Find all scalar triple products of the unit vectors vec…
Find all scalar triple products of the unit vectors $\vec i,\vec j,\vec k$ of a right-handed coordinate system.
Explanation of Method
All scalar triple products of the coordinate unit vectors.
Step 1: Non-zero products
\[ [\vec i,\vec j,\vec k]=(\vec i\times\vec j)\cdot\vec k=\vec k\cdot\vec k=1,\qquad [\vec j,\vec i,\vec k]=(-\vec k)\cdot\vec k=-1,\ \ldots \]
| $(\vec u,\vec v,\vec w)$ | $[\vec u,\vec v,\vec w]$ |
|---|---|
| $(\vec i,\vec j,\vec k),\ (\vec j,\vec k,\vec i),\ (\vec k,\vec i,\vec j)$ | $+1$ (cyclic) |
| $(\vec j,\vec i,\vec k),\ (\vec i,\vec k,\vec j),\ (\vec k,\vec j,\vec i)$ | $-1$ (anti-cyclic) |
| any two equal (21 triples) | $0$ |
Out of the $3^3=27$ ordered triples, $6$ are permutations of $(\vec i,\vec j,\vec k)$ and the other $21$ contain a repeated vector.
\[[\vec u,\vec v,\vec w]=\begin{cases}+1&\text{cyclic}\\-1&\text{anti-cyclic}\\\ \ 0&\text{repeated vector}\end{cases}\]
Geometrically, $\pm1$ is the signed volume of the unit cube; the sign gives the orientation.
Result
\[\boxed{\,[\vec u,\vec v,\vec w]=\begin{cases}+1&\text{cyclic}\\-1&\text{anti-cyclic}\\\ \ 0&\text{repeated vector}\end{cases}\,}\]