# Question on relative angular acceleration

I want someone to kindly check whether I am doing (understanding) the math correctly or not.

So, let's consider two bodies with constant angular velocities $\omega_1 \hat y$ and y $\omega_2 \hat x$ so the relative angular velocity of body 2 with respect to 1 is, $\vec \omega=\omega_2 \hat x -\omega_1 \hat y$

Now, the understanding (checking) part begins.

Let's calculate the relative angular velocity of body 2 w.r.t 1,

So now I'm going to consider the $\hat x$ unit vector rotating about the $\hat y$ vector. ( $\hat y$ vector remains stationary ) Thus when taking the time derivative of $\vec \omega$, the $\omega_1 \hat y$ term vanishes (constant) but the derivative of $\omega_2 \hat x$ is $\omega_2\dot{\hat x}$.

The $\dot{\hat x}$ has its direction towards the negative $z-axis$ and it's value is $\omega_1$ so, $$\dot{\hat x}=-\omega_1 \hat z$$

giving us, $$\dot {\vec \omega}=-\omega_1 \omega_2 \hat z$$

Am I right ?