1. **State the problem:** We need to find the volume of the parallelepiped defined by the vectors from the origin to the points \((3, -3, 2)\), \((3, -1, 1)\), and \((3, -7, 5)\). 2. **Find the vectors:** Let \(\mathbf{a} = (3, -3, 2)\), \(\mathbf{b} = (3, -1, 1)\), and \(\mathbf{c} = (3, -7, 5)\). 3. **Volume formula:** The volume \(V\) of the parallelepiped is the absolute value of the scalar triple product: $$V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$$ 4. **Calculate \(\mathbf{b} \times \mathbf{c}\):** $$\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & -1 & 1 \\ 3 & -7 & 5 \end{vmatrix} = \mathbf{i}((-1)(5) - (1)(-7)) - \mathbf{j}(3 \cdot 5 - 1 \cdot 3) + \mathbf{k}(3 \cdot (-7) - (-1) \cdot 3)$$ $$= \mathbf{i}(-5 + 7) - \mathbf{j}(15 - 3) + \mathbf{k}(-21 + 3) = 2\mathbf{i} - 12\mathbf{j} - 18\mathbf{k}$$ So, \(\mathbf{b} \times \mathbf{c} = (2, -12, -18)\). 5. **Calculate \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\):** $$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 3 \cdot 2 + (-3) \cdot (-12) + 2 \cdot (-18) = 6 + 36 - 36 = 6$$ 6. **Find the volume:** $$V = |6| = 6$$ **Final answer:** The volume of the parallelepiped is \(6\).