1. **State the problem:** We are given the vector equation of a line: $$(x, y, z) = (3, -1, 2) + r(3, 2, -4)$$ We need to find the correct Cartesian form of this line from the given options. 2. **Recall the formula:** The vector equation of a line in 3D is: $$\vec{r} = \vec{r_0} + r\vec{v}$$ where $\vec{r_0} = (x_0, y_0, z_0)$ is a point on the line and $\vec{v} = (a, b, c)$ is the direction vector. The Cartesian form is: $$\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$$ 3. **Identify components:** From the given equation: - Point on the line: $(3, -1, 2)$ - Direction vector: $(3, 2, -4)$ 4. **Write the Cartesian form:** Substitute values: $$\frac{x - 3}{3} = \frac{y - (-1)}{2} = \frac{z - 2}{-4}$$ which simplifies to: $$\frac{x - 3}{3} = \frac{y + 1}{2} = \frac{z - 2}{-4}$$ 5. **Match with options:** The denominator for $z$ is $-4$, but the options have $4$ in the denominator. To write with positive denominator, multiply numerator and denominator by $-1$: $$\frac{z - 2}{-4} = \frac{-(z - 2)}{4} = \frac{-z + 2}{4}$$ 6. **Final Cartesian equation:** $$\frac{x - 3}{3} = \frac{y + 1}{2} = \frac{-z + 2}{4}$$ This matches option D. **Answer:** Option D is correct.