Given a line and a point ), in slopeintercept form, the equation
of 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 ,
If the line is represented by the endpoints of a Vector and , then the Perpendicular
Vector 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
A given point on the line is

(13) 
so the pointline distance is
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, PointLine Distance3D
© 19969 Eric W. Weisstein
19990525