Rotating a point in 3-space through an angle about some vector

[Eclair_de_XII]

TL;DR Summary

Find the matrix of the rotation in 3-space through the angle $\alpha$ around the vector $(1,2,3)^T$. We assume that the rotation is counterclockwise if we sit at the tip of the vertex and are looking at the origin.

It is alright to express the matrix as a product of matrices.

Denote $v=(1,2,3)^T$, $\theta=\arctan(2)$, and $\phi=\arctan(\frac{3}{\sqrt{5}})$.The way that I attempted this was by performing the following steps:

(1) Rotate $v$ about the z-axis $-\theta$ degrees, while keeping the z-coordinate constant.
(2) Rotate $v$ about the y-axis $-\phi$ degrees, while keeping the y-coordinate constant.
(3) Rotate $v$ about the x-axis $\alpha$ degrees, while keeping the x-coordinate constant.
(4) Rotate $v$ about the y-axis back $\phi$ degrees.
(5) Rotate $v$ about the z-axis back $\theta$ degrees.

I've attached images for visualization. I sort of drew the second projection incorrectly in the second image, so the slope of the line should actually be steeper and be at $x=\sqrt{15}$ instead of some arbitrary point after $x=3$.


Denote the transformation $R_{(u,\beta)}:\mathbb{R}^3\rightarrow \mathbb{R}^3$ by the counterclockwise rotation of some $v\in \mathbb{R}^3$ through some angle $\beta$ about the $u$-axis, where $u\in \{x,y,z\}$ and $\beta\in [0,2\pi]$. Then the desired transformation is given by:

$T=R_{(z,\theta)}R_{(y,\phi)}R_{(x,\alpha)}R_{(z,-\theta)}R_{(y,-\phi)}$

Anyway, does this look right? If not, did I misunderstand the problem?

 

[FactChecker]

You have to reverse the order of the $\theta$ and $\phi$ transformations when you are moving the vector back to its original position.

 

[Eclair_de_XII]

Thanks for checking my work.