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## Summary:

- I am looking for an algebraic proof that the Euler angles can define a proper rotation matrix

## Main Question or Discussion Point

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied.

However, now I can state my question properly (due to the last few responses).

Go to this page and scroll down to the matrix for sixth row of the proper Euler angles:

https://en.wikipedia.org/wiki/Euler_angles

Given ANY proper rotation matrix, I can compute the single angle that informs the third row, third column.

Good: one down.

Then, I proceed UP that third column and I find two equations for the first angle.

s1s2 = #

-c1s2 = different number

So assuming a superficial ignorance, I would have objected with: "there is a relationship between the sine and cosine, so those two terms cannot be any value."

BUT then I realize that there are restrictions on a proper rotation matrix: it must be orthogonal.

So if I wrote out the equations of orthogonality for a 3 by 3 matrix, I get nine equations: three that are equal to 1, and six that are equal to 0.

However, the six equal to 0, are not unique due to the transpose. So I really have only six equations for 9 numbers.

This means that 3 numbers can uniquely define a proper rotation matrix. So that is encouraging.

However, now I want to know why the Proper Euler Angles (or for that matter, the Tait-Bryan) can describe any proper rotation matrix.

But here is the issue: I do NOT want to "know" this by "observing" the rotations, geometrically (as everyone has shown me so far, and for which I am grateful). I want to know it with my eyes closed.

I am looking for a ALGEBRAIC proof that any matrix defined by six equations of orthogonality can be obtained by the trigonometric equations that result from the Proper Euler or Tait angles.

(And I am aware of the issue of gimbal lock; so do not focus on that.)

However, now I can state my question properly (due to the last few responses).

Go to this page and scroll down to the matrix for sixth row of the proper Euler angles:

https://en.wikipedia.org/wiki/Euler_angles

Given ANY proper rotation matrix, I can compute the single angle that informs the third row, third column.

Good: one down.

Then, I proceed UP that third column and I find two equations for the first angle.

s1s2 = #

-c1s2 = different number

So assuming a superficial ignorance, I would have objected with: "there is a relationship between the sine and cosine, so those two terms cannot be any value."

BUT then I realize that there are restrictions on a proper rotation matrix: it must be orthogonal.

So if I wrote out the equations of orthogonality for a 3 by 3 matrix, I get nine equations: three that are equal to 1, and six that are equal to 0.

However, the six equal to 0, are not unique due to the transpose. So I really have only six equations for 9 numbers.

This means that 3 numbers can uniquely define a proper rotation matrix. So that is encouraging.

However, now I want to know why the Proper Euler Angles (or for that matter, the Tait-Bryan) can describe any proper rotation matrix.

But here is the issue: I do NOT want to "know" this by "observing" the rotations, geometrically (as everyone has shown me so far, and for which I am grateful). I want to know it with my eyes closed.

I am looking for a ALGEBRAIC proof that any matrix defined by six equations of orthogonality can be obtained by the trigonometric equations that result from the Proper Euler or Tait angles.

(And I am aware of the issue of gimbal lock; so do not focus on that.)