Rotation axes and shifting origin

Change of Axis : Rotation in axes and shifting in origin.

1.Shifting of origin without Rotation of Axes :

Initial : 

           Origin =  O( 0 , 0)

           X-axis  = OX

           Y-axis  = OY    

          Point  =  P( x, y )

shift origin without rotation

After Shifting :

         Origin =  O’ (a , b )

           X-axis  = O’X’

           Y-axis  = O’Y’    

          Point  =  P( x’, y’ )

Here you can see OX and OY  are parallel to O’X’  and

O’Y’

Then  ,

  x = x’ + a 

  y = y’ + b

2. Rotation of Axes without Shifting the origin :

Initial : 

           Origin =  O( 0 , 0)

           X-axis  = OX

           Y-axis  = OY    

          Point  =  P( x, y )

rotation of axes without changing the origin

After rotation :

         Origin =  O (0 , 0 ).  Not changed

           X-axis  = OX’

           Y-axis  = OY’

           Rotation angle =     θ  ( anti-clockwise )

X’OX = YOY’ =  θ

          Point  =  P( x’, y’ )

Then  ,

  x = x’ cos θ  – y’sin θ  

  y = x’ sin θ  +  y’cos θ   

3.Shifting of origin and  Rotation of Axes :

Initial : 

           Origin =  O( 0 , 0)

           X-axis  = OX

           Y-axis  = OY    

          Point  =  P( x, y )

After rotation and shifting :

         Origin =  O’ (a , b ).

           X-axis  = O’X’

           Y-axis  = O’Y’

           Rotation angle =     θ ( anti-clockwise )

X’OX = YOY’ =  θ

          Point  =  P( x’, y’ )

Then  ,

  x =  a  +  x’ cos θ    y’sin θ  

  y = b  +  x’ sin θ  + y’cos θ  

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