Editing Angle (section)

==Dot product and generalisations{{anchor|Dot product}}==
In the [[Euclidean space]], the angle ''θ'' between two [[Euclidean vector]]s '''u''' and '''v''' is related to their [[dot product]] and their lengths by the formula

<math display="block"> \mathbf{u} \cdot \mathbf{v} = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their [[normal vector]]s and between [[skew lines]] from their vector equations.

===Inner product===
To define angles in an abstract real [[inner product space]], we replace the Euclidean dot product ( '''·''' ) by the inner product <math> \langle \cdot , \cdot \rangle </math>, i.e.

<math display="block"> \langle \mathbf{u} , \mathbf{v} \rangle = \cos(\theta)\ \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

In a complex [[inner product space]], the expression for the cosine above may give non-real values, so it is replaced with

<math display="block"> \operatorname{Re} \left( \langle \mathbf{u} , \mathbf{v} \rangle \right) = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

or, more commonly, using the absolute value, with

<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> spanned by the vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> correspondingly.

===Angles between subspaces===
The definition of the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> given by

<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| </math>

in a [[Hilbert space]] can be extended to subspaces of finite dimensions. Given two subspaces <math> \mathcal{U} </math>, <math> \mathcal{W} </math> with <math> \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l </math>, this leads to a definition of <math>k</math> angles called canonical or [[principal angles]] between subspaces.

===Angles in Riemannian geometry===
In [[Riemannian geometry]], the [[metric tensor]] is used to define the angle between two [[tangent]]s. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'',

<math display="block">
\cos \theta = \frac{g_{ij} U^i V^j}{\sqrt{ \left| g_{ij} U^i U^j \right| \left| g_{ij} V^i V^j \right|}}.
</math>

===Hyperbolic angle===
A [[hyperbolic angle]] is an [[argument of a function|argument]] of a [[hyperbolic function]] just as the ''circular angle'' is the argument of a [[circular function]]. The comparison can be visualized as the size of the openings of a [[hyperbolic sector]] and a [[circular sector]] since the [[area]]s of these sectors correspond to the angle magnitudes in each case.<ref>[[Robert Baldwin Hayward]] (1892) [https://archive.org/details/algebraofcoplana00haywiala/page/n5/mode/2up ''The Algebra of Coplanar Vectors and Trigonometry''], chapter six</ref> Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as [[infinite series]] in their angle argument, the circular ones are just [[alternating series]] forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by [[Leonhard Euler]] in ''[[Introduction to the Analysis of the Infinite]]'' (1748).