Rotation of Axis: Let [katex]xy[/katex]-coordinate system be given. We rotate [katex]Ox[/katex] and [katex]Oy[/katex] and about the origin through an angle [katex] \theta \left(0<\theta<90^\circ\right) [/katex] so that the new axes are [katex]OX[/katex] and [katex]OY[/katex] as shown in the figure. Let a point [katex]P[/katex] have coordinates [katex]\left(x,y\right)[/katex] referred to the [katex]xy[/katex]-system of coordinates. Suppose [katex]P[/katex] has coordinates [katex](X, Y)[/katex] referred to the [katex]XY[/katex]-coordinate system. We have to find [katex]X, Y[/katex] in terms of the given coordinates [katex]x, y[/katex]. Let a be measure of the angle that [katex]OP[/katex] makes with [katex]O[/katex]
From [katex]P[/katex], draw [katex]PM[/katex] perpendicular to [katex]Ox[/katex] and PM’ perpendicular to [katex]OX[/katex]. Let [katex]\left|OP\right|=r[/katex] ,
[katex]OM’=X=r\;\cos\left(\alpha-\theta\right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)[/katex].
From the right triangle [katex]OPM'[/katex], we have
and
[katex]M’P=Y=r\;\sin\left(\alpha-\theta\right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)[/katex]
Also from triangle [katex]OPM[/katex] , we have
[katex]x=r\;\cos\alpha\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;y=r\;sin\alpha[/katex]
equations (1) and (2) may be re-written as:
[katex]X=r\;\cos\alpha\;\cos\theta+\;r\;\sin\alpha\;\sin\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)[/katex].
[katex]Y=r\;sin\alpha\;\cos\;\theta\;+\;r\;\cos\alpha\;\sin\;\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(4)[/katex]
Substituting [katex]x=r\;\cos\alpha\;\;\;and\;\;\;y=r\;sin\alpha[/katex], we get
[katex]X=x\cos\theta+y\sin\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(5)[/katex].
[katex]Y=y\cos\theta-x\sin\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6)[/katex].
[katex]i.e,\;\;\;\;\;\;(X,Y)\;=\;(x\cos\theta+y\sin\theta\;\;\;,\;\;\;y\cos\theta-x\sin\theta\;)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[/katex].
are the coordinates of P referred to the new axes [katex]OX[/katex] and [katex]OY[/katex].
Example Rotation of Axis:
The [katex]xy[/katex]-coordinate axes are rotated about the origin through an angle of [katex]30^\circ[/katex] If the [katex]xy[/katex]-coordinates of a point are [katex](5, 7)[/katex], find its [katex]XY[/katex]-coordinates, where [katex]OX[/katex] and [katex]OY[/katex] are the axes obtained after rotation.
Solution:
Let [katex](X, Y)[/katex] be the coordinates of [katex]P[/katex] referred to the [katex]XY[/katex]-axes.
since, we have
[katex]X=x\cos\theta+y\sin\theta[/katex].
[katex]Y=y\cos\theta-x\sin\theta[/katex].
using values, [katex](x,y)=(5,7) \;\;\;\;\;\;and\;\;\;\;\;\; \theta = 30^\circ[/katex], we get
[katex]X=5\cos30^\circ+7\sin30^\circ[/katex].
[katex]Y=7\cos30^\circ-5\sin30^\circ[/katex].
So, [katex]X={\textstyle\frac{5\sqrt3}2}+{\textstyle\frac72}\;\;\;\; and\;\;\;\;\; Y={\textstyle\frac{7\sqrt3}2}-{\textstyle\frac52}[/katex].
[katex]i.e, \;\;\;\;\;\boxed{(X,Y)=({\textstyle\frac{5\sqrt3}2}+{\textstyle\frac72}\;\;,{\textstyle\frac{7\sqrt3}2}-{\textstyle\frac52})}[/katex].
Rotation of Axes Example:
The [katex]xy[/katex]-axes are rotated about the origin through an angle of [katex]\arctan\frac43[/katex] lying in the first quadrant. The coordinates of a point [katex]P[/katex] referred to the new axes [katex]OX[/katex] and [katex]OY[/katex] are [katex]P (-1, -7)[/katex]. Find the coordinates of [katex]P[/katex] referred to the [katex]xy[/katex]-coordinate system.
Solution:
Let [katex]P(x, y)[/katex] be the coordinates of [katex]P[/katex] referred to the [katex]xy[/katex]-coordinate system.
Angle of rotation is given by [katex]\tan\theta =\frac43[/katex] and therefore [katex]\sin\theta=\frac45[/katex] and [katex]\cos\theta=\frac35[/katex] also we have [katex](X,Y)=(-1, -7)[/katex].
since, we have
[katex]X=x\cos\theta+y\sin\theta[/katex].
[katex]Y=y\cos\theta-x\sin\theta[/katex].
by using [katex](X,Y)=(-1, -7),\;\;\;\; \sin\theta=\frac45\;\;\;\; and\;\;\;\; \cos\theta=\frac35[/katex], we get
[katex]-1=x\frac35+y\frac45[/katex].
[katex]-7=y\frac35-x\frac45[/katex].
or
[katex]3x+4y+5=0 \;\;\;and \;\;\;-4x+3y+35=0[/katex]
Solving these equations, we have
[katex]\frac x{125}=\frac y{-125}=\frac z{25}[/katex]
[katex]⇒x=5 \;\;\;\;and \;\;\;\;y=-5[/katex]
Thus coordinates of [katex]P[/katex] referred to the [katex]xy[/katex]-system are [katex](5, -5)[/katex].
After Rotation of Axes must-see translation of axes, for more information about the rotation of axis visit Wikipedia