3.7 KB · ⇊ 3 parts · ⤓ Micron 3.7 KB · ⤓ HTML 31.1 KB

top

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

• Formula and proof

• See also

• References

• External links

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 = m x + b 1 {\displaystyle y=mx+b_{1}\,}

y = m x + b 2 , {\displaystyle 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 . {\displaystyle y=-x/m\,.}

This distance can be found by first solving the linear systems

{ y = m x + b 1 y = − − x / m , {\displaystyle {\begin{cases}y=mx+b_{1}\\y=-x/m\,,\end{cases}}}

and

{ y = m x + b 2 y = − − x / m , {\displaystyle {\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

( x 1 , y 1 ) = ( − − b 1 m m 2 + 1 , b 1 m 2 + 1 ) , {\displaystyle \left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,,}

and

( x 2 , y 2 ) = ( − − b 2 m m 2 + 1 , b 2 m 2 + 1 ) . {\displaystyle \left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right)\,.}

The distance between the points is

d = ( b 1 m − − b 2 m m 2 + 1 ) 2 + ( b 2 − − b 1 m 2 + 1 ) 2 , {\displaystyle 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 = | b 2 − − b 1 | m 2 + 1 . {\displaystyle d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.}

When the lines are given by

a x + b y + c 1 = 0 {\displaystyle ax+by+c_{1}=0\,}

a x + b y + c 2 = 0 , {\displaystyle ax+by+c_{2}=0,\,}

the distance between them can be expressed as

d = | c 2 − − c 1 | a 2 + b 2 . {\displaystyle d={\frac {|c_{2}-c_{1}|}{\sqrt {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 Akgebra. 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