
Defines agonic lines
Defines skew lines