# Thornton & Marion, Classical Dynamics, 5th Edition

## Chapter 1. Matrices, Vectors, and Vector Calculus

### Problem 11. Triple scalar product

#### The problem asks you to

• Triple scalar product can be written as

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

• The product is unaffected by an interchange of operators or by a change in the order.

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

• A geometrical interpretation, which is the volume of the parallelepiped.

#### The problem gives

• the definition of the triple scalar product $(\vec{A} \times \vec{B}) \cdot \vec{C}$