4. Scalar triple product invariance under cyclic permutation

Let $\vec a,\vec b,\vec c\in\mathbb{R}^3$. The scalar triple product is defined by

$$\vec a\cdot(\vec b\times\vec c).$$

One way to see invariance under cyclic permutation is to recall that the scalar triple product equals the determinant of the $3\times3$ matrix whose rows are the components of $\vec a,\vec b,\vec c$ (or whose columns are those vectors). That is,

$$\vec a\cdot(\vec b\times\vec c) = \det\begin{pmatrix}a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\end{pmatrix}.$$

A cyclic permutation of the rows of a determinant leaves the determinant unchanged (up to the permutation parity; specifically a cyclic permutation of rows is an even permutation and preserves the determinant). Therefore

$$\vec a\cdot(\vec b\times\vec c) = \vec b\cdot(\vec c\times\vec a) = \vec c\cdot(\vec a\times\vec b).$$