Rotation of matrices

Question

I am doing rotation of matrices at the moment, I know that if I want to rotate a point, let's say (2,1) 90 degrees clockwise, I have to multiply the matrix [ 2 1 ] * [0 1, -1 0] , but how do I find these points? if Iam asked for instance to rotate it 54 degrees anticlock wise, what matrix would I have to multiply it for? is there any formula for that?

Answer 1

You would use the following rotation matrix, $A_\theta = \left[ \begin{matrix} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{matrix} \right]$, where $\theta$ represents an anticlockwise rotation. $(2,1)$ can be represented as a vector $[\begin{matrix} 2 \\ 1\end{matrix}]$ and multiplied on the right by the rotation matrix to get your answer.

Answer 2

Use http://en.wikipedia.org/wiki/Rotation_matrix and plug in the appropriate angle (in radians).

Answer 3

Yes, you would use the matrix

$$ \left[ \begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right] $$