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The distance between two parallel lines in the plane is the minimum distance between any two points.

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

$y=mx+b_1$

$y=mx+b_2,$

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

$y=-x/m.$

This distance can be found by first solving the system of linear equations

$\{\begin{array}{c}y=mx+b_1\\ y=-x/m,\end{array}$

and

$\{\begin{array}{c}y=mx+b_2\\ y=-x/m,\end{array}$

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

$(x_1,y_1)=(\frac{-b_{1}m}{m^2+1},\frac{b_1}{m^2+1}),$

and

$(x_2,y_2)=(\frac{-b_{2}m}{m^2+1},\frac{b_2}{m^2+1}).$

The distance between the points is

$d=\sqrt{{\left(\frac{b_{1}m-b_{2}m}{m^2+1}\right)}^2+{\left(\frac{b_2-b_1}{m^2+1}\right)}^2},$

which reduces to

$d=\frac{|b_2-b_1|}{\sqrt{m^2+1}}.$

When the lines are given by

$ax+by+c_1=0$

$ax+by+c_2=0,$

the distance between them can be expressed as

$d=\frac{|c_2-c_1|}{\sqrt{a^2+b^2}}.$

More generally, when the coefficients of $x$ and $y$ are different, i.e. the parallel lines are given by the $a_{1}x+b_{1}y+c_1=0$ and the $a_{2}x+b_{2}y+c_2=0$ equations where $(a_1,b_1)\left|\right|(a_2,b_2)$, the distance can be expressed as

$d=\frac{\sqrt{(a_1{c}_2-a_2{c}_1{)}^2+(b_1{c}_2-b_2{c}_1{)}^2}}{|a_1{b}_1+a_2{b}_2|}$

• 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 Algebra. Diesterweg, 1988, ISBN 3-425-05301-9, p. 298 (German)

• Florian Modler: Vektorprodukte, Abstandsaufgaben, Lagebeziehungen, Winkelberechnung – Wann welche Formel?, pp. 44-59 (German)
• A. J. Hobson: “JUST THE MATHS” - UNIT NUMBER 8.5 - VECTORS 5 (Vector equations of straight lines), pp. 8-9