Scalar Triple Product
Definition
The following expression is called the scalar triple product.
$$ \mathbf{A}\cdot (\mathbf{B} \times \mathbf{C} ) $$
Explanation
A scalar triple product is an operation involving the product of three vectors, where the result is a scalar. The operation resulting in a vector is called vector triple product. To get a scalar result, one must first cross multiply two vectors to produce another vector and then dot multiply it with a different vector.
By the commutative property below, the following notation is also used, known as the Grassmann symbol1.
$$ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C} ) = [\mathbf{A}, \mathbf{B}, \mathbf{C}] = [\mathbf{A} \mathbf{B} \mathbf{C}] $$
Parallelepiped
The magnitude of the scalar triple product equals the volume of the parallelepiped formed by the three vectors.
$$ \mathbf{A}\cdot (\mathbf{B} \times \mathbf{C} )=|\mathbf{A}||\mathbf{B}\times \mathbf{C}| \cos\theta $$
$ |\mathbf{B} \times \mathbf{C}|$ is the area of the base of the parallelepiped, and $|\mathbf{A} \cos\theta|$ is the height. Hence, the scalar triple product, being the product of the area of the base and the height, represents the volume of the parallelepiped.
A closer examination of this feature reveals that whatever the order of operation, the same value should result. This is because the parallelepiped formed by the three vectors is unique. Therefore, the following equation is valid.
Commutability
The value of the scalar triple product can be cyclically commuted.
$$ \mathbf{A}\cdot (\mathbf{B} \times \mathbf{C} ) = \mathbf{B}\cdot (\mathbf{C} \times \mathbf{A} ) =\mathbf{C}\cdot (\mathbf{A} \times \mathbf{B} ) $$
It can be proven simply using the Levi-Civita symbol.
$$ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C} ) = A_{i} (B \times C)_{i} =A_{i} \epsilon_{ijk} B_{j}C_{k} =\epsilon_{ijk}A_{i}B_{j}C_{k} $$
$$ \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A} ) = B_{i} (C \times A)_{i} =B_{i} \epsilon_{ijk} C_{j}A_{k} =\epsilon_{ijk}B_{i}C_{j}A_{k} $$
$$ \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B} ) = C_{i} (A \times B)_{i} =C_{i} \epsilon_{ijk} A_{j}B_{k} =\epsilon_{ijk}C_{i}A_{j}B_{k} $$
Due to the properties of the Levi-Civita symbol, it is understood that the above three expressions yield the same value. Whether it’s ABC, BCA, or CAB, as long as the order is correct, the calculation will yield the same result. Conversely, if the order is different, the result will differ because the direction is crucial as the result of the cross product involved in the operation is a vector.
$$ \mathbf{A}\cdot (\mathbf{B} \times \mathbf{C} ) \neq \mathbf{A}\cdot (\mathbf{C} \times \mathbf{B} ) $$
$$ \mathbf{A}\cdot (\mathbf{B} \times \mathbf{C} ) \neq \mathbf{B}\cdot (\mathbf{A} \times \mathbf{C} ) $$
The scalar triple product can also be represented in the form of a determinant. In an orthogonal coordinate system, it looks as follows.
$$ \mathbf{A}\cdot (\mathbf{B} \times \mathbf{C} ) = \epsilon_{ijk}A_{i}B_{j}C_{k}=\begin{vmatrix} A_{i} & A_{j} & A_{k} \\ B_{i}&B_{j}&B_{k} \\ C_{i}&C_{j}&C_{k} \end{vmatrix}=\begin{vmatrix} A_{x} & A_{y} & A_{z} \\ B_{x}&B_{y}&B_{z} \\ C_{x}&C_{y}&C_{z} \end{vmatrix} $$