Let's say the vector field $\vec A_1 = \begin{pmatrix} r\\ 0 \\ z\end{pmatrix}$ in cylindrical coordinates is given and I want to calculate $\vec A_1 \cdot \vec e_r$, where $\vec e_r = \begin{pmatrix} \cos(\phi) \\ \sin(\phi) \\ 0\end{pmatrix}$
1) $ \begin{pmatrix} r \\ 0 \\ z\end{pmatrix} \cdot \begin{pmatrix} \cos(\phi) \\ \sin(\phi) \\ 0\end{pmatrix} = r \cos(\phi) $
2) $\vec A_1 = \begin{pmatrix} r\\ 0 \\ z\end{pmatrix} = r \cdot\vec e_r + z \cdot \vec e_z$ thus $\vec A_1 \cdot \vec e_r = r \cdot\vec e_r \cdot \vec e_r + z \cdot \vec e_r \cdot \vec e_z= r$
Which one is right and where is the mistake?