The
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between two
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
lines in the
plane is the minimum distance between any two points.
Formula and proof
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
:
:
the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line
:
This distance can be found by first solving the linear systems
:
and
:
to get the coordinates of the intersection points. The solutions to the linear systems are the points
:
and
:
The distance between the points is
:
which reduces to
:
When the lines are given by
:
:
the distance between them can be expressed as
:
See also
*
Distance from a point to a line
References
*''Abstand'' In: ''Schülerduden – Mathematik II''. Bibliographisches Institut & F. A. Brockhaus, 2004, , 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
''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
Euclidean geometry
Distance