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

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}

We should know about

  • Scalar and vector products of vectors and their geometrical interpretations.
  • Some properties of the determinant (for matrix calculation).

Solution

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