Distance between two parallel lines: Difference between revisions

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{{Short description|Problem in coordinate geometry}}

{{unreferenced|date=April 2013}}

{{redirect-distinguish|Distance between two lines|Distance between two skew lines}}

The '''[[distance]] between two [[Parallel (geometry)|parallel]] [[Line (geometry)|lines]]''' in the [[plane (geometry)|plane]] is the minimum distance between any two points.

:''This article considers two lines in a plane. For two lines not in the same plane, see [[Skew lines#Distance]].''

== Formula and proof ==

The '''[[distance]] between two [[Line (geometry)|straight lines]]''' in the [[plane (geometry)|plane]] is the minimum distance between any two points lying on the lines. In case of intersecting lines, the distance between them is zero, because the minimum distance between them is zero (at the point of intersection); whereas in case of two [[Parallel (geometry)|parallel]] lines, it is the [[perpendicular]] distance from a [[Point (geometry)|point]] on one line to the other line.

Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines

:<math>y = mx+b_1\,</math>

==See also==

:<math>y = mx+b_2\,,</math>

the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line

:<math>y = -x/m \, .</math>

This distance can be found by first solving the [[linear systems]]

:<math>\begin{cases}

y = mx+b_1 \\

y = -x/m \, ,

\end{cases}</math>

and

:<math>\begin{cases}

y = mx+b_2 \\

y = -x/m \, ,

\end{cases}</math>

to get the coordinates of the intersection points. The solutions to the linear systems are the points

:<math>\left( x_1,y_1 \right)\ = \left( \frac{-b_1m}{m^2+1},\frac{b_1}{m^2+1} \right)\, ,</math>

and

:<math>\left( x_2,y_2 \right)\ = \left( \frac{-b_2m}{m^2+1},\frac{b_2}{m^2+1} \right)\, .</math>

The distance between the points is

:<math>d = \sqrt{\left(\frac{b_1m-b_2m}{m^2+1}\right)^2 + \left(\frac{b_2-b_1}{m^2+1}\right)^2}\,,</math>

which reduces to

:<math>d = \frac{|b_2-b_1|}{\sqrt{m^2+1}}\,.</math>

When the lines are given by

:<math>ax+by+c_1=0\,</math>

:<math>ax+by+c_2=0,\,</math>

the distance between them can be expressed as

:<math>d = \frac{|c_2-c_1|}{\sqrt {a^2+b^2}}.</math>

==See also==

*[[Distance from a point to a line]]

*[[Skew lines#Distance]]

==References==

*''Abstand'' In: ''Schülerduden – Mathematik II''. Bibliographisches Institut & F. A. Brockhaus, 2004, {{ISBN|3-411-04275-3}}, pp. 17-19 (German)

*Hardt Krämer, Rolf Höwelmann, Ingo Klemisch: ''Analytische Geometrie und Lineare Akgebra''. Diesterweg, 1988, {{ISBN|3-425-05301-9}}, p. 298 (German)

==External links ==

*Florian Modler: [http://www.emath.de/Referate/Zusammenfassung-wichtiger-Formeln.pdf ''Vektorprodukte, Abstandsaufgaben, Lagebeziehungen, Winkelberechnung – Wann welche Formel?''], pp. 44-59 (German)

*A. J. Hobson: [https://archive.uea.ac.uk/jtm/8/Lec8p5.pdf ''“JUST THE MATHS” - UNIT NUMBER 8.5 - VECTORS 5 (Vector equations of straight lines)''], pp. 8-9

[[Category:Euclidean geometry]]

*[[Skew lines#Category:Distance]]