# Volume of a pyramid as a determinant?

I have three given points, A, B and C, each of them is a corner of a pyramid. Another corner is located in origo.

The task is to set up a determinant to describe the pyramids volume.

Unfortunately, my book and Wikipedia won´t agree on how to do this, that´s why I´m asking you guys.

PS. The follow up question is if the volume would be any different if the position vectors (a, b and c) and origo all where located in the same plane?

Any help would be very appreciated!

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How do your book and wikipedia disagree? –  Hagen von Eitzen Sep 25 '12 at 20:16
If you have only the four corners of a pyramid, then how do you know how tall it is? It looks like you don't have enough information to locate the apex. –  Henning Makholm Sep 25 '12 at 20:22
@HenningMakholm: Well, it's probably a tetrahedron. –  Javier Sep 25 '12 at 20:26

## 1 Answer

The parallelepiped spanned by $\mathbf a, \mathbf b, \mathbf c$ has (oriented) volume $(\mathbf a\times \mathbf b)\cdot \mathbf c$ (or with any permutation thereof). The pyramid has $\frac16$ of this volume. The expression can also be written as $$V = \frac16 \det(\mathbf a, \mathbf b, \mathbf c)$$ i.e. one sixth of the determinant of the matrix made from the three given vectors. If all vertoces of the pyramid are coplanar, the volume is obviously 0 (and that is also what the determinant gives).

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Ahh, great explanation, thank you. Does it matter ifa, b and c are rows or columns in the determinant? Because it´s always rows in matrices, so this confused me. –  marsrover Sep 25 '12 at 20:56
No, it doesn't matter. The determinant of the transpose of a matrix is the same as the determinant of the original matrix. –  Hagen von Eitzen Sep 25 '12 at 20:57
Ahh, ok, thank you for taking your time. Appreciate it. –  marsrover Sep 25 '12 at 21:37