Thornton & Marion, Classical Dynamics, Fifth Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 11. Triple scalar product

show that
1) triple scalar product can be written as

$(\vec{A} \times \vec{B}) \cdot \vec{C} = \left| \begin{array}{ccc} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{array} \right|$

2) the product is unaffected by an interchange of operators or by a change in the order.

$(\vec{A} \times \vec{B}) \cdot \vec{C} = \vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) = (\vec{C} \times \vec{A}) \cdot \vec{B}$

3) a geometrical interpretation, which is the volume of the parallelepiped.
The problem gives
the definition of the triple scalar product $(\vec{A} \times \vec{B}) \cdot \vec{C}$$(\vec{A} \times \vec{B}) \cdot \vec{C}$
1) scalar and vector products of vectors and its geometrical interpretations.
2) some properties of the determinant (for matrix calculation).