Yesenia Sherman

2022-06-29

I have something starting at (50, 10) it then rotates counter clockwise by 30 degrees, around the point at (50, 0), essentially mapping out an arc of a circle. How do I find the point it now lies on?

### Answer & Explanation

Colin Moran

You apply a rotation matrix to the vector from the center of rotation to the point that is rotating. Here the vector is $\left(50-50,10-0\right)=\left(0,10\right)$. Then you multiply that by $\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$, getting
$\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 10\end{array}\right]=\left[\begin{array}{c}10\mathrm{sin}{30}^{\circ }\\ 10\mathrm{cos}{30}^{\circ }\end{array}\right]=\left[\begin{array}{c}5\\ 5\sqrt{3}\end{array}\right]$
and add that to the center, getting $\left(50+5,0+5\sqrt{3}\right)=\left(55,5\sqrt{3}\right)$

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