Pre-Quantum Electrodynamics

Scalar triple product

\(| {\bf A} \cdot ({\bf B} \times {\bf C}) |\) is the volume of the parallelepiped subtended by \({\bf A}, {\bf B}\) and \({\bf C}\).

The scalar triple product is preserved under cyclic reordering:

\[ {\bf A} \cdot ({\bf B} \times {\bf C}) = {\bf B} \cdot ({\bf C} \times {\bf A}) = {\bf C} \cdot ({\bf A} \times {\bf B}) \label{VectAn:TripleProduct} %\label{Gr(1.15)} \]

Useful form:

\[ {\bf A} \cdot ({\bf B} \times {\bf C}) = \left| \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right| \label{VectAn:TripleProductAsDet} %\label{Gr(1.16)} \]

The order can be interchanged:

\[ {\bf A} \cdot ({\bf B} \times {\bf C}) = ({\bf A} \times {\bf B}) \cdot {\bf C} \label{Gr(1.16)} \]

Vector triple product

\[ {\bf A} \times ({\bf B} \times {\bf C}) = {\bf B} ({\bf A} \cdot {\bf C}) - {\bf C} ({\bf A} \cdot {\bf B}) \label{Gr(1.17)} \]

Nota bene:

\[ ({\bf A} \times {\bf B}) \times {\bf C} = -{\bf A} ({\bf B} \cdot {\bf C}) + {\bf B} ({\bf A} \cdot {\bf C}) \]

is a different vector. The cross-product is {\bf not} associative in general but obeys the more general relation

\[ {\bf A} \times ({\bf B} \times {\bf C}) + {\bf B} \times ({\bf C} \times {\bf A}) + {\bf C} \times ({\bf A} \times {\bf B}) = 0 \]

All higher vector products can be reduced to combinations of single vector product terms.