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+1vectors.

If there is a symmetrical configuration of n+1vectors 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 {\displaystyle \tau =\arccos \left(-{\frac {1}{3}}\right)\approx 109{,}47^{\circ }}.

See also

  • Types of angles

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