Distance between two parallel lines: Difference between revisions

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{{Short description|Problem in coordinate geometry}}

{{multiple issues|

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

{{Orphan|date=September 2014}}

{{unreferenced|date=April 2013}}

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The '''[[distance]] between two [[Parallel (geometry)|parallel]] [[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 [[Parallelplane (geometry)|parallelplane]] lines, it is the perpendicularminimum distance frombetween aany [[Pointtwo (geometry)|point]] on one line to the other linepoints.

== Formula and proof ==

Line 12 ⟶ 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 30 ⟶ 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 54 ⟶ 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]]