Defining a vector

In the first picture, you have

$|OB|=|BA|=b.$

Hence, $OAB$ is isosceles. Therefore, $\angle OAB=\angle AOB=\theta.$

Therefore, $\angle ABO=\pi-2\theta.$

Now we use the law of sines to find $|OA|.$ That is,

$\dfrac{\sin(\theta)}{b}=\dfrac{\sin(\pi-2\theta)}{|OA|}.$

Solve for $|OA|.$

Finally, you can use the expression

$OA=|OA|\langle\sin(\theta),\cos(\theta)\rangle$ to find the final vector $OA.$

In the second problem, you just have the usual

$OA=r\langle\cos(\theta),\sin(\theta)\rangle$ . Or do you need to express the vectors using different variables?