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Rotation on a plane 
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#1
Jan2413, 04:22 AM

P: 223

1. The problem statement, all variables and given/known data
On a XY plane we have a square with its 4 corners A(3,1) B(7,3) C(2,6) D(0,2). We are to rotate the rest of the square around the point A clockwise by 70 degrees. 2. Relevant equations (I am not sure how they are called in English) The rotation matrix 2x2 1st row: cosa ,sina 2nd row: sina, cosa  call it G(a) so that X' = G(a) * X 3. The attempt at a solution I know how to use this matrix transposition or conversion (not sure how it is called) when I am rotating a vector around the 0point, but I don't know how to rotate a vector around a point on the plane. EDIT: Just as I posted this I got a revelation  I will Rotate the vector AB using the rotation matrix and then add point A's x and y coordinate respectively to the product of the matrixes. And all the same with the other corners  construct vector AC AD and deja vu. Now there is a question: When I do the product of G(a) * A , where A is the vector matrix  the vector spins counterclockwise, but when I do the product of A^{T} * G(a)  the vector spins clockwise. I don't understand why  does it mean that A^{T} * G(a) = G(a) * A? 


#2
Jan2413, 06:03 AM

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P: 39,503

Yes, the rotation matrix,
[tex]\begin{bmatrix}cos(\theta) & sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}[/tex] is orthogonal its columns or rows, considered as vectors, are "orthonormal", perpendicular and of length 1. Also, the transpose is, as you say, equal to the reverse rotation: [tex]\begin{bmatrix}cos(\theta) & sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}= \begin{bmatrix}cos(\theta) & sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}[/tex] because cosine is an "even" function and sine is an "odd" function: [itex]cos(\theta)= cos(\theta)[/itex] and [itex]sin(\theta)= sin(\theta)[/itex]. 


#3
Jan2413, 06:54 AM

P: 234




#4
Jan2413, 08:42 AM

P: 223

Rotation on a plane
Yes I know it's not a square :D But that's what the assignment said, it's not important though. I guess you can call it a polygonia or however it is called in English :/
Also thanks HallsofIvy  read some about it and played around with the numbers a bit and I understand how it works :) 


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