Why is this rotation "incorrect"?

Question

I've been trying to use the following formula for the rotation of a point around the origin:

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

Now, I'm trying to apply this formula to the coordinate $(5,3)$ and rotating it $90$ degrees clockwise, and I ended up with the following result:

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos{90} & -\sin{90} \\ \sin{90} & \cos{90} \end{bmatrix} \begin{bmatrix} 5 \\ 3 \end{bmatrix} \\ = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 5 \\ 3 \end{bmatrix} \\ = \begin{bmatrix} 0(5) -1(3) \\ 1(5) + 0(3) \end{bmatrix} \\ = \begin{bmatrix} -3 \\ 5 \end{bmatrix} $$

I ended up with the rotated coordinates $(-3,5)$. Unfortunately, this was wrong. Can anyone tell me what I'm doing wrong, and how I can do it correctly? I tried this method on other coordinate points, and all of them were wrong as well.

Answer 1

It seems right to me. Ah, maybe you're thinking of a clockwise rotation. That matrix gives a counterclockwise rotation through an angle $\theta$.

Answer 2

For future reference, or for anyone who happens to stumble upon this question looking for help, here's the two proper equations for rotation, where $\theta$ is a positive number in the range $0\rightarrow360$, and $x$ and $y$ are the $x$ and $y$ values for your point.

Clockwise

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

Counterclockwise

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

Alternatively, you can just use positive and negative rotation values for $\theta$ as well.

Answer 3

It is correct. What makes one think that it is wrong? You mean the convention that CCW is positive?

EDIT1:

What I meant is with Clockwise rotation I obtain $(-3,5)$ by matrix multiplication and Counterclockwise rotation gives $(3,-5)$ for a diametrically opposite point. So only association with sign convention of rotation could be the source of error.