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

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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 $(\tilde{x}_1,\tilde{x}_2)$.

enter image description here

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: enter image description here

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

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Updated on March 17, 2020

Comments

  • user122358
    user122358 over 3 years

    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?

    enter image description here

    • polfosol
      polfosol almost 7 years
      Consider $\tilde{x}_1$ and $\tilde{x}_2$ as two vectors. Then each of them is the sum of two components...
    • user122358
      user122358 over 6 years
      Each of them is the sum of two components...?? Uhmmmm, I think I understand what you say halfway through... I know you are saying vector addition...