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 )
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 )
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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