Matrix associated to rotation around Oz real axis

As we have seen before, we need just to find

$$\mathcal{R}_{\alpha}e_1=\cos\alpha e_1+\sin \alpha e_2+0e_3$$
$$\mathcal{R}_{\alpha}e_2=-\sin\alpha e_1+\cos \alpha e_2+0e_3$$
$$\mathcal{R}_{\alpha}e_3=0e_1+0e_2+e_3$$

$ R_{\alpha}=\left [\begin{array}{ccc}<br />
\cos\alpha&-\sin\alpha&0\\</p>
<p>\sin\alpha&\cos\alpha&0\\<br />
0&0&1<br />
\end {array}\right ] \) $

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