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 formula of Pythagoras.

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

Distance between Point and 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 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