line in plane

Equation of a line

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

$$ax+by+c=\mathrm{\hspace{0.33em}0},$$

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 parallel to the $y$-axis. The $x$-intercept is defined similarly.

If $b\ne 0$, then the above equation of the line can be rewritten as

$$y=mx+d.$$

This is called the slope-intercept form 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 $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ in ${ℝ}^2$ are collinear if and only if the following determinant vanishes:

Therefore, the line through distinct points $(x_1,y_1)$ and $(x_2,y_2)$ has equation

or more simply

$$(y_1-y_2)x+(x_2-x_1)y+y_2{x}_1-x_2{y}_1=0.$$

Line segment

Let $p_1=(x_1,y_1)$ and $p_2=(x_2,y_2)$ be distinct points in ${ℝ}^2$. The closed line segement generated by these points is the set

$$\{p\in {ℝ}^2\mid p=t{p}_1+(1-t)p_2,\mathrm{\hspace{0.33em}0}\le t\le 1\}.$$

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