Let's say we have a vector $D=(\sqrt2, 0, 0)$ and we do the following operations on it.
1) Rotate it in the clockwise direction against the $X$ axis for $\pi/4$
2) Rotate it in the counter clockwise direction against the $Y$ axis for $\pi/4$
that's the vector $D1$, we then perform steps 1 & 2 but we reverse the order, i.e. 1 would be counter clockwise and 2 would be clockwise; that's vector $D2$.
What's the angle that $D$ forms with $D1$ and $D2$?
From my calculations if we rotate it for the same amount in both axis it would still retain the $\pi/4$ span against both cases would it not?