Request: My dynamics book does not contain the vector method to obtain the radial and cross-radial velocity and acceleration. Does anybody give me the proof, please? I have searched the internet but found nothing useful.

For dimension two, the calculation is as simple as to write the velocity in polar coordinates. In dimension three, playing with the velocity ($\dot{\vec r}$) and the position vector, we can write
$\vec v_r=\left(\dfrac{\vec r}{r}·\dot{\vec r}\right)\dfrac{\vec r}{r}=\dfrac{1}{2r^2}\dfrac{d(r^2)}{dt}\vec r$
$\vec v_{cr}=\dot{\vec r}-\vec v_r$
$\vec a_r=\left(\dfrac{\vec r}{r}·\ddot{\vec r}\right)\dfrac{\vec r}{r}=\dfrac{1}{r^2}\left(\dfrac{d(\vec r·\dot{\vec r})}{dt}-\dot r^2\right)\vec r$
$\vec a_{cr}=\ddot{\vec r}-\vec a_r$