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单词 LineInPlane
释义

line in plane


Equation of a line

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

y=mx+d.

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:

|x1x2x3y1y2y3111|=0.

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

|x1x2xy1y2y111|=0,

or more simply

(y1-y2)x+(x2-x1)y+y2x1-x2y1=0.

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}.
Titleline in plane
Canonical nameLineInPlane
Date of creation2013-03-22 15:18:29
Last modified on2013-03-22 15:18:29
Ownermatte (1858)
Last modified bymatte (1858)
Numerical id17
Authormatte (1858)
Entry typeDefinition
Classificationmsc 53A04
Classificationmsc 51N20
Synonymy-intercept
Synonymx-intercept
Related topicLineSegment
Related topicSlopeAngle
Related topicLineInSpace
Related topicSlope
Related topicAnalyticGeometry
Related topicFanOfLines
Related topicPencilOfConics
Definesy-intercept
Definesx-intercept
Definesslope-intercept form
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更新时间:2025/5/5 1:17:40