This question involves the computation of the Laplacian of $\frac{1}{|\mathbf{r}|}$, where $\mathbf{r} = x\hat{i} + y\hat{j} + 3\hat{k}$ and $|\mathbf{r}| = \sqrt{x^2 + y^2 + 3^2} = \sqrt{x^2 + y^2 + 9}$.

Step-by-step solution:

1. Expression for the Laplacian:
   The Laplacian $\nabla^2 \left(\frac{1}{|\mathbf{r}|}\right)$ in three dimensions is given by: $\nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) = \frac{\partial^2}{\partial x^2} \left(\frac{1}{|\mathbf{r}|}\right) + \frac{\partial^2}{\partial y^2} \left(\frac{1}{|\mathbf{r}|}\right) + \frac{\partial^2}{\partial z^2} \left(\frac{1}{|\mathbf{r}|}\right).$

2. General formula for $\nabla^2 \left(\frac{1}{|\mathbf{r}|}\right)$: The Laplacian of $\frac{1}{|\mathbf{r}|}$ for $\mathbf{r} \neq 0$ is a well-known result: $\nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) = 0, \quad \text{for } |\mathbf{r}| \neq 0.$

   Since the question does not involve a delta function at the origin (distribution theory), we conclude: $\nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) = 0.$

Final Answer:

$\boxed{(d) \, 0}$

Would you like additional details or have further questions? Here are some related problems you might find interesting:

1. What is the divergence of the vector field $\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3}$?
2. Compute $\nabla^2 |\mathbf{r}|$ for $\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}$.
3. Evaluate $\nabla \cdot \mathbf{F}$ for $\mathbf{F} = \nabla \left(\frac{1}{|\mathbf{r}|}\right)$.
4. Determine $\nabla^2 \frac{1}{|\mathbf{r}|^n}$ for $n > 1$.
5. Verify if $\mathbf{F} = -\nabla \left(\frac{1}{|\mathbf{r}|}\right)$ is irrotational.

Tip: The Laplacian often simplifies greatly for symmetric functions like $\frac{1}{|\mathbf{r}|}$. Learning its standard results can save significant time!