Rotation of a vector proof

Question

How would I prove that multiplying a vector by two rotation matrices of different angles is equivalent to multiplying the vector by a rotation matrix of the angles added together? I understand how to prove it for two specific angles, but I'm not sure how to generalize it for all angles.

Answer 1

Note that in the 2d-plane rotations are equivalent to multiplication by a complex on the unit circle.

$z=x+iy\longleftrightarrow \vec u=(x,y)^T$

$ze^{i\theta}=(x+iy)(\cos\theta+i\sin \theta)=(x\cos\theta-y\sin\theta)+i(x\sin\theta+y\cos\theta)$

$R(\theta).\vec u=\begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x\cos\theta -y\sin\theta\\ x\sin\theta+y\cos\theta\end{bmatrix}$

So $\quad z\,e^{i\theta}\longleftrightarrow R(\theta).\vec u$

Now it is straightforward that chaining rotations is adding angles since $z\,e^{i\theta_1}\,e^{i\theta_2}=z\,e^{i(\theta_1+\theta_2)}$