1. **State the problem:** We need to find the area of a parallelogram formed by two vectors $ \mathbf{a} = [-5,1,3] $ and $ \mathbf{b} = [-2,3,-4] $. 2. **Formula:** The area of a parallelogram formed by vectors $ \mathbf{a} $ and $ \mathbf{b} $ is given by the magnitude of their cross product: $$\text{Area} = \| \mathbf{a} \times \mathbf{b} \|$$ 3. **Calculate the cross product:** $$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -5 & 1 & 3 \\ -2 & 3 & -4 \end{vmatrix} = \mathbf{i}(1 \cdot (-4) - 3 \cdot 3) - \mathbf{j}(-5 \cdot (-4) - 3 \cdot (-2)) + \mathbf{k}(-5 \cdot 3 - 1 \cdot (-2))$$ Simplify each component: $$= \mathbf{i}(-4 - 9) - \mathbf{j}(20 + 6) + \mathbf{k}(-15 + 2)$$ $$= \mathbf{i}(-13) - \mathbf{j}(26) + \mathbf{k}(-13)$$ So, $$\mathbf{a} \times \mathbf{b} = [-13, -26, -13]$$ 4. **Calculate the magnitude:** $$\| \mathbf{a} \times \mathbf{b} \| = \sqrt{(-13)^2 + (-26)^2 + (-13)^2} = \sqrt{169 + 676 + 169} = \sqrt{1014}$$ 5. **Approximate the value:** $$\sqrt{1014} \approx 31.82$$ 6. **Round to the nearest whole number:** $$\boxed{32}$$ **Final answer:** The area of the parallelogram is approximately 32.