Distance between two parallel lines

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

:$y\; =\; mx+b\_1\backslash ,$

:$y\; =\; mx+b\_2\backslash ,,$

the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line

:$y\; =\; -x/m\; \backslash ,\; .$

This distance can be found by first solving the linear systems

:$\backslash begin\{cases\}$

y = mx+b_1 \\

y = -x/m \, ,

\end{cases}

and

:$\backslash begin\{cases\}$

y = mx+b_2 \\

y = -x/m \, ,

\end{cases}

to get the coordinates of the intersection points. The solutions to the linear systems are the points

:$\backslash left(\; x\_1,y\_1\; \backslash right)\backslash \; =\; \backslash left(\; \backslash frac\{-b\_1m\}\{m^2+1\},\backslash frac\{b\_1\}\{m^2+1\}\; \backslash right)\backslash ,\; ,$

and

:$\backslash left(\; x\_2,y\_2\; \backslash right)\backslash \; =\; \backslash left(\; \backslash frac\{-b\_2m\}\{m^2+1\},\backslash frac\{b\_2\}\{m^2+1\}\; \backslash right)\backslash ,\; .$

The distance between the points is

:$d\; =\; \backslash sqrt\{\backslash left(\backslash frac\{b\_1m-b\_2m\}\{m^2+1\}\backslash right)^2\; +\; \backslash left(\backslash frac\{b\_2-b\_1\}\{m^2+1\}\backslash right)^2\}\backslash ,,$

which reduces to

:$d\; =\; \backslash frac$

When the lines are given by

:$ax+by+c\_1=0\backslash ,$

:$ax+by+c\_2=0,\backslash ,$

the distance between them can be expressed as

:$d\; =\; \backslash frac$

• Distance from a point to a line

• 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)

• 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