# Rotation on a plane

by lendav_rott
Tags: plane, rotation
 P: 221 1. The problem statement, all variables and given/known data On a X-Y 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 0-point, 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 AT * G(a) - the vector spins clockwise. I don't understand why - does it mean that AT * G(a) = G(-a) * A?
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,896 Yes, the rotation matrix, $$\begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}$$ 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: $$\begin{bmatrix}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{bmatrix}= \begin{bmatrix}cos(-\theta) & -sin(-\theta) \\ sin(-\theta) & cos(-\theta)\end{bmatrix}$$ because cosine is an "even" function and sine is an "odd" function: $cos(-\theta)= cos(\theta)$ and $sin(-\theta)= -sin(\theta)$.
P: 221
 Quote by lendav_rott On a X-Y plane we have a square with its 4 corners A(3,1) B(7,3) C(2,6) D(0,2).
Funny looking square!

P: 221

## 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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