Distance between two straight lines

From Wikipedia, the free encyclopedia
This article considers two lines in a plane. For two lines not in the same plane, see Skew lines#Distance.

The distance between two straight lines in the plane is the minimum distance between any two points lying on the lines. In case of intersecting lines, the distance between them is zero, because the minimum distance between them is zero (at the point of intersection); whereas in case of two parallel lines, it is the perpendicular distance from a point on one line to the other line.

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

Retrieved from "https://en.wikipedia.org/w/index.php?title=Distance_between_two_straight_lines&oldid=803601183"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Distance_between_two_straight_lines
This page is based on the copyrighted Wikipedia article "Distance between two straight lines"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA