1. **State the problem:** We are given vectors $\mathbf{u} = [-1, 3, -2]$, $\mathbf{v} = [4, 0, -1]$, and $\mathbf{w} = [-3, -1, 2]$. We need to compute the vector $3\mathbf{u} - 2\mathbf{v}$. 2. **Formula and rules:** Scalar multiplication of a vector means multiplying each component of the vector by the scalar. Vector subtraction means subtracting corresponding components of the vectors. 3. **Calculate $3\mathbf{u}$:** $$3\mathbf{u} = 3 \times [-1, 3, -2] = [3 \times -1, 3 \times 3, 3 \times -2] = [-3, 9, -6]$$ 4. **Calculate $2\mathbf{v}$:** $$2\mathbf{v} = 2 \times [4, 0, -1] = [2 \times 4, 2 \times 0, 2 \times -1] = [8, 0, -2]$$ 5. **Compute $3\mathbf{u} - 2\mathbf{v}$:** $$3\mathbf{u} - 2\mathbf{v} = [-3, 9, -6] - [8, 0, -2] = [-3 - 8, 9 - 0, -6 - (-2)] = [-11, 9, -4]$$ **Final answer:** The vector $3\mathbf{u} - 2\mathbf{v}$ is $[-11, 9, -4]$.