Thornton & Marion (5th Edition), Chapter 01, Exercise Problem 11

Thornton & Marion, Classical Dynamics, Fifth Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 11. Triple scalar product

The problem asks you to
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}
We should know about
1) scalar and vector products of vectors and its geometrical interpretations.
2) some properties of the determinant (for matrix calculation).

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