Point-Line Distance--2-Dimensional

The equation of a line in slope-intercept form is given by

(1)

so the line has slope . Now consider the distance from a point to the line. 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)
			(7)
			(8)
			(9)
			(10)
			(11)

If the line is specified by two points and , then a vector perpendicular to the line is given by

(12)

Let be a vector from the point to the first point on the line

(13)

then the distance from to the line is again given by projecting onto , giving

(14)

As it must, this formula corresponds to the distance in the three-dimensional case

(15)

with all vectors having zero -components, and can be written in the slightly more concise form

(16)

where denotes a determinant.

The distance between a point with exact trilinear coordinates and a line is

(17)

(Kimberling 1998, p. 31).

Wolfram Web Resources