Classical Dynamics of Particles and Systems · Problem 1-11

Newtonian Mechanics—Single Particle — Marion & Thornton

Physics — Mechanics · triple-scalar-product

Problem

Show that the triple scalar product

(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}

can be written as

(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C} = \begin{vmatrix} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{vmatrix}

Show also that the product is unaffected by an interchange of the scalar and vector product operations or by a change in the order of

\mathbf{A}

\mathbf{B}

\mathbf{C}

, as long as they are in cyclic order; that is,

(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C} = \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A}) = (\mathbf{C} \times \mathbf{A}) \cdot \mathbf{B}, \quad \text{etc.}

We may therefore use the notation

\mathbf{ABC}

to denote the triple scalar product. Finally, give a geometric interpretation of

\mathbf{ABC}

by computing the volume of the parallelepiped defined by the three vectors

\mathbf{A}

\mathbf{B}

\mathbf{C}