Properties of the Scalar Triple Product

Theorem 6.2. For any three vectors $\vec a,\vec b,\vec c$: $(\vec a\times\vec b)\cdot\vec c=\vec a\cdot(\vec b\times\vec c)$. Proof: both equal the same determinant $\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}$, up to the row swaps $R_1\leftrightarrow R_3$ then $R_2\leftrightarrow R_3$ which cancel each other's sign flip.

Because of Theorem 6.2, dot and cross can be freely interchanged inside a scalar triple product as long as their relative order (which vectors are crossed, which is dotted) and the cyclic order of the three vectors is kept — dot lies outside the bracket, cross lies between the first two vectors inside it. Chaining this with the fact that the dot product is commutative gives:

Notation. $(\vec a\times\vec b)\cdot\vec c$ is written $[\vec a,\vec b,\vec c]$ — read "box $\vec a,\vec b,\vec c$" — because $|[\vec a,\vec b,\vec c]|$ is the volume of a box (rectangular parallelepiped); it is also called the box product.

Properties.

1. Cyclic invariance: $[\vec a,\vec b,\vec c]=[\vec b,\vec c,\vec a]=[\vec c,\vec a,\vec b]$ — permuting the three vectors cyclically never changes the value.
2. Odd-permutation sign flip: swapping any TWO of the three vectors multiplies the value by $-1$: $[\vec a,\vec b,\vec c]=-[\vec b,\vec a,\vec c]=-[\vec a,\vec c,\vec b]=-[\vec c,\vec b,\vec a]$.
3. Linearity in each slot (Theorem 6.3): $[\vec a+\vec b,\vec c,\vec d]=[\vec a,\vec c,\vec d]+[\vec b,\vec c,\vec d]$ and $[\lambda\vec a,\vec b,\vec c]=\lambda[\vec a,\vec b,\vec c]$, and similarly in the 2nd and 3rd slots — the scalar triple product distributes over vector addition and pulls out scalar multiples, exactly like a determinant's row-linearity.

Theorem 6.4 (coplanarity test). $[\vec a,\vec b,\vec c]=0$ for non-zero $\vec a,\vec b,\vec c$ if and only if $\vec a,\vec b,\vec c$ are coplanar. (Reason: $[\vec a,\vec b,\vec c]=0\iff\vec c\perp(\vec a\times\vec b)\iff\vec c$ lies in the plane parallel to both $\vec a,\vec b$.)

Theorem 6.5. $\vec a,\vec b,\vec c$ are coplanar iff there exist scalars $r,s,t$, not all zero, with $r\vec a+s\vec b+t\vec c=\vec 0$ (a genuine linear dependence). …