How can I represent the rotation of a point in $\mathbb{R}^2$ graphically?

Question

Let $P = (x, y)$ be a point in the plane; now if I want to rotate it by an angle $\theta$ the point $P$ will be moved from $(x, y)$ to $(X,Y)$ where:

$$X = x(\cos \theta) - y(\sin \theta)$$ $$Y = x(\sin \theta) + y(\cos \theta)$$

Why is this true, how can I represent it graphically? Can somebody explain me how to get the representative matrix of the rotation function?

Answer 1

For any linear transformation $T: V_1 \rightarrow V_2$, where $V_1$ and $V_2$ are vector spaces, its matrix representation consists of the images under $T$ of the basis vectors of $V_2$. Since $V_1 = \mathbb{R}^2$, we have that the matrix for a counterclockwise rotation of $\theta$ is

$$\left[\begin{array}{ll} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$

Answer 2

Prove the parallelogram rule: if $(x,y)$ and $(x',y')$ are any two points, then the four points $(0,0)$, $(x,y)$, $(x',y')$ and $(x+x',y+y')$ form a parallelogram.

Now: Where does $(1,0)$ go when rotated by $\theta$? Where does $(0,1)$ go? (Use the definitions of sine and cosine.)

Where does $(x,0)$ go when rotated by $\theta$? Where does $(0,y)$ go? (Use the results of the last paragraph.)

Where does $(x,y)$ go when rotated by $\theta$? (Hint: use the parallelogram rule on the points $(x,0)$ and $(0,y)$. Notice that when you rotate a parallelogram, you get another parallelogram. Also notice that a rectangle is a special type of parallelogram.)