line in plane

Equation of a line

Suppose a,b,c. Then the set of points (x,y) in the plane that satisfy

ax+by+c= 0,

where a and b can not be both 0, is an (infinite) line.

The value of y when x=0, if it exists, is called the y-intercept. Geometrically, if d is the y-intercept, then (0,d) is the point of intersection of the line and the y-axis. The y-intercept exists iff the line is not parallelMathworldPlanetmathPlanetmathPlanetmath to the y-axis. The x-intercept is defined similarly.

If b0, then the above equation of the line can be rewritten as


This is called the slope-intercept formMathworldPlanetmath of a line, because both the slope and the y-intercept are easily identifiable in the equation. The slope is m and the y-intercept is d.

Three finite points (x1,y1), (x2,y2), (x3,y3) in 2 are collinearMathworldPlanetmath if and only if the following determinantMathworldPlanetmath vanishes:


Therefore, the line through distinct points (x1,y1) and (x2,y2) has equation


or more simply


Line segment

Let p1=(x1,y1) and p2=(x2,y2) be distinct points in 2. The closed line segement generated by these points is the set

{p2p=tp1+(1-t)p2, 0t1}.
Title line in plane
Canonical name LineInPlane
Date of creation 2013-03-22 15:18:29
Last modified on 2013-03-22 15:18:29
Owner matte (1858)
Last modified by matte (1858)
Numerical id 17
Author matte (1858)
Entry type Definition
Classification msc 53A04
Classification msc 51N20
Synonym y-intercept
Synonym x-intercept
Related topic LineSegment
Related topic SlopeAngle
Related topic LineInSpace
Related topic Slope
Related topic AnalyticGeometry
Related topic FanOfLines
Related topic PencilOfConics
Defines y-intercept
Defines x-intercept
Defines slope-intercept form