Register to reply

Rotation on a plane

by lendav_rott
Tags: plane, rotation
Share this thread:
Jan24-13, 04:22 AM
P: 223
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?
Phys.Org News Partner Science news on
Security CTO to detail Android Fake ID flaw at Black Hat
Huge waves measured for first time in Arctic Ocean
Mysterious molecules in space
Jan24-13, 06:03 AM
Sci Advisor
PF Gold
P: 39,345
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].
Jan24-13, 06:54 AM
P: 229
Quote Quote by lendav_rott View Post
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!

Jan24-13, 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 :)

Register to reply

Related Discussions
Co-ordinate plane rotation Linear & Abstract Algebra 6
Axis of rotation, plane of reflection Linear & Abstract Algebra 1
Rotation of plane tangential to sphere General Math 2
Rotation of a chair-o-plane Introductory Physics Homework 1
Rotation in Vertical Plane Introductory Physics Homework 3