1. Statement of the problem. We are given point sources at positions $\mathbf{r}_i$ with strengths $\theta_i$, and an observation point $\mathbf{r}$, and we want the total electric field by superposition. 2. Define relative vectors and unit vectors. Define the relative vector from source $i$ to the field point as $\mathbf{s}_i = \mathbf{r}-\mathbf{r}_i$. Let $S_i = |\mathbf{s}_i|$ and the unit vector $\hat{\mathbf{s}}_i = $\frac{\mathbf{s}_i}{S_i}$. 3. Single-source field by Coulomb's law. For a single source with strength $\theta_i$, Coulomb's law gives the field at $\mathbf{r}$ as $$\mathbf{E}_i(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\,\theta_i\,\frac{\hat{\mathbf{s}}_i}{S_i^2}$$ 4. Substitute the unit vector. Replace $\hat{\mathbf{s}}_i$ by $\frac{\mathbf{s}_i}{S_i}$ to obtain $$\mathbf{E}_i(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\,\theta_i\,\frac{\mathbf{s}_i}{S_i^3}$$ 5. Superpose all contributions. Summing over all sources $i=1\ldots N$ gives $$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^N \theta_i\,\frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3}$$ 6. Interpretation and final answer. This vector expression gives the field at $\mathbf{r}$ as the vector sum of each source's contribution directed along $\mathbf{r}-\mathbf{r}_i$ and decaying as the inverse cube of the separation in the vector form. Final answer: $$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^N \theta_i\,\frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3}$$