Let $\vec{v_1}$ and $\vec{v_2}$ given:
$\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\
\overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + \phi_2\hat{u_\phi} \\$
where $r_n\theta_n\phi_n$ are known constants.
My first question is whether $\vec{v_1}$ and $\vec{v_2}$ are functions? I think they are functions since for each point in coordinate system, the result of $\vec{v_1}$ and $\vec{v_2}$ is different since $\hat{u_r} \hat{u_\theta} \hat{u_\phi}$ changes direction. This is little weird when compared to cartesian coordinate system.
Second, Can we dot product in a similar manner done in cartesian coordinate system, i.e does dot product equal to $\vec{v_1} \cdot \vec{v_2}=r_1 \cdot r_2 + \theta_1 \cdot \theta_2 + \phi_1 \cdot \phi_2$? By similar arguments at question 1 It seems to me that it is only possible when they are defined at same point. Otherwise, i think we need to convert vectors to cartesian coordinate system.
