Definition:Scalar Triple Product

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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.


Sources

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