# How to recover three successive rotations of a vector

I have a vector, which I rotated with respect to $x$, $y$ and $z$ axes, respectively.

Now I want to recover this operation, that means I want to bring it to the previous position by rotating it with $-\theta$, $-\alpha$ and $-\beta$, where $\theta$, $\alpha$ and $\beta$ are the amounts of initial rotation, in radians/degrees.

I tried to do it by computing the dot product of this vector with axis vectors ($(1,0,0)$ for $x$-axis, $(0,1,0)$ for $y$-axis and $(0,0,1)$ for $z$-axis).

However, this did not produce the right result possibly because It was rotated in 3d, thus the dot product was resulting in a different value that it should be.

What I should do in order to perform this operation? Thanks.

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do you mean you rotated using Euler angles? or did you rotate around the fixed axes? – nbubis Aug 15 '12 at 7:03
I'm not really sure what you're asking. Do you mean that you know the position $x$ of your point before rotating, the position $x'$ of the point after rotating, and then you want to find the angles $\theta,\alpha,\beta$ such that $x'$ gets send back to $x$? Or do you already know $\theta, \alpha,\beta$ and you just want to know how to use these to create a rotation that sends $x'$ back to $x$? – Lieven Aug 15 '12 at 7:12
@Lieven The latter one. I know $\theta$,$\alpha$ and $\beta$, and I want to find the rotation that sends $x'$ back to $x$. – user13791 Aug 15 '12 at 8:00
@nbubis using Euler angles. – user13791 Aug 15 '12 at 8:31
As @nbubis mentioned, Euler angles is what you are looking for. – Sait Aug 15 '12 at 8:34

If you used Euler angles, simply multiply your vector by the rotation matrices in reverse order. If you used $\alpha$ around $\hat{x}$, then $\beta$ around $\hat{y}$, and finally $\gamma$ around $\hat{z}$ to get at a vector $v$, then the original vector $v_0$ is given by: $$v_0 = Z(-\gamma)Y(-\beta)X(-\alpha)v$$ Where $X,Y,Z$ are the rotation matrices.