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Summary:

Find the matrix of the rotation in 3space 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.
Main Question or Discussion Point
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 zaxis ##\theta## degrees, while keeping the zcoordinate constant.
(2) Rotate ##v## about the yaxis ##\phi## degrees, while keeping the ycoordinate constant.
(3) Rotate ##v## about the xaxis ##\alpha## degrees, while keeping the xcoordinate constant.
(4) Rotate ##v## about the yaxis back ##\phi## degrees.
(5) Rotate ##v## about the zaxis 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?
(1) Rotate ##v## about the zaxis ##\theta## degrees, while keeping the zcoordinate constant.
(2) Rotate ##v## about the yaxis ##\phi## degrees, while keeping the ycoordinate constant.
(3) Rotate ##v## about the xaxis ##\alpha## degrees, while keeping the xcoordinate constant.
(4) Rotate ##v## about the yaxis back ##\phi## degrees.
(5) Rotate ##v## about the zaxis 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?