Point-Line Distance

Result

If line L has the normalized direction vector l, and A is a point on L, the distance from L to a point P is:

    dist(P,L) = |AP x l|

where AP is the vector from A to P, x is the cross product, and |v| is the magnitude of the vector v.

Derivation

     P
     |
     |d = dist(P,L)
 A___|______ 
       L

L is defined by the normalized direction vector l and A is a point on L.

By d we denote the distance from P to L.

     . 
  v /|
   / |d
  /__|_____ l

We can also define the vector v = AP.

If we define a parallelogram spanned by v and l, the area is:

    area = |v x l|     (x is the cross product)

The area of the parallelogram can also be computed as the magnitude of the base
times the height (= d), which, since l is normalized, is:

    area = |l| * d = d

So we have:

    d = area = |v x l|

References

• softsurfer.com: About Lines and Distance of a Point to a Line (2D & 3D)