MatheAss 9.0Geometry 3D

Distances between Points, Lines and Planes.

Distance between two Points:

Given  A(2|1|-7),  B(5|5|5)

Distance between  A  and  B :      
  d(A,B) = 13


It's calculated by the Pythagorean formula.


d = √( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2)

Distance between a point and a line:

Given  P(2|0|3)

    −>  ⎧ 1 ⎫     ⎧ 1 ⎫
g : x = ⎪ 1 ⎪ + s·⎪ 0 ⎪ 
        ⎩ 0 ⎭     ⎩-2 ⎭

Distance between P and g:
  d(P,g) = 2,4494897


Take the plane E in normal form with the point P as position vector and the direction of the line g as normal vector. Determine the point of intersection point S between this plane and the line g. The distance between S and P is the distance between the point and the line.

Distance between Point and Plane:

Given  P(0|0|0)

E : x + y = 1

Distance between P and E :          
  d(P,E) = -0,70710678


Intersect the plane by the perpendicular from the point to the plane and determine the distance between the point of intersection and the given point.

Distance between two Lines:

See Intersection of two Lines

Distance between Plane and Line:

See Intersection of Plane and Line

Distance between two Planes:

See Intersection of two Planes