Definition:Scalar Triple Product

Definition
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:

where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$.

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


 * $\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = \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 known as
The scalar triple product is also known as the triple scalar product.

The notation $\sqbrk {\mathbf a, \mathbf b, \mathbf c}$ can often be found.

Also see

 * Scalar Triple Product equals Determinant: justification for this definition