Distance between two parallel lines: Difference between revisions

Revision as of 03:13, 29 April 2013

Latest revision as of 19:44, 14 August 2023

(46 intermediate revisions by 28 users not shown)

Line 1:

{{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 [[Line (geometry)|straight lines]]''' in the 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.

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

== Formula and proof ==

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>

:<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 26 ⟶ 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 50 ⟶ 52:

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

==See also==

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

==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]]

[[Category:Distance]]