## Point-Line Distance--2-D

Given a line and a point ), in slope-intercept 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 ,       (6)

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   (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.