Vector – Inner Product

 

 

Motivation:

                              

                                   Let   AOB =   .   cos  = ?

                            By using the diagram shown on the left,  it can be shown that

                                               cos  = 

 

                                  ² = a² + b² ,    ² = c² + d² ,   ² = ( a-c )² + ( b-d )²

 

                               Thus,

                                                      cos  =

                                   

                                          If we define  inner product:    = (a,b)  (c,d) = ac+bd

 

                                 Then

                                                    cos  = 

 

 

 

 

 

Definition:   For two dimensional  vectors   =(a,b)  and  =(c,d) ,  the inner product of   and   is

                                          

                                                                   = ac + bd

 

 

 

 

 

 

 

Property 1:                                                  cos  =

 

 

Property 2 ( Projection )

                      The projection of vector   on    is 

                                              

                                             =  (  cos  ) 

                                                 =  (  )

                                                 =  (  )

 

 

                           Please notice that   is a unit vector on the direction of   .

 

 

 

 

 

Property 3:   If    and   are non-zero vectors.

                        and   are perpendicular to each other if and only if   = 0 .

 

                     ( cos  =  cos90⁰ = 0 )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Property 4: ( Normal Vector )

                     For  a line with equation   ax+by=c ,  the vector (a,b) is perpendicular to this line.

 

                Proof:    

                                Let (x₀, y₀) be a point on the line.

                                Thus, it satisfies the line equation:

                                                          ax₀ + by₀ = c

 

                                For any other points (x,y) on this line, 

                                the vector  (x-x­­_0, y-y₀) is parallel to this line.

                              

                                Consider the inner product:

 

                                            (a,b)  (x-x­₀, y-y₀) = a(x-x­₀)+b(y-y₀)

                                                                         = ax+by-ax₀-by₀

                                                                         = c – c

                                                                         = 0

 

                                So, (a,b) is perpendicular  to the line ax+by=c .