“Point-Line Distance--2-D”的概念、定义、习题解题方法及翻译-数学辞典

单词

Point-Line Distance--2-D

释义

Point-Line Distance--2-D

Given a line and a point ), in slope-intercept form, the equationof the line is

(1)

so the line has Slope

. Points on the line have the vector coordinates

(2)

Therefore, the Vector

(3)

is Parallel to the line, and the Vector

(4)

is Perpendicular to it. Now, a Vector from the point to the line is given by

(5)

Projecting

onto



			(6)

If the line is represented by the endpoints of a Vector

and

, then the PerpendicularVector is

(7)

(8)

where

(9)

so the distance is

(10)

The distance from a point (,) to the line can be computed using Vector algebra. Let be a Vector in the same direction as the line

			(11)
			(12)

A given point on the line is

(13)

so the point-line distance is






			(14)

Therefore,

(15)

This result can also be obtained much more simply by noting that the Perpendicular distance is just

times the vertical distance

. But the Slope

is just

, so

(16)

and

(17)

The Perpendicular distance is then

(18)

the same result as before.

See also Line, Point, Point-Line Distance--3-D

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