Euler angles - geometry

They don't have to be, but convention has it that consecutive rotations in Euler angle schemes be orthogonal to each other.

The orientation is set by a sequence of three rotations, each being perpendicular to the previous rotation such that the influence of one angle to all the other rotations is zero.

And since the sequence $mathrm{R} = mathrm{R}_z mathrm{R}_x mathrm{R}_y$ consists of three mutually orthogonal rotations, the resulting rotational speed is a sequence of mutually orthogonal relative rotations

$$ vec{omega} = hat{z}_0 dot{psi} + mathrm{R}_z left( hat{x}_1 dot{theta} + mathrm{R}_x (hat{y}_2 dot{phi}) right) $$

In this case, $hat{z}_0$ is perpendicular to $mathrm{R}_z hat{x}_1$, since $hat{z}_0$ is perpendicular to $hat{x}_0$ and the rotation $mathrm{R}_z$ does not change that. Also, $hat{x}_1$ is perpendicular to $mathrm{R}_x hat{y}_2$, since $hat{x}_1$ is perpendicular to $hat{y}_1$ and the rotation $mathrm{R}_x$ does not change that also.