Is the rotation in 2D done by vector addition, project, or rotation?

Question

Looking at the two equations in the picture below, the clockwise rotation matrix is applied to them. However, the picture says the axes are rotated anti-clockwise by $ \theta$, Is this done by vector projection, vector addition, or rotation?

Answer 1

The rotation matrix $$\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$$ rotates a vector clockwise by $\theta$. But that is equivalent as rotating the axis counterclockwise by $\theta$ and use the new coordinate system.

Consider a simple example, the vector $(1,0)$ becomes $(0,-1)$ if rotated clockwise by $90$ degrees. But that is equivalent as rotating the $xy$-axis counterclockwise (in which $x$ is pointing upwards, and $y$ is pointing to the left), and look at the original point now in the new $xy$-system, which is $(0,-1)$.

The following picture shows a clockwise rotation of vector $(x_1,x_2)$ to $(\tilde{x}_1,\tilde{x}_2)$.

We have $$\tilde{x}_1=r\cos(\phi-\theta)=r\cos\phi\cos\theta+r\sin\phi\sin\theta=x_1\cos\theta+x_2\sin\theta,\\ \tilde{x}_2=r\sin(\phi-\theta)=r\sin\phi\cos\theta-r\cos\phi\sin\theta=-x_1\sin\theta+x_2\cos\theta.$$

Now as I said, this is equivalent as rotating the axis counterclockwise:

In the new system $\tilde{x}_1\tilde{x}_2$, the original points have the coordinate just as above calculated.