Rotating a point around the orgin?

Question

Can some explain how to calculate how to rotate point (x,y) n radians around the orgin clockwise or counterclockwise?

Answer 1

I suppose we have $(x,y) \in \mathbb{R^2}$. We can multiply it by the rotation matrix $ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin\theta & \cos\theta \end{pmatrix}$ to make it rotate through an angle $\theta$ (in radians) counter-clockwise about the origin.

Answer 2

By multiplication by the rotation matrix: $$\pmatrix{\cos\alpha& -\sin\alpha\\ \sin\alpha& \cos\alpha}\cdot\pmatrix{x\\y}.$$ This yields to the point $(\cos\alpha\cdot x-\sin\alpha\cdot y,\ \sin\alpha\cdot x+\cos\alpha\cdot y)$.