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 [[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.

== 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]]