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]].''

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

== Formula and proof ==

Line 11 ⟶ 10:

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

the distance between the two lines is the distance between the two interceptsintersection 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}

Line 29 ⟶ 28:

\end{cases}</math>

to get the coordinates of the interceptintersection 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>

Line 55 ⟶ 54:

==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]]