# Definition:Scalar Triple Product

## Definition

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

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

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

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

• Results about scalar triple product can be found here.