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

  • #1
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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.

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 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##.

IMG_0489.JPG IMG_0490.JPG
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?
 

Answers and Replies

  • #2
FactChecker
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You have to reverse the order of the ##\theta## and ##\phi## transformations when you are moving the vector back to its original position.
 
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  • #3
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Thanks for checking my work.
 

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