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

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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    $\begingroup$ We cannot help you with homework or check your work. Such questions are considered off-topic. Thanks. $\endgroup$
    – joseph h
    Jan 14 at 21:52
  • $\begingroup$ 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 :( $\endgroup$ 2 days ago