How the electric field of those dipoles created in that directions? (configuration added)

I am a little bit confused how the electric field produced by dipole $P_1$ at $vec{r_2}=(ell,0,0)$ is equal to $boldsymbol{E}_{1}left(boldsymbol{r}_{2}right)=-frac{k p_{1}}{l^{3}} hat{z}$

and the electric field produced by dipole $P_2$ at $vec{r_1}=(0,0,0)$ is equal to :$boldsymbol{E}_{2}left(boldsymbol{r}_{1}right)=frac{2 k p_{2}}{l^{3}} hat{x}$

According to my understanding the electric field of dipoles are $vec{E}= frac{k p}{r^{3}}(2 hat{r} cos theta+hat{theta} sin theta)$. I do not understand how it can yield from that formula. Am I supposed to work with spherical coordinates? and if yes I think that $P_2$ should have component in $hat y$ as well.