Rotation on a Plane | Using Rotation Matrix and Point Rotation

[lendav_rott]

Homework Statement

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.

Homework 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

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 A^T * G(a) - the vector spins clockwise. I don't understand why - does it mean that A^T * G(a) = G(-a) * A?

 

[HallsofIvy]

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 $sin(-\theta)= -sin(\theta)$.

 

[skiller]

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!

 

[lendav_rott]

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 :)