---- - Let [[inner product space]] be an [[inner product space|inner product space]] over $\mathbb{R}$ or $\mathbb{C}$. > [!definition] Definition. ([[angle between vectors]]) > The **angle $\theta$ between nonzero vectors $x,y \in V$** w.r.t. $\langle \cdot, \cdot \rangle$ having induced [[norm]] $\|\cdot\|$ is defined by $\cos \theta := \frac{|\langle x,y \rangle |}{\|x\| \ \|y\|} \in [0, \frac{\pi}{2}).$ > [!justification] Motivation. > We can derive this property for the [[Euclidean inner product]] using [[Euclidean inner product and cos|Euclidean inner product and cos]]. Then we generalize it as a *definition*. ---- #### ---- #### 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 > ```