# Definition:Scalar Triple Product

## Definition

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

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

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

### Definition 1

The scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$ is defined and denoted as:

$\sqbrk {\mathbf a, \mathbf b, \mathbf c} := \mathbf a \cdot \paren {\mathbf b \times \mathbf c}$

where:

$\cdot$ denotes dot product
$\times$ denotes vector cross product.

### Definition 2

The scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$ is defined and denoted as:

$\sqbrk {\mathbf a, \mathbf b, \mathbf c} := \begin {vmatrix} a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \\ \end {vmatrix}$

where $\begin {vmatrix} \ldots \end {vmatrix}$ is interpreted as a determinant.

## Also known as

The scalar triple product is also known as the triple scalar product.

Some sources denote the scalar triple product as $\sqbrk {\mathbf a \mathbf b \mathbf c}$.

## Also see

• Results about scalar triple product can be found here.