Geometry question on $S_3$

Consider the equilateral triangle above with the vertices laid out

So I take two motions $R_1$ and $R_2$ both are reflections about their subscripted vertices and I take their composition

$R_1 R_2$ Now apparently this yields $R(120)$, a rotation of 120 cow

I don't follow this. After $R_2$ has been applied, if we reflect about vertex 1's line of symmetry (the new position), we should get $R(240)$. I computed the actual cycle permutation and it disagrees with my answer.

This is really unintuitive. I eventually figured out that the book is using the Identity's line of reflection. Why is this so confusing?

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You must fix the lines of reflection beforehand. – Andrew Dec 3 '12 at 3:38
What do you call "reflection about a vertex" to?! – DonAntonio Dec 3 '12 at 3:39
Reflection about a vertex make little sense (or rather none). As Andrew mentions it you should have a line about which you want to take a reflection. The only way to choose which line should you consider is either by a convention or by a priority order in case of nothing being mentioned. – Aseem Dua Dec 3 '12 at 3:43
A rather off topic question, but also related. Any tips on how to fill out the entire $S_3$ table quickly other than computing the transformations in your head? – Hawk Dec 3 '12 at 4:14

I'm going to use $(v_1,v_2,v_3)$, where the $v_i's$ are the numbers at the bottom left, upper and bottom right vertices respectively. At first we have $(1,3,2)$. After we apply $R_2$ we have $(3,1,2)$ and after $R_1$ we have $(2,1,3)$. This is a rotation of $120^{\circ}$ clockwise ($240^{\circ}$ counterclockwise). Not sure if you meant counter or "regular" clockwise when you typed "cow".