In the first picture, you have
$\displaystyle |OB|=|BA|=b.$
Hence, $\displaystyle OAB$ is isosceles. Therefore, $\displaystyle \angle OAB=\angle AOB=\theta.$
Therefore, $\displaystyle \angle ABO=\pi-2\theta.$
Now we use the law of sines to find $\displaystyle |OA|.$ That is,
$\displaystyle \dfrac{\sin(\theta)}{b}=\dfrac{\sin(\pi-2\theta)}{|OA|}.$
Solve for $\displaystyle |OA|.$
Finally, you can use the expression
$\displaystyle OA=|OA|\langle\sin(\theta),\cos(\theta)\rangle$ to find the final vector $\displaystyle OA.$
In the second problem, you just have the usual
$\displaystyle OA=r\langle\cos(\theta),\sin(\theta)\rangle$. Or do you need to express the vectors using different variables?