Angle at a point

Builds on Angle between two vectors Euclidean space

Definition

Let $A$ , $B$ and $C$ be points of a Euclidean space $\mathbb{R}^n$ with $A \neq B$ and $C \neq B$ . The angle $\angle ABC$ is the angle between the vectors $A - B$ and $C - B$ . Thus it is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that $\cos \theta = \frac{(A - B) \cdot (C - B)}{\|A - B\| \, \|C - B\|}.$ The point $B$ is the vertex of the angle.

Remarks

The angle is symmetric: $\angle ABC = \angle CBA$ . The angle is $\pi$ if and only if $B$ lies strictly between $A$ and $C$ on a line. It is $0$ if and only if $A$ and $C$ are on the same ray from $B$ . Mathlib's EuclideanGeometry.angle uses the same formula for all points. It gives $\pi / 2$ when $A = B$ or $C = B$ . The angle is a right angle when $\theta = \pi / 2$ , acute when $\theta < \pi / 2$ and obtuse when $\theta > \pi / 2$ .

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Tag	`0ZDDNBQSC4`
URI	`m2://0ZDDNBQSC4`
Kind	definition
Writing	W0 stub
MSC	51M05
Defines	Angle at a point
Uses	Angle between two vectors, Euclidean space

Angle at a point

Definition

Remarks

Reading path

Further reading

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