In the plane $Oxy,$ a rotation of a vector around the origin by $\theta$ is a linear map. How to find the rotation expression and the rotation matrix?

Question

In the plane $Oxy,$ one a rotation of a vector around the origin by $\theta$ is a linear map (check the image below). How to find the rotation expression and the rotation matrix ?

Applied example. Given the triangle $MNP$ with $M\left ( 1, 1 \right ), N\left ( 1, 2 \right ), P\left ( 3, 3 \right ).$ Find the image of the triangle $MNP$ by using that rotation with $\theta= \frac{\pi}{4},$ then draw a diagram for the example.
I follow the reasoning in Qiaochu Yuan's answer- https://math.stackexchange.com/a/1293/822157
My attempt. Given a vector $\left ( x, y \right )\in\mathbb{R}^{2}.$ Let $\alpha$ be the angle determined by $\left ( x, y \right ),$ the $x-$axis and $r= \sqrt{x^{2}+ y^{2}}$ its length. Then, of course $$x, y= r\cos\alpha, r\sin\alpha$$ If you rotate $\left ( x, y \right )$ an angle $\theta,$ you'll obtain the vector $\left ( {x}', {y}' \right ):$ $${x}', {y}'= r\cos\left ( \alpha+ \theta \right ), r\sin\left ( \alpha+ \theta \right )$$ Now you apply those sum and difference formulas and get $${x}'= r\cos\alpha\cos\theta- r\sin\alpha\sin\theta= x\cos\theta- y\sin\theta$$ $${y}'= r\sin\alpha\cos\theta+ r\cos\alpha\sin\theta= x\sin\theta+ y\cos\theta$$ Which is the same as saying that $\left ( {x}', {y}' \right )$ is obtained from $\left ( x, y \right )$ multiplying with the matrix $$\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}$$ Since the rotation $\left ( x, y \right )\mapsto\left ( {x}', {y}' \right )$ is the same as multiplication by a matrix, it is a linear transformation.
Then, what should I do for the applied example ? Hope you can help.

Answer 1

Since rotation is a linear transformation, it maps straight lines into straight lines and so maps a triangle into a triangle. Determine the points that (1, 1), (1, 2), and (3, 3) are mapped into. The triangle with vertices (1, 1), (1, 2), (3, 3) is mapped into the triangle with those new vertices.