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

Thornton & Marion, Classical Dynamics of Particles and Systems, 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

$latex (\\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.

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

We should know about

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


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