PlanetMath: line in plane

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line in plane

(Definition)

Equation of a line

Suppose $a,b,c\in \mathbbmss{R}$ . Then the set of points

in the plane that satisfy

$\displaystyle ax + by = c,$

where

and

can not be both 0, is an (infinite) line.

The value of when , if it exists, is called the -intercept. Geometrically, if is the -intercept, then is the point of intersection of the line and the -axis. The -intercept exists iff the line is not parallel to the -axis. The -intercept is defined similarly.

If $b\neq0$ , then the above equation of the line can be rewritten as

$\displaystyle y = mx + d.$

This is called the slope-intercept form of a line, because both the slope and the

-intercept are easily identifiable in the equation. The slope is

and the

-intercept is

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:

$\displaystyle \left\vert \begin{array}{ccc} x_1 & x_2 &x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1& 1\end{array} \right\vert=0.$

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

$\displaystyle \left\vert \begin{array}{ccc} x_1 & x_2 &x \\ y_1 & y_2 & y \\ 1 & 1& 1\end{array} \right\vert=0,$

or more simply

$\displaystyle (y_1-y_2)x+(x_2 - x_1)y + y_2 x_1-x_2 y_1=0.$

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

$\displaystyle \{ p\in \mathbbmss{R}^2 \mid p=t p_1+(1-t) p_2,\; 0\leq t\leq 1\}.$

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"line in plane" is owned by matte. [ full author list (4) ]

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Other names:

y-intercept, x-intercept

Also defines:

-intercept, $x$

-intercept, slope-intercept form

This object's parent.

Attachments:: line through an intersection point (Topic) by pahio

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Cross-references: generated by, closed, vanishes, determinant, collinear, finite, slope, equation, parallel, iff, intersection, line, infinite, plane, points
There are 5 references to this entry.

This is version 13 of line in plane, born on 2005-05-23, modified 2007-09-14.
Object id is 7108, canonical name is LineInThePlane.
Accessed 6345 times total.

Classification:

AMS MSC:	51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
	53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

None.

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