Definition:Scalar Triple Product

Definition
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a vector space of $3$ dimensions:


 * $\mathbf a = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k$
 * $\mathbf b = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k$
 * $\mathbf c = c_i \mathbf i + c_j \mathbf j + c_k \mathbf k$

where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of the vector space in question.

The scalar triple product, denoted as $\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right)$, is defined as:


 * $\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right) = \begin{vmatrix}

a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \\ \end{vmatrix}$

where:
 * $\begin{vmatrix} \ldots \end{vmatrix}$ is interpreted as a determinant
 * $\mathbf a \cdot \mathbf b$ denotes dot product
 * $\mathbf a \times \mathbf b$ denotes vector cross product.

Also see

 * Scalar Triple Product equals Determinant: justification for this definition