Vector angles are the usual oriented angles of Euclidean plane geometry. Thanks to the resources of naive set theory, they can be defined purely algebraically using an equivalence relation and the vectorial rotations of the plane. The operation of composing rotations is then transported to the addition of angles.

1 Orientated angles of half-lines: vector angles

1.1 The intuition of angles with an orientation

In The Trigonometric Circle, we considered oriented angles intuitively, without defining them rigorously. An oriented angle is intuitively an ‘opening’ between two segments (for example in a figure) or, in general, two half-lines, based at the same point. However, this opening is considered with its orientation, clockwise or anti-clockwise: if \(D_1\) and \(D_2\) are two half-lines based at the same point \(M\), the oriented angle between \(D_1\) and \(D_2\) is not the same as the oriented angle between \(D_2\) and \(D_1\). An oriented angle being determined by two half-lines, by a translation we can always consider that the half-lines are based at the origin of the plane, the point \(O=(0,0)\).

1.2 Replacing half-lines with vectors

Now, a pair of half-lines based at $O$ is determined by the respective points of intersection $M_1$ and $M_2$ of $D_1$ and $D_2$ with the circle $S^1$. But points in the Euclidean plane can also be thought of as vectors, so the trigonometric circle $S^1$ is both the set of points $M$ in $\mathbb R^2$ such that $OM=1$ and the set of unit vectors in the plane. A vector $\vec u=(x,y)\in\mathbb R^2$ is said to be unitary if it has (Euclidean) norm $1$, i.e. if $||\vec u||=\sqrt{x^2+y^2}=1$. The pair $(D_1,D_2)$ can therefore be replaced by the two unit vectors $\vec{u_1}=M_1$ and $\vec{u_2}=M_2$. In other words, we could have defined the oriented angles as angles between two unit vectors, and this is the point of view we are now adopting.

1.3 Principle of a definition by equivalence relation

Mathematically, oriented angles are rigorously defined by an equivalence relation, as for the construction of $\mathbb Z$ from $\mathbb N$, of $\mathbb Q$ from $\mathbb Z$ or of $\mathbb R$ from $\mathbb Q$. Since an oriented angle is determined by two unit vectors, we consider that a pair $(\vec u,\vec v)$ of such vectors represents such an angle. But two pairs of vectors $(\vec u,\vec v)$ and $(\vec{u’‘’},\vec{v’’’})$ can represent the same angle, when intuitively the angle between $\vec u$ and $\vec v$ is the same as the angle between $\vec{u’}$ and $\vec{v’}$. Precisely, this is the case exactly when there is a rotation which transforms one into the other. We therefore consider oriented angles as equivalence classes of pairs of unit vectors.

1.4 A rigorous mathematical definition

Let us now give an exact definition of oriented angles or vector angles. Consider the set $E$ of all pairs $(\vec u,\vec v)$ of unit vectors. We define an equivalence relation on $E$ by stating that two elements of $E$, i.e. two pairs $(\vec u,\vec v)$ and $(\vec{u‘},\vec{v’})$ of unit vectors are equivalent if there exists a vector rotation $r$ which ‘transforms $\vec u$ into $\vec {u’}$ and $\vec v$ into $\vec{v‘}$, i.e. such that $r(\vec u)=\vec {u’}$ and $r(\vec v)=\vec {v’}$. An oriented angle or angle of vectors is then, by definition, an equivalence class of pairs $(\vec u,\vec v)$ of unit vectors for this relation.
In other words, it is the set of pairs $(\vec{u’},\vec{v’})$ in $E$ equivalent to a given pair $(\vec u,\vec v)$. We denote $[(\vec u,\vec v)]$ such an angle, and any of these pairs is a representation of the angle $[(\vec u,\vec v)]$.

2 The cosine and sine of a vector angle

2.1 Standard representation of a vector angle

If $\vec\alpha=[(\vec u,\vec v)]$ is a vector angle, there are by definition many representations of it. Among these, a pair can be distinguished, which is of the form $(\vec i,\vec {v’})$, where $(\vec i$) is the vector $(1,0)$ which directs the half-line of positive abscissas. This is because the pair $(\vec u,\vec v)$ is a representation of $\vec\alpha$, and there is always a unique rotation that sends any point of $S^1$ onto any other given point of $S^1$, so a unit vector onto a given unit vector. Let’s consider the rotation $r$ which sends $\vec u$ onto $\vec i=(0,1)$. This rotation transforms $\vec v$ into a unit vector $\vec{v’}$, and so we have $\vec\alpha=[(\vec i,\vec{v’})]$. We will call $(\vec i,\vec{v’})$ the standard representation of the (oriented) angle $\vec\alpha$.

2.2 Circular coordinates of a vector angle

Given a vector angle $\alpha$, we can now choose a standard representation of it, in the form of the pair $(\vec i\vec v)$, with $\vec i=(1,0)$ and $\vec v=(a,b)$ say. Since, by definition, the vector $\vec v$ has norm $1$, it also determines a point on the trigonometric circle, so let’s call it $M=(a,b)$ to emphasise the duality between points and vectors. But with the trigonometric circle we have introduced the circular coordinates of an oriented angle as that of the point it determines on this circle. In the same way, we can define the circular coordinates of an angle of vectors as the coordinates of the point that its standard representation determines on the trigonometric circle: with the present notations, by definition the cosine of $\alpha$, denoted $\cos\alpha$, is the abscissa of $M$, i.e. the real number $a$, while the sine of $\alpha$, denoted $\sin\alpha$, is by definition the ordinate of $M$, i.e. the real number $b$. The cosine and sine of an oriented angle can therefore be obtained, from any representation, by means of a rotation which allows the standard representation to be identified.

2.3 Scalar product and determinant

Given the vector angle $\alpha=[(\vec u,\vec v)]$, with $\vec u=(a,b)$ and $\vec v=(c,d)$ unitary, how can its cosine and sine be determined? This can be done in a purely analytical way using the standard representation $\alpha=[(\vec i,\vec {v’})]$, with $\vec {v’}=r(\vec i)$, $r$ being the only rotation that sends $u$ onto $v$. The rotation $r$ being of the form $(x,y)\in\mathbb R^2\mapsto (\lambda x-\mu y,\lambda y+\mu x)$ for unique real numbers $\lambda,\mu$ (see Vector rotations of the plane), these coefficients must satisfy $r(\vec u)=\vec v$, i.e. the following system in $\lambda$ and $\mu$: \begin{eqnarray} \lambda a-\mu b &= &c\\
\lambda b+\mu a&=&d,\end{eqnarray} whose only solution is $(\lambda=ac+bd,\mu=ad-bc)$. Now, we recognise here the scalar or dot product $\vec u\cdot\vec v=ac+bd$ and the determinant $det(\vec u,\vec v)=ad-bc$ of the vectors $\vec u$ and $\vec v$ ! In other words, to calculate the cosine and sine of an angle of vectors $[(\vec u,\vec v)]$, all you have to do is calculate their dot product and determinant.

2.4 Vector rotations and oriented angles

The very construction of vector angles shows that oriented angles and vector rotations are closely related: they are in fact essentially the same objects, presented differently, which is what group theory allows us to conceptualise. It suffices here to point out that rotations ‘preserve angles’: if $r$ is a vector rotation, and if $\alpha=[(\vec u,\vec v)]$ is a vector angle, then the vector angle $[(r(\vec u),r(\vec v)]$ is still the angle $\alpha$: this is true precisely by definition of vector angles, which were in fact conceived in this way from the idea of rotation.

3 The group of vector angles

3.1 Bijection between rotations and angles

There is a bijection between the set $\mathcal R$ of vector rotations of the plane and the trigonometric circle $\mathcal S^1$ (see Vector rotations of the plane: the ‘analytical’ approach), which associates to a rotation $r$ described as $r(x,y)=(ax-by,bx+ay)$ the point or vector $(a,b)$. It is therefore also possible to associate a vector angle with such a rotation: by considering the vector $\vec u=r(\vec i)$, or $\vec u=(a,b)$ (the point of $S^1$ determined by $r$…) we obtain the angle $[(\vec i,\vec u)]$. From the bijection between $\mathcal R$ and $S^1$, we can easily demonstrate that we obtain a new bijection $f$ between the set of vector rotations and the set $\mathcal A$ of vector angles. If $r$ is a rotation, the corresponding angle, $f(r)$, is by definition the angle of the rotation $r$.

3.2 Addition and subtraction of vectors

Now, geometric intuition tells us that we can add and subtract vector angles. To define this addition rigorously, we can go back to the definition and use equivalence classes. However, it is possible to proceed by means of what is known as ‘structure transport’. Since the set $\mathbb R$ of plane vector rotations is a commutative group, we can use the bijection $f$ between $\mathcal R$ and $\mathcal A$ to define the addition of angles. If $\vec\alpha$ and $\vec\beta$ are two oriented angles, there exist two uniquely determined rotations $r$ and $s$ of angles $\vec\alpha$ and $\vec\albeta$ respectively, i.e. such that $f(r)=\vec\alpha$ and $f(s)=\vec\beta$. We then define the sum $\vec\alpha+\vec\beta$ as the angle of the compound rotation $r\circ s$ (do $s$, then $r$), i.e. $f(r\circ s)$. Since the rotations ‘’commute‘ with each other (the rotations $r\circ s$ and $s\circ r$ are equal), this sum is also $f(s\circ r)$, i.e. $\vec\beta+\vec\alpha$.
The inverse transformation of a rotation is a rotation, and so the opposite of $\vec\alpha$ is the angle, $-\vec\alpha$, of the rotation $r^{-1}$. This means that angles can be subtracted: $\vec\alpha-\vec\beta=\vec\alpha +(-\vec\beta)$. The zero angle, denoted $\vec 0$ and equal to $[(\vec u,\vec u)]$ for any unit vector $\vec u$, is that of the identity of the plane, and we always have $\vec 0+\vec 0=\vec 0+\vec alpha=\vec alpha$ and $\vec alpha+(-\vec alpha)=\vec 0$. In other words, the null angle is the ‘zero’ of the addition of angles, and the set $\mathcal A$ is thus a commutative group. By construction, the bijection $f$ between $\mathcal R$ and $\mathcal A$ is then a group isomorphism. This gives us a third representation of the same essential mathematical object: the group $\mathcal R$ of plane vector rotations, the group $S^1$ of complex numbers of modulus $1$ or trigonometric circle, and the group $\mathcal A$ of vector angles.

4 Conclusion: the concept of angle is algebraic

As we noted in connection with vector rotations in the plane, these are defined purely algebraically (the use of the ‘analytic’ method, using coordinates, should not be confused with real ‘analysis’ as the theory of functions from $\mathbb R$ into $\mathbb R$). The attentive reader will have noticed that the definition of the vector angles, as well as their operations, only uses as a ‘structure’ the duality between algebra (vectors) and geometry (points) in the plane, and vector rotations.
This means that, as in the case of vector rotations, everything that has been done here can be reproduced in the plane $K^2$ defined on a sub-field $K$ of $\mathbb R$. Thus, contrary to the usual intuition of angles, which are conceived as ‘continuous’ objects, the modern rigorous conceptualisation of angles in Euclidean geometry, possible here thanks to naive set theory, does not involve real analysis. Measuring these angles, on the other hand, requires the use of trigonometric functions, but that is another story.