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What happens when you rotate a vector $\vec{a} (0,1)$ in $\mathbb{R}^2$ around the x-axis? Shouldn't it just become $\vec{a}(0,-1)$, or have I got a completely wrong idea about rotations around axes?

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    That clears up my confusion! I'm sorry, I mixed up reflection and rotation. Cheers!2012-10-24

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As has been discussed in the comments, the vector $(0,1)$ in $\mathbb R^2$ is transformed into the vector $(0,-1)$ by a reflection in the $x$ axis. However, one can also consider $\mathbb R^2$ canonically embedded in $\mathbb R^3$; in that case, $(0,-1)$ could also be regarded as the result of rotating $(0,1)$, embedded as $(0,1,0)$, through $\pi$ about either the $x$ axis or the $z$ axis.