Triple products | Improve Society

Dot and cross multiplication of three vectors $\bold{A}$ , $\bold{B}$ and $\bold{C}$ may produce meaningful products of the form $(\bold{A}\cdot\bold{B})\bold{C}$ , $\bold{A}\cdot(\bold{B}\times\bold{C})$ and $\bold{A}\times(\bold{B}\times\bold{C})$ . The following laws are valid:

$(\bold{A}\cdot\bold{B})\bold{C} \ne \bold{A}(\bold{B}\cdot\bold{C})$ in general

$\bold{A}\cdot(\bold{B}\times\bold{C}) = \bold{B}\cdot(\bold{C}\times\bold{A}) = \bold{C}\cdot(\bold{A}\times\bold{B})$ is volume of a parallelepiped having $\bold{A}$ , $\bold{B}$ , and $\bold{C}$ as edges, or the negative of this volume according as $\bold{A}$ , $\bold{B}$ and $\bold{C}$ do or do not form a right-handed system. If $\bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k}$ , $\bold{B} = B_1\bold{i} + B_2\bold{j} + B_3\bold{k}$ and $\bold{C} = C_1\bold{i} + C_2\bold{j} + C_3\bold{k}$ , then
$\displaystyle \bold{A}\cdot(\bold{B}\times\bold{C}) = \left|\begin{array}{ccc} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{array}\right|\cdots (6)$

$\bold{A} \times (\bold{B} \times \bold{C}) \ne (\bold{A} \times \bold{B}) \times \bold{C}$

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

The product $\bold{A} \cdot (\bold{B} \times \bold{C})$ is sometimes called the scalar triple product or box product and may be denoted by $\left[\bold{ABC}\right]$ . The product $\bold{A} \times (\bold{B} \times \bold{C})$ is called the vector triple product.

In $\bold{A} \cdot (\bold{B} \times \bold{C})$ parentheses are sometimes omitted and we write $\bold{A} \cdot \bold{B} \times \bold{C}$ . However, parentheses must be used in $\bold{A} \times (\bold{B} \times \bold{C})$ . Note that $\bold{A} \cdot (\bold{B} \times \bold{C}) = (\bold{A} \times \bold{B}) \cdot \bold{C}$ . This is often expressed by stating that in a scalar triple product the dot and the cross can be interchanged without affecting the result.