Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* ---- > [!definition] Definition. ([[orientation of euclidian space]]) > The 'positive' **orientation** of $\rrn$ is the collection of [[frame|right-handed n-frames]] in $\rrn$. The 'negative' **orientation** of $\rrn$ is the collection of [[frame|left-handed n-frames]] in $\rrn$. > [!generalization] > [[orientation of a vector space]] ^generalization > [!basicexample] > **$n=1$**. Every [[frame]] has the form $a \in \rr$, $a \neq 0$. And $\det a =a$, hence the [[orientation of euclidian space|positive orientation of]] $\rrn$ is merely $\rr_+$, while the negative [[orientation of euclidian space|orientation]] is merely $\rr_-$. \ $n=2$. Consider a $2$-frame $(a,b)$ where $a,b \in \rr^{2}$. ![[CleanShot 2023-01-03 at 17.38.21.jpg|200]] $\det [a_{1} \ \ \ a_{2}] > 0$ (and hence $(a,b)$ is right-handed) **iff** one must rotate $a_{1}$ counterclockwise through an angle less than $\pi$ to obtain a [[vector]] in the direction of $a_{2}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```