Euclidean space
Angle

The (non-oriented) angle θ between two nonzero vectors x and y in E → {\displaystyle {\overrightarrow {E}}} is

θ = arccos (x ⋅ y | x | | y |) {\displaystyle \theta =\arccos \left({\frac {x\cdot y}{|x|\,|y|}}\right)}

where arccos is the principal value of the arccosine function. By Cauchy–Schwarz inequality, the argument of the arccosine is in the interval. Therefore θ is real, and 0 ≤ θ ≤ π (or 0 ≤ θ ≤ 180 if angles are measured in degrees).

Angles are not useful in a Euclidean line, as they can be only 0 or π.

In an oriented Euclidean plane, one can define the oriented angle of two vectors. The oriented angle of two vectors x and y is then the opposite of the oriented angle of y and x. In this case, the angle of two vectors can have any value modulo an integer multiple of 2 π. In particular, a reflex angle π < θ < 2 π equals the negative angle − π < θ − 2 π < 0.

The angle of two vectors does not change if they are multiplied by positive numbers. More precisely, if x and y are two vectors, and λ and μ are real numbers, then

angle (λ x, μ y) = { angle (x, y) if λ and μ have the same sign π − angle (x, y) otherwise. {\displaystyle \operatorname {angle} (\lambda x,\mu y)={\begin{cases}\operatorname {angle} (x,y)\qquad \qquad {\text{if }}\lambda {\text{ and }}\mu {\text{ have the same sign}}\\\pi -\operatorname {angle} (x,y)\qquad {\text{otherwise}}.\end{cases}}}

If A, B, and C are three points in a Euclidean space, the angle of the segments AB and AC is the angle of the vectors A B → ({\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {(}{}}}} and A C →. ({\displaystyle {\overrightarrow {AC}}.{\vphantom {\frac {(}{}}}} As the multiplication of vectors by positive numbers do not change the angle, the angle of two half-lines with initial point A can be defined: it is the angle of the segments AB and AC, where B and C are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point.

The angle of two lines is defined as follows. If θ is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either θ or π − θ. One of these angles is in the interval, and the other being in. The non-oriented angle of the two lines is the one in the interval. In an oriented Euclidean plane, the oriented angle of two lines belongs to the interval.
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