1. **Problem statement:** Given two vectors in 3D space $\vec{u} = (2, -1, 3)$ and $\vec{v} = (1, 2, -2)$, find the coordinates of the vector $\vec{w} = [\vec{u}, \vec{v}]$, which is the cross product of $\vec{u}$ and $\vec{v}$. 2. **Formula for cross product:** $$\vec{w} = \vec{u} \times \vec{v} = \left( u_y v_z - u_z v_y,\; u_z v_x - u_x v_z,\; u_x v_y - u_y v_x \right)$$ 3. **Calculate each component:** - $w_x = (-1) \times (-2) - 3 \times 2 = 2 - 6 = -4$ - $w_y = 3 \times 1 - 2 \times (-2) = 3 + 4 = 7$ - $w_z = 2 \times 2 - (-1) \times 1 = 4 + 1 = 5$ 4. **Result:** $$\vec{w} = (-4, 7, 5)$$ 5. **Answer choice:** This matches option A. Thus, the cross product vector is $\boxed{(-4, 7, 5)}$.