Vector – Representation of the Points on a Line

 

Motivation:

                        Given a line segment  ,  if we know a point C on  ,

                      and  we have   :  = m : n ,  can we try to know more about the

                      point C ?

                     

 

                                          

                              Let’s consider the relationship between the vectors  and .

                      There is no doubt that

 

                                          =  

 

                             And assume that we have a reference point O so that the coordinate of

                    A and B is known by using this reference O as the origin:

 

                                                

 

                             From our basic concept for vector, we have

 

                                             =    +     =  -

                                             =    +      =  -

 

                         So, 

                                      (  -  ) =   (  -  )

                                    =   + 

 

                        But in general,   the choice of reference point will not affect the form of the result.   For simplicity,  we have the following representation:

                                      =   -

                                      =   - 

 

 

                       Thus,

                                      =  + 

                                       

                       We write it as a theorem as follows.

 

Theorem    If C is on  and  :  = m:n , then

                                      =  + 

 

 

                  What if  C is on , but not on  ?

                  Let’s consider in the following way.

 

                           If  m > n,  the position of C should be as follows:

                                              

                                    then

                                                 =  

                                            -  =   (  -  )

                                            =   + 

 

                           If  m < n,  the position of C is on the other side:

                                                

                                    then

                                                    =  ( -  ) =

                                                           =

                                          =   + 

 

                           The results for both cases are the same.

 

                           As a matter of fact,  you can just  replace  either m by (-m)  or n by (-n)

                          in the theorem above to get this result.

 

                           We write down another theorem from the conclusion here.

 

 

Theorem:    If C is on   but C  , and  :  = m:n , then

                           

                                    =   + 

 

 

 

                     And from the two theorems,  we notice that

                          the sum of coefficients belonging to   and   is 1.

                     So,  we have the following theorem for any points on .

 

 

Theorem: For any point C on ,  it can be represented as

                                 = t  + (1-t)  ,  t  R .

 

                  This can also be used as a line equation.

 

                  Please notice that the results here are independent from the choice of

                   the reference point.  Furthermore,  it does not limit you to put this line

                   into a 2-dimensional coordinate system or 3-dimensional coordinate system.