**1**. Calculate the work that charge $q_O$ do to $q_t$ when  $q_t$ moves from $P_1$ to $P_2$  in the first picture, and here is its formula

$$W=\int^{P_1}_{P_2}\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2}\cdot[\vec u_RdR+\vec u_\theta Rd\theta]$$

I want to ask why can we just write $\cdot [\vec u_RdR+\vec u_\theta Rd\theta] $ to describe this irregular path ?
[![enter image description here][1]][1]

In the second picture,we have to calculate the work that charge $q_O$ do to $q_t$ when  $q_t$ moves from $P_1$ to $P_2$ along the red line path, the radius of inner circle and outer circle is $r_1$ and $r_2$ and here is its formula

$$W=W_1+W_2$$

$$W_1=\int^{r_2}_{r_1}\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2} \cdot \vec u_R dR$$

$$W_2=\int^{\frac{\theta}{2}}_{\theta=0}\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2} \cdot \vec u_\theta Rd\theta$$

in the second picture,i want ask why should write 

1.$\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2}$, not  $\vec u_\theta \frac{q_oq_t}{4\pi\epsilon_{0}R^2}$,why must the direction be R,not $\theta$ ?

2.why should we $\cdot \vec u_\theta Rd\theta$ in the $W_2$, instead of $\cdot \vec u_R Rd\theta$   
[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/ye7bm.png
  [2]: https://i.sstatic.net/BCCK4.png