1. **State the problem:** We need to evaluate the double integral over the entire xy-plane: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \begin{pmatrix}0 \\ y \\ x \end{pmatrix} \, dx \, dy$$ 2. **Understand the integral:** The integrand is a vector function with components $0$, $y$, and $x$. The integral is taken over all $x$ and $y$ from $-\infty$ to $\infty$. 3. **Integral of each component:** The integral of a vector function is the vector of the integrals of each component: $$\int \int \begin{pmatrix}0 \\ y \\ x \end{pmatrix} dx dy = \begin{pmatrix} \int \int 0 \, dx dy \\ \int \int y \, dx dy \\ \int \int x \, dx dy \end{pmatrix}$$ 4. **Evaluate each component:** - First component: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} 0 \, dx dy = 0$$ - Second component: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} y \, dx dy = \int_{-\infty}^{\infty} y \left( \int_{-\infty}^{\infty} dx \right) dy$$ The inner integral $$\int_{-\infty}^{\infty} dx$$ diverges to infinity, so the integral does not converge. - Third component: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x \, dx dy = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} x \, dx \right) dy$$ The inner integral $$\int_{-\infty}^{\infty} x \, dx$$ also diverges. 5. **Conclusion:** Since the integrals of the second and third components diverge, the double integral does not converge to a finite value. **Final answer:** The integral diverges and does not have a finite value.