Rotation of Rectangular Co-ordinates ( X-Y plane about Z-axis )

 

 X - Y Rectangular Co-Ordinates

 
  x=  
  y=  
  Angle of Rotation (z in degree)  =  
  Find  in  X1 - Y1 Rectangular Co-Ordinates       
  Angle of Rotation (angle Z in Radian)  
  Rotation Matrix  K = (k11) (k12)
  k11=cosz; k12=sinz;k21=-sinz;k22=cosz (k21) (k22)
  Determinant & Trace
  x1 (xcosz+ysinz) =  
  y1 (-xsinz+ycosz) =  
  Length in X-Y  
  Length in X1-Y1  
  Polar Co-ordinate r= √( x2 + y2 )  
  Polar Co-ordinate φ (in radian) = tan-1 (y/x)  
  Polar Co-ordinate φ (in degree)  
  Rotation Matrix Kp (polar to  polar co-ordinates) =    (kp11)     (kp12)
  (kp11=1; kp12=0; kp21=0 ; kp22=1-  (z/φ)    (kp21)   (kp22)
  Determinant & Trace
  r1(r1=r)  
  φ1in radian(φ1=z-φ)  
  φ1in degree  
  Transformation Matrix kpk from Polar Co-ordinate (r1, φ1 ) to Rectangular Co-ordinate (x1.y1) (kpr11) (kpr12)
  kpr11=cosφ1 ; kpr12=0;kpr21=sinφ1; kpr22=0 (kpr21) (kpr22)
  Determinant & Trace
  x1=  
  y1=  
  Transformation Matrix kpp from Rectangular Co-ordinate ( x,y) to Polar  Co-ordinate (r, φ) (kpp11) (kpp12)
  kpp11=√( x2 + y2 ) / x ; kpp12=0 ;kpp21=0; kpp22=(1/y)*tan-1 (y/x) (kpp21) (kpp22)
  Determinant & Trace
  r =  
  φ =  
  Overall transformation Matrix:(x,y) to (x1,y1) (ot11)    (ot12)
  ot11=cos(z-φ)*√( x2 + y2 ) / x  = kpr11*kpp11 ; ot12=0; ot21=sin(z-φ)*√( x2 + y2 ) / x =kpr21*kpp11; ot22=0; ot=kpk*kp*kpp (ot21)    (ot22)
  Determinant & Trace
  x1=  
  y1=  
       
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