Obtuse angle

An angle α $\alpha$ is called obtuse if it is greater than 90° and less than 180° (in degree measure), respectively if π $\pi /2<\alpha <\pi$ (in radians) applies.

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

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

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

If there is a symmetrical configuration of n+1 vectors in $\mathbb {R} ^{n}$ , then the angle φ $\varphi$ between each two (different) vectors is: φ $\varphi =\arccos \left(-{\frac {1}{n}}\right)$ .

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

From this we obtain directly the tetrahedron angle τ $\tau =\arccos \left(-{\frac {1}{3}}\right)\approx 109{,}47^{\circ }$ .

Obtuse angle

See also

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