Point rotating in a coordinate system

  • Thread starter Karol
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  • #1
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The point P rotates with angle α to point P'. the coordinates of the old P are x1 and x2 and for P': x'1 and x'2.
Prove that:
$$x'_1=x_1\cos\alpha+x_2\sin\alpha$$
$$x'_2=x_2\cos\alpha-x_1\cos\alpha$$

I drew on the left the problem and on the right my attempt. the line OA, which is made of ##x_1\cos\alpha## plus ##x_2\sin\alpha## which is the blue line is indeed x'1 but i don't see the congruent triangles.
 

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  • #2
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##P(x_1,x_2)## and If we rotate α degree we get new coordinates ##P'(x'_1,x'_2)##.Now Let make a vector which initial point Origin and terminal point P.This vector has magnitude R.Now we can show this vector in like this P=R(cosβ+sinβ) so ##x_1=Rcosβ## and ##x_2=Rsinβ##.Now we want to rotate this coordinate α degree.

This will lead us ##x'_1=Rcos(β+α)##
and ##x'_2=Rsin(β+α)##.Think this way.
 
  • #3
1,380
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Thanks RyanH, i solved
 

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