Vector

 

Motivation:

 

                      The concept of vector is from Physics.  The rigorous definition of Vector Space will be found in more advanced mathematics.  Here, a simple example is used to introduce this concept.

 

                                 

 

                        If a person moves from A to B,  then this person moves from B to C.  As a result,  this person moves from A to C if you do not care about the places he travels before he arrives at C.

 

                     There should be some mechanisms to represent this behavior like

                                                 

                                                             +  =

 

                        Consider the following diagram:

 

                                           

                                 The coordinate of A is ( 1,2 ) and B is at (3,2) – but this is the relationship with respect to the point O.   For the people standing at A, they would like to use the place they stand as the origin for the new coordinate system.  If the new coordinate system is with the same unit length and both axes are with the same directions as the old ones.  Then the position of B with respect to A is (2,0 ).

 

  

                                                   = ( 1,2 )      = (3,2)   = (2,0) 

                                Thus,

                                                    +   = 

 

                                This system is fit to describe the situation we mention earlier.   At this moment, you can think  vectors as ordered collection of n components associated with a scalar field.  For the case

                          of n=2:

                                              = ( a, b )     = ( c,d )    = ( e,f )

                                   

                                         +   is defined as  ( a+c, b+d )

                                      s  is defined as ( sa, sb )           s is an element in scalar field.

 

 

  Distance between two points:

                           Given the points  A(a,b)  and  B(c,d) ,  the distance between the two points can be found via their x-components and y-components as shown below:

 

                                

 

                          x-axis and y-axis are perpendicular to each other;  that means the distance between A and B can be found via  Pythagorean Theorem

                                                                       

 

                                      distance between A and B =