A parallelepiped has the edges a = i + j, b = j + k and c = i + 4 k, in metres…

Step 1. The first box. $

= 1(4 - 0) - 1(0 - 1) + 0 = 5$. The volume is$5$ cubic metres.

Step 2. The new edges. $\vec a + \vec b = (1, 2, 1)$, $\vec b + \vec c = (1, 1, 5)$ and $\vec c + \vec a = (2, 1, 4)$.

Step 3. The second determinant. $

= 1(4 - 5) - 2(4 - 10) + 1(1 - 2) = -1 + 12 - 1 = 10$.

Step 4. The reason. Expanding the triple product, the terms with a repeated vector vanish. Only $[\vec a\ \vec b\ \vec c] + [\vec b\ \vec c\ \vec a]$ remains, and these are equal. So the second box always has twice the volume of the first.