Scalar Triple Product

The second triple product is the scalar product of two vectors, of which one is itself a vector product ... This sort of product has a scalar value and consequently is often called the scalar triple product.*

-E.B Wilson, *Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics Founded upon the Lectures of J. Willard Gibbs*, 1901

The scalar triple product of the vectors u , v , and w is denoted

$$ \left(\mathbf{u}\quad \mathbf{v}\quad \mathbf{w}\right) = \mathbf{u}\cdot\left(\mathbf{v}\times\mathbf{w}\right) $$

Engineering Context

One important context that is applicable to all disciplines of engineering is that the scalar triple product can be used to determine whether three vectors are linearly independent. Let's take the example of the unit vectors, î , ĵ , and k̂ . We know that these vectors are mutually orthogonal to one another and will therefore be linearly independent. Applying the scalar triple product, we find

$$ \begin{align} \left(\hat{i}\quad \hat{j}\quad \hat{k}\right) &= \hat{i}\cdot\left(\hat{j}\times\hat{k}\right)\\ &= \begin{bmatrix}1& 0& 0\end{bmatrix}\cdot\begin{vmatrix}\hat{i}& \hat{j}& \hat{k}\\ 0& 1& 0\\ 0& 0& 1\end{vmatrix}\\ &= \begin{bmatrix}1& 0& 0\end{bmatrix}\cdot\begin{bmatrix}1& 0& 0\end{bmatrix}\\ &=1 \end{align} $$

Since the scalar triple product of these vectors is non-zero, these three vectors are linearly independent.

MAE:

The relationship between vorticity and circulation in fluid mechanics is given by
$$ \left(\nabla \times \mathbf{V}\right) \cdot \mathbf{n} = -\frac{d\Gamma}{dS} $$

which is a scalar triple product.

The orbit equation for dynamics of space flight is derived using a scalar triple product as

$$ r + \overline{r}\cdot\frac{\left(\overline{\dot{r}} \times \overline{h}\right)}{\mu} = \frac{h^2}{\mu} $$

ECE:

BE:

CEE:

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