Scalar Triple Product

The scalar triple product (sometimes called the box product) is an operation combining the Dot Product and the Cross Product for three Vectors. Since it involves the cross product, which only applies to vectors in 3D Real Space, ${\mathbb{R}}^3$, it too can only apply in 3D. It returns a scalar number.

One of the properties of the scalar triple product is that we can

Definition §

The most common way to obtain it is to represent all three vectors as row matrices and calculate the Determinant of the final matrix.

Formula §

$$\vec{u}\cdot (\vec{v}\times \vec{w})=\det (\left[\begin{array}{ccc}{u}_1& u_2& u_3\\ v_1& v_2& v_3\\ w_1& w_2& w_3\end{array}\right])$$

• $\vec{u},\vec{v},\vec{w}$ = Three vectors in ${\mathbb{R}}^3$
• $\det ()$ = Determinant operator

Geometry §

The scalar triple product can be geometrically seen as the signed volume of the parallelepiped defined by the three vectors:

$$\text{Volume}=|\vec{a}\cdot (\vec{b}\times \vec{c})|$$

Properties §

#todo https://en.wikipedia.org/wiki/Triple_product