Rotation Transformation

$$\text{rot}(\vec{x})=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\vec{x}=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]\vec{x},a^2+b^2=r^2$$

• $\theta$ is the rotation from the x-axis toward the y-axis
• $r$ is the scaling factor of the vector, with 1 being no scaling
• One can imagine the transformation as scaling and rotating the vector$\left[\begin{array}{c}1\\ 0\end{array}\right]$to$\left[\begin{array}{c}a\\ b\end{array}\right]$, transforming the curve/coordinate plane together as well