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If `(x , y)` and `(x ,y)` are the coordinates of the same point referred to two sets of rectangular axes with the same origin and it `u x+v y ,` where `u` and `v` are independent of `xa n dy` , becomes `V X+U Y ,` show that `u^2+v^2=U^2+V^2dot`

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Let the axes rotate at angle `theta`. If `(x,y)` is the point with respect to the old axes and (x,y) are the coordinates with respect to the new axes, then
`{(x,=,Xcostheta-Ysintheta,,),(y,=,Xsintheta+Ycostheta,,):}`
Then `ux+vy=u(Xcostheta-Ysintheta)+v(Xsintheta + Y costheta`
`=(ucostheta +vsintheta )X+(-usintheta +vcos theta )Y`
But given new exression is `VX+UY`. Then,
`VX+UY=(ucostheta +vsintheta )X+(-usintheta +vcos theta) Y`
On comparing the coefficients of X and Y, we get
`ucostheta +vsintheta=V` (1)
and `-usintheta +vcostheta=U` (2)
Squareing and adding (1) and (2) , we get
`u^2+v^2=U^2+V^2`
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