Definition:Scalar Triple Product

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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.