Matrix – Rotation matrix

 

Motivation:

 

                        Consider to rotate the axes of the coordinate for    . 

 

                             

                        We use the unit vectors    ,   ,    ,    to represent those axes.  Then

 

                                     =  cos     +  sin  

                                     =  - sin   +  cos  

 

                                   ( The length of unit vector is 1. )

 

                       For a fixed point,  we have different way to represent it by choosing different coordinate systems:

 

                                   = a  + b  =   +    

 

                       What is the relationship between  (a,b)  and  ( ,   ) ?  

                       We can use the fact that   and   are perpendicular to each other so that  the projection from one to the other is zero.  Thus,

 

                         a  + b  =    (cos     +  sin   ) +   (  - sin   +  cos   )

                                           =  (  cos  + (-  )sin  )  +  (  sin  +  cos  )

 

                  So, we have

                             a =  (  cos  + (-  )sin  )

                             b =  (  sin  +  cos  )

 

                  In Matrix notation,

 

                               = 

                              

 

 

Theorem (  coordinate rotation )

               Let B₁ and B₂  be two coordinate systems such that B₂  is obtained by rotating

               B₁  with the angle  .   Assume  a point  is represented as ( a,b ) under B₁ and

                 ( ,   ) under B₂ .    Then we have

 

                             = 

             and

                             =

 

               

              Proof:

                              The first one has been proved by the previous introduction.

                             For the second equality,   people can use similar approach to represent

                              ,   in terms of   ,   .

 

                             However,  the problem can be considered in this way:  B₁ can be 

                             obtained  by  rotating B₂  with  -  .  Thus, the matrix turns out to be

 

                               =

 

                             So,

                                             =

 

 

                         This theorem can be covered by more generalized theorem for the change of basis .  That will be mentioned in more advanced course - Linear Algebra.