Distance between two parallel lines: Difference between revisions

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{{Short description|Problem in coordinate geometry}}

'''[[Distance]]''' between two [[Parallel (geometry)|parallel]] [[Line (geometry)|lines]] is the distance from a [[Point (geometry)|point]] of the first line to the another line.

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

The '''[[Distancedistance]]''' between two [[Parallel (geometry)|parallel]] [[Line (geometry)|lines]]''' isin the distance from a [[Pointplane (geometry)|pointplane]] ofis the firstminimum linedistance tobetween theany anothertwo linepoints.

== Formula and proof ==

Because athe lines are parallel, linethe isperpendicular adistance linebetween thatthem hasis ana equalconstant, distanceso withit thedoes oppositenot line,matter therewhich point is achosen uniqueto distance betweenmeasure the two parallel linesdistance. 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 can be found by solving the linear systems

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>

theThis distance between the two lines 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>

:<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>

theirthe 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]]

==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]]