\begin{equation} \vec{ \textbf{r}} = x \, \hat{\textbf{i}} + y \, \hat{\textbf{j}} + z \, \hat{\textbf{k}} \tag{4.1} \end{equation} \begin{equation} \vec{ \textbf{v}}_{avg} = \frac{ \vec{ \textbf{r}}_2 - \vec{ \textbf{r}}_1} {t_2 - t_1} = \frac{\Delta \vec{\textbf{r}}}{\Delta t} \tag{4.2} \end{equation} \begin{equation} \vec{ \textbf{v}} = \lim_{\Delta t\to 0} \frac{ \Delta \vec{ \textbf{r}}} {\Delta t} = \frac{d \vec{ \textbf{r}}}{dt} \tag{4.3} \end{equation} \begin{equation} v_x = \frac{dx}{dt} \qquad v_y = \frac{dy}{dt} \qquad v_z = \frac{dz}{dt} \tag{4.4} \end{equation} \begin{equation} \vec{ \textbf{v}} = \frac{d \vec{\textbf{r}} }{dt} = \frac{dx}{dt} \hat{\textbf{i}} + \frac{dy}{dt} \hat{\textbf{j}} + \frac{dz}{dt} \hat{\textbf{k}} \tag{4.5} \end{equation} \begin{equation} | \vec{ \textbf{v}} | = v = \sqrt{v_x^2 + v_y^2 + v_z^2 } \tag{4.6} \end{equation} \begin{equation} a = \lim_{\Delta t\to 0} \frac{\Delta v}{ \Delta t} = \frac{dv}{dt} \tag{4.7} \end{equation} \begin{equation} a_x = \frac{dv_x}{dt} \qquad a_y = \frac{dv_y}{dt} \qquad a_z = \frac{dv_z}{dt} \tag{4.8} \end{equation} \begin{equation} \vec{ \textbf{a}} = \frac{d \vec{\textbf{v}} }{dt} = \frac{dv_x}{dt} \hat{\textbf{i}} + \frac{dv_y}{dt} \hat{\textbf{j}} + \frac{dv_z}{dt} \hat{\textbf{k}} \tag{4.9} \end{equation} \begin{equation} a_{rad} = \frac{v^2}{R} \tag{4.10} \end{equation} \begin{equation} a_{rad} = \frac{ 4 \pi^2 R}{T^2} \tag{4.11} \end{equation}
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