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Given a parallelepiped in $\mathbb{R}^3$ with the three adjacent vectors corresponding to three adjacent edges of the parallelepiped,

to find the volume, we just take any two vectors $\vec{u},\vec{v}$ from the three adjacent vectors and take the crossproduct $\vec{u}\times\vec{v}$ and then take the dot product with the other vector $\vec{w}$, that is, $(\vec{u}\times\vec{v})\cdot\vec{w}$.

And then taking the absolute value, we have the volume $|(\vec{u}\times\vec{v})\cdot\vec{w}|$.

The reason for the above argument is because

  1. if we take any two adjacent vectors from the given three adjacent vectors, they form a base of a parallelepiped
  2. and by taking the dot product with the other one, we have the volume or the negative the volume of the parallelepiped. So we take the absolute value.

Is this the correct argument?

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    $\begingroup$ Parallelogram (note spelling) is two-dimensional. Do you mean parallelipiped? $\endgroup$ – Gerry Myerson Feb 10 at 4:00
  • $\begingroup$ @GerryMyerson Thanks! I just corrected the question and body. $\endgroup$ – Junpyo Choi Feb 10 at 4:12
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    $\begingroup$ parallelepiped You're missing the "r" (and we both had the wrong vowel between the "l" and the "p"). $\endgroup$ – Gerry Myerson Feb 10 at 4:29
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    $\begingroup$ As for your question, the whole problem is, why is your statement 2 true? That's what needs to be established, not just asserted. $\endgroup$ – Gerry Myerson Feb 10 at 4:31
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Enclosed volume is given by the scalar triple product and is unaffected when their order is unchanged.

$$|(\vec{u}\times\vec{v})\cdot\vec{w}|=|\vec{u}\cdot(\vec{v}\times\vec{w})|$$

When order is interchanged, the volume becomes negative.

Btw, triclinic mineral crystals form in nature with such a 3d edge geometry.

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