:PROPERTIES: :ID: ec07501d-0eb3-4433-82b4-1b37bbee6f21 :mtime: 20231017010618 20231014013223 :ctime: 20231014013142 :END: #+title: vector triple product #+filetags: :public:project: * Definition Given [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][vectors]] $\vec{a}, \vec{b}, \vec{c}$, the *vector triple product* is defined as $\vec{a} \times (\vec{b} \times \vec{c})$ * Properties of the vector triple product Note that $\vec{a} \times (\vec{b} \times \vec{c})$ must lie in the plane of $\vec{b}, \vec{c}$. * Warning about non-associativity WARNING: \[\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}\] * As Components Because $\vec{a} \times (\vec{b} \times \vec{c})$ must lie in the plane of $\vec{b}$ and $\vec{c}$, then we can write \[\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\] * Algebraic Properties of the vector triple product - By [[id:51f7af57-31ed-4224-ae23-3e8230d908e7][dot product]]: $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}$ * Example Problem 1 Calculate $(\vec{a} \times \vec{b}) \times (\vec{c } \times \vec{r})$ \[ (\vec{a} \times \vec{b}) \times (\vec{c } \times \vec{r}) = ((\vec{c} \times \vec{r}) \cdot \vec{a}) \vec{b} - ((\vec{c} \times \vec{r}) \cdot \vec{b}) \vec{a} \]