I really need help with this question, thank you in advance.
Let $e_1= \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} $, $e_2= \begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix} \in \mathbb{R}^3. $ For a non-zero vector v $ \in \mathbb{R}^3$ and a real number $\theta \in \mathbb{R}$, let us denote by $R(v,\theta)$ the space rotation around the axis v with the angle $\theta$.
(1) Let u $\in \mathbb{R}$ be a vector with |u| $= 1$. Show that there exist $ \alpha, \beta \in \mathbb{R}$ such that $$R(e_2,\beta)R(e_1,\alpha)u=e_1.$$
Answer to Question (1):
Relationship
$$R(e_2,\beta)R(e_1,\alpha)u=e_1$$
can be written into an equivalent form:
$$u=\underbrace{R(e_1,\alpha)^{-1}}_{R_1}\underbrace{R(e_2,\beta)^{-1}}_{R_2}e_1$$
which can be transformed into:
$$\begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}=\underbrace{\begin{pmatrix}1&0&0\\0&\cos(\alpha)&\sin(\alpha)\\0&-\sin(\alpha)&\cos(\alpha)\end{pmatrix}}_{R_1}\underbrace{\begin{pmatrix}\cos(\beta)&0&\sin(\beta)\\0&1&0\\-\sin(\beta)&0&\cos(\beta)\end{pmatrix}}_{R_2} \begin{pmatrix}1\\0\\0\end{pmatrix}$$
$$\begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}=\begin{pmatrix}\cos(\beta)\\ -\sin(\alpha)\sin(\beta)\\-\cos(\alpha)\sin(\beta)\end{pmatrix}$$
(do you recognize spherical coordinates, but not in the usual order ? In fact $\beta$ can be at once recognized as the classical longitude).
As coordinate $|u|=1 \implies u_1 \in (-1,1)$, we can deduce that :
$$\beta = \operatorname{acos}u_1 \ \ \text{and} \ \ \alpha=\operatorname{atan}\tfrac{u_2}{u_3}$$
with a particular case for the forbidden value $u_3=0$ for which $\alpha=\pi/2.$
Up to you for the other questions.