:PROPERTIES: :ID: e72ab120-e332-4c9d-bf88-ea02dcd4eb99 :mtime: 20231017010547 20231014012016 :ctime: 20231014011945 :END: #+title: scalar triple product #+filetags: :public:project: * Definition Given vectors $\vec{a}, \vec{b}, \vec{c}$, the scalar triple product is given by $\vec{a} \cdot (\vec{b} \times \vec{c})$. * Notation Sometimes, the scalar triple product is written as $\vec{a} \cdot (\vec{b} \times \vec{c}) = [\vec{a}, \vec{b} , \vec{c}]$. * How to Compute the Scalar Triple Product You can compute the scalar triple product using the [[id:3aadb278-8b51-4ce7-9ef2-8a27fae2349f][determinant of a matrix]]. \[\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y } & b_{z} \\ c_{x} & c_{y} & c_{z} \end{vmatrix}\] * Algebraic Properties of the Scalar Triple Product - If one or more of $\vec{a}, \vec{b}, \vec{c}$ are parallel, then $[\vec{a}, \vec{b}, \vec{c}] = 0$ - If you make a swap of $\vec{a}, \vec{b}, \vec{c}$, then the scalar triple product becomes negated. - $\vec{a} \cdot (\vec{b} \times \vec{c}) = |\vec{a}| |\vec{b}| |\vec{c}| \cos \theta$ * Volume of a Parallelepiped One physical intuition of the scalar triple product is that $[\vec{a}, \vec{b}, \vec{c}]$ represents the (signed) volume of a parallelepiped whose edges are parallel to $\vec{a}, \vec{b}$, and $\vec{c}$.