Angular velocity of a particle moving in 3-d [closed]

$$\newcommand{\bl}{\boldsymbol{#1}} \newcommand{\e}{\bl=} \newcommand{\p}{\bl+} \newcommand{\m}{\bl-} \newcommand{\x}{\bl\times} \newcommand{\vp}{\vphantom{\dfrac{a}{b}}} \newcommand{\tl}{\tag{#1}\label{#1}}$$

I was trying to find the equation of angular velocity of a particle moving arbitrarily in 3-d space using the spherical coordinate system . For an instant there is a radius vector $$\,\mathbf r\,$$ of the particle and velocity vector $$\,\mathbf v\,$$ in an arbitrary direction . Now these two vectors make a 2-d plane whose perpendicular direction is found from the cross product of those two vectors and this direction is same as the direction of angular velocity of that particle . Now we can use $$\begin{equation} \bl\omega\e\dfrac{\mathbf r\x\mathbf v}{r^2} \tl{01} \end{equation}$$ to find $$\,\bl\omega$$ . Here $$\,\mathbf v\,$$ is given by $$\begin{equation} \mathbf v\e\dot{r}\,\mathbf{\hat{r}}\p r\:\,\dot{\!\!\theta}\,\bl{\hat{\theta}}\p r\,\dot{\!\varphi}\,\sin\theta\,\bl{\hat{\varphi}} \tl{02} \end{equation}$$

And finally I found $$\,\bl\omega\,$$ given by $$\begin{equation} \bl\omega\e\:\dot{\!\!\theta}\,\bl{\hat{\varphi}} \m \dot{\!\varphi}\,\sin\theta\:\bl{\hat{\theta}}\vp \tl{03} \end{equation}$$

Now just tell if I am right or wrong . If I'm wrong then please elucidate elaborately.

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• We cannot help you with homework or check your work. Such questions are considered off-topic. Thanks. Jan 14 at 21:52
• It's not a homework . Please tell me wether the equation of angular velocity given above correct or not . Because I am too much confused :( 2 days ago