# line in plane

## Equation of a line

Suppose $a,b,c\in\mathbbmss{R}$. 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 parallel to the $y$-axis. The $x$-intercept is defined similarly.

If $b\neq 0$, 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 $\mathbbmss{R}^{2}$ are collinear if and only if the following determinant vanishes:

 $\left|\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\ y_{1}&y_{2}&y_{3}\\ 1&1&1\end{array}\right|=0.$

Therefore, the line through distinct points $(x_{1},\,y_{1})$ and $(x_{2},\,y_{2})$ has equation

 $\left|\begin{array}[]{ccc}x_{1}&x_{2}&x\\ y_{1}&y_{2}&y\\ 1&1&1\end{array}\right|=0,$

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 $\mathbbmss{R}^{2}$. The closed line segement generated by these points is the set

 $\{p\in\mathbbmss{R}^{2}\mid p=tp_{1}+(1-t)p_{2},\;0\leq t\leq 1\}.$
 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