Obtuse angle

An angle is called obtuse if it is larger than 90 ° and smaller than 180 ° (in degrees ), or if it applies (in radians ). ${\displaystyle \alpha }$ ${\displaystyle \pi /2<\alpha <\pi }$

In linear algebra , a family of vectors is called obtuse if the angle between any two of these (different) vectors is obtuse. The formal definition is as follows:

Let be a family of vectors and the standard scalar product . Then S is called obtuse , if for . ${\displaystyle S=\{v_{1},v_{2},...,v_{k}\}\subset \mathbb {R} ^{n}}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \langle v_{i},v_{j}\rangle <0}$ ${\displaystyle 1\leq i<j\leq k}$

It can be shown that an obtuse-angled family can contain at most vectors. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n+1}$

If a symmetrical configuration of vectors in the front, the following applies for the angle (different) vectors between two: . ${\displaystyle n+1}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi =\arccos \left(-{\frac {1}{n}}\right)}$

In the case, for example, a symmetrical configuration of four vectors of equal length describes a regular tetrahedron . ${\displaystyle n=3}$

The tetrahedron angle is obtained directly from this . ${\displaystyle \tau =\arccos \left(-{\frac {1}{3}}\right)\approx 109{,}47^{\circ }}$

See also

• Types of angles