2
$\begingroup$

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!

$\endgroup$
  • 2
    $\begingroup$ How do your book and wikipedia disagree? $\endgroup$ – Hagen von Eitzen Sep 25 '12 at 20:16
  • $\begingroup$ 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. $\endgroup$ – hmakholm left over Monica Sep 25 '12 at 20:22
  • $\begingroup$ @HenningMakholm: Well, it's probably a tetrahedron. $\endgroup$ – Javier Sep 25 '12 at 20:26
2
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – marsrover Sep 25 '12 at 20:56
  • $\begingroup$ No, it doesn't matter. The determinant of the transpose of a matrix is the same as the determinant of the original matrix. $\endgroup$ – Hagen von Eitzen Sep 25 '12 at 20:57
  • $\begingroup$ Ahh, ok, thank you for taking your time. Appreciate it. $\endgroup$ – marsrover Sep 25 '12 at 21:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.