# Rotate a vector around an axis

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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How many degrees do you rotate by ? why is the $a$ missing in the second expression $(0,-1)$ ? – Belgi Oct 24 '12 at 10:33
Isn't rotating around the x-axis sufficient? Like, a complete rotation? Or not? – JohnPhteven Oct 24 '12 at 10:35
From what you have written, $(0,-1)$ is a reflection about the $x-$axis of the point $(0,1)$. – Daryl Oct 24 '12 at 10:49
That clears up my confusion! I'm sorry, I mixed up reflection and rotation. Cheers! – JohnPhteven Oct 24 '12 at 10:55

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.