:PROPERTIES: :ID: dded15e1-eb8c-4576-8c9f-818ea7b80ac1 :mtime: 20231017010157 20231007013244 :ctime: 20231007013234 :END: #+title: Vector Equation for a Line #+filetags: :public:project: * 2 Vector Form Given 2 vectors $\vec{a}, \vec{b}$, we can express the line as $\forall \lambda \in \mathbb{R} \qquad \vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a})$ where this line will go through the points given by position vectors $\vec{a}, \vec{b}$. ** Warning $\vec{a}$ and $\vec{b}$ should be distinct. * Component Form of a Line $\frac{x-a_{x}}{b_x-a_{x}}=\frac{y-a_{y}}{b_y-a_{y}}=\frac{z-a_{z}}{b_z-a_{z}}$ * Cross Product Equation $(\vec{r} - \vec{a}) \times (\vec{b} - \vec{a}) = \vec{0}$