Relationship of Two Lines in 3-dimensional Space

 

Motivation:

                      Given two lines in 3-dimensional space,  how many possible scenarios can be found between these two lines?

 

                    I.  the two lines can be intersecting at one point:                                  

                                               

                    II.  the two line can be parallel to each other :

                                             

 

III.               the two lines are not parallel to each other,

and the two lines are not intersecting at any point.

 

 

                          For the first two cases,  the two lines are obviously reside on the same plane.  For the last case,  no plane can contain the two lines.  We are particular interested in this case.   Let P be a sliding point on L and Q be another sliding point on M. What is the minimum value of  ?

 

                         And furthermore, we would like to call this “minimum value” as the “distance” between line L and line M.

 

                         It is still very difficult to imagine the situation of the two lines to find some good properties.   You can imagine the two line are situated as   and .

                            

                                               

 

 

                 

                        To utilize the scenario we are familiar with, we can create a plane that containing the line M, but this plane does not intersect with the line L at any point.

 

                                      

 

                        Then we project line L on this plane. The projection is parallel to the line L, and it also intersects the line M.  Furthermore, the distance between the two parallel lines is the distance from the line L to the plane.   And the normal vector of the plane should be perpendicular to the directional vectors of L and M respectively.

 

                        Similarly, we paraphrase this question by using equations.

 

 

 

 

 

 

 

Question ( Distance between two lines in the space )

                    Let the lines L and M be with the following equations:

 

                                L  :  

                                M :   

                    What is the shortest distance between the two lines ?

 

         Sol:

 

                        Let’s try to find the plane equation that contains M and its normal vector perpendicular to L.

                        The normal vector  is perpendicular to the two directional vectors of the two lines. So, we can set the normal vector as

 

                             = (2,1,3)  ( 1,1,1 ) =  (-2, 1, 1)

 

                       Thus, the plane equation is

                            E:     (x-1, y-2, z-1) = 0

                          -2(x-1) + (y-2) + (z-1) = 0

                         -2x+y+z-1 = 0

 

                   

                     Now,  we will find out the distance from any point in L to the plane.

 

 

                     Recall that we have the result for the distance from a point to a plane. We can just use that result directly: we have a point P(0,2,0) on line L, and the equation of plane E.

 

                Distance between P and E =

                                                          =

                                                         

 

                This method can also be used to check if two lines are intersecting to each other or not. Usually, it is very difficult to do that by just looking at their equations.

 

 

 

 

 

 

 

 

 

Example:   Describe the relationship between the two lines L and M with the following equations respectively:

 

                    L  :

                    M :

 

         Sol:

                        Use similar approach as above,  the plane equation containing M with normal vector perpendicular to L is

 

                           E: -2x+y+z-1 = 0

 

                       Choose a point P on L and calculate the distance from P to E.

                       You can find out the distance is always 0.

 

                        If  P(0,1,0), then

                                                      d(P, E) = 0

 

 

                         In general,

                                            

 

                                                            x = 2t

                                                            y = t + 1

                                                            z = 3t

 

                         Substitute into the equation E,

 

                                          -2(2t) + (t+1) + 3t -1 = 0

 

                          Every point on L satisfies the equation E.