Q:

Kinematic Modelling with clockwise angles defined as positive

It seems when modeling a system for which an clockwise angles is defined as positive, the resulting equations become wrong. 
For example : 
In the picture below $\theta_x$, the orientation of the ball, is measured positive in clockwise rotation around the $x$ axis. 
When I want to compute the velocity $v_a$ of the point on the ball touching the ground it should be 0. 
However:
$
v_a = \begin{bmatrix} 0 \\ \dot{\theta_x}*r_s\\ 0\end{bmatrix} + \begin{bmatrix} \dot{\theta_x}\\0 \\0 \end{bmatrix} \times \begin{bmatrix} 
0 \\ 0 \\ -r_s
\end{bmatrix} = \begin{bmatrix} 0 \\ 2*r_s*\dot{\theta_x} \\0 \end{bmatrix}
$
Is there a mistake in my reasoning ? 

A:

Remember that the definition of the angular velocity vector is
$$\vec{\omega} = \frac{\vec{r}\times\vec{v}}{r^2}$$
When you change which direction corresponds to positive rotation, you are also changing the direction that $\vec{\omega}$ points relative to the $z$-axis.  To be consistent, if you haven't changed which direction the $z$-axis points on your diagrams, you must instead change the definition of the cross product to use the "left-hand rule" instead of the right-hand rule, which gives you the result of the opposite sign.