1. Stating the problem: Given vectors \(\vec{A} = (1,3,3)\), \(\vec{B} = (0,1,2)\), and \(\vec{C} = (5,3,1)\) which form adjacent edges of a parallelepiped, find its volume. 2. The volume of the parallelepiped is given by the scalar triple product: $$\text{Volume} = |\vec{A} \cdot (\vec{B} \times \vec{C})|$$ 3. Find the cross product \(\vec{B} \times \vec{C}\): $$\vec{B} = (0,1,2), \quad \vec{C} = (5,3,1)$$ $$\vec{B} \times \vec{C} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 1 & 2 \\ 5 & 3 & 1 \end{vmatrix} = \mathbf{i}(1 \cdot 1 - 2 \cdot 3) - \mathbf{j}(0 \cdot 1 - 2 \cdot 5) + \mathbf{k}(0 \cdot 3 - 1 \cdot 5)$$ $$ = \mathbf{i}(1 - 6) - \mathbf{j}(0 - 10) + \mathbf{k}(0 - 5) = -5\mathbf{i} + 10\mathbf{j} - 5\mathbf{k} = (-5, 10, -5)$$ 4. Compute the dot product \(\vec{A} \cdot (\vec{B} \times \vec{C})\): $$\vec{A} = (1,3,3), \quad \vec{B} \times \vec{C} = (-5, 10, -5)$$ $$\vec{A} \cdot (\vec{B} \times \vec{C}) = 1 \times (-5) + 3 \times 10 + 3 \times (-5) = -5 + 30 - 15 = 10$$ 5. Volume is the absolute value: $$\text{Volume} = |10| = 10$$ Thus, the volume of the parallelepiped is 10. Final answer: c) 10