An orientation of some [[Triangle]] $\triangle PQR$ is some [[../Math/Vector|Vector]] $\vec n$ that is [[../Math/Orthogonal|Orthogonal]] to the [[Triangle]], in other words The [[Normal Vector|Normal]] $\vec n$ of the [[Planes|Plane]] $\alpha$ where $\triangle PQR \in \alpha$ in $\R^{n}, n>2$. An orientation [[../Math/Vector|Vector]] for a [[Triangle]] points away from the front face of the triangle. Note there are an [[Infinity|Infinite]] amount of [[../Math/Normal Vector|Normals]] for any given [[Triangle]], as any *positive* multiple of $\vec n$ To determinate what is the "Front" or "Back" of the [[Triangle]], we specify a cyclic ordering for the [[Triangle]] [[Vertex|Vertices]]. For some triangle $\triangle PQR$ with [[../Math/Counter-Clockwise|Counter-Clockwise]] ordering, the normal vector will follow a "*right hand rule*". >[!info] Right Hand Rule >When you point at the triangle, point your thumb of your right hand in the direction of the orientation [[../Math/Vector|Vector]] and curl your fingers [[../Math/Counter-Clockwise|Counter-Clockwise]], ![[../../00 Asset Bank/Pasted image 20250121091415.png|invert_Sepia]] >[!tip] [[Equivalent|Equivalence]] In respect to a [[../Math/Counter-Clockwise|Counter-Clockwise]] [[Triangle Orientation|Vertex Order]] $\circlearrowleft \ba{\triangle v_{1}v_{2}v_{3}}$: >$ \huge >\ang{v_{1},v_{2},v_{3}} \cong >\ang{v_{2},v_{3},v_{1}} \cong >\ang{v_{3},v_{1},v_{2}} >$ >[!example]- >$ P=\pa{4,0,-1}, Q=\pa{9,1,-2}, R=\pa{8,2,0}$ >To find an orientation $\vec n$ for $\triangle PQR$: >$ >\begin{align} >(Q-P)\times (R-P) = \vec n, \vec n \perp \alpha \subseteq \triangle PQR >\end{align}$