Exercise 7

The position vector is given by \[\vec{r}(t) = \left( \begin{matrix} r_x(t) \\ r_y(t) \end{matrix} \right) = \left( \begin{matrix} R \cos \omega t \\ R \sin \omega t \end{matrix} \right)\]

The velocity vector is the derivative of the position vector with respect to time:

\[\vec{v}(t) = \left( \begin{matrix} v_x(t) \\ v_y(t) \end{matrix} \right) = \left( \begin{matrix} \dot{r}_x(t) \\ \dot{r}_y(t) \end{matrix} \right) = \left( \begin{matrix} -R \omega \sin \omega t \\ R \omega \cos \omega t \end{matrix} \right)\]

Two vectors are orthogonal, if their dot product equals to zero:
\[ \vec{r}(t) \cdot \vec{v}(t) = r_x(t) v_x(t) + r_y(t) v_y(t) = -R^2 \omega \cos \omega t \sin \omega t + R^2 \omega \sin \omega t \cos \omega t = 0 \]