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PARALLELEPIPED VOLUME
Nigel Nettheim
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This item first appeared in Mathematics Magazine,
Vol 43 No. 4, September 1970, pp. 229-230.
Published by the Mathematical Association of America.
**

Problem 750. [January, 1970] *Proposed by Charles W. Trigg, San Diego,
California.*

| | a^{2} + b^{2} - c^{2} - d^{2} |
2bc - 2ad |
2bd + 2ac |
| |

| | 2bc + 2ad |
a^{2} - b^{2} + c^{2} - d^{2} |
2cd - 2ab |
| |

| | 2bd - 2ac |
2cd + 2ab |
a^{2} - b^{2} - c^{2} + d^{2} |
| |

*Solution by Nigel F. Nettheim, Bureau of the Census, Washington, D. C.*

and the squared length of each of the remaining vectors is the same by symmetry. The inner product between the first pair of vectors is 0 and the inner product between each of the remaining pairs is the same by symmetry. Therefore the parallelepiped is a cube with volume

since the coefficient of

By the same method more general results could be obtained. For example, let

let

then, for