Angle formed by summing $n$ unit vectors

Question

I'm interested in the angle formed by the sum of $n$ unit vectors. Said angle must be a function of the angles of the $n$ unit vectors.

Specifically, suppose that the $i$-th unit vector's angle is $\theta_i$. Then the angle of the sum is

$$f(\theta_1,\dots,\theta_n) = \tan^{-1}\Big(\frac{\sin(\theta_1) + \cdots + \sin(\theta_n)}{\cos(\theta_1) + \cdots + \cos(\theta_n)}\Big)$$

which is easy to solve for $n=1,2$

$$f(\theta_1) = \theta_1$$

$$f(\theta_1,\theta_2) = \frac{1}{2}(\theta_1 + \theta_2)$$

Can anyone solve for $n=3$? When

$$f(\theta_1,\theta_2,\theta_3) = \tan^{-1}\Big(\frac{\sin(\theta_1) + \sin(\theta_2) + \sin(\theta_3)}{\cos(\theta_1) + \cos(\theta_2) + \cos(\theta_3)}\Big)$$

Here's a graphical representation for $n=3$ (except the vectors are each about 40 units long) where an expression for the angle of the red line is desired.

And can one solve for $n$ in general?

Answer 1

The case of $n=2$ can simplify thanks to

$$e^{i\alpha}+e^{i\beta}=(e^{i(\alpha-\beta)/2}+e^{-i(\alpha-\beta)/2})e^{i(\alpha+\beta)/2}=2\cos(\alpha-\beta)e^{i(\alpha+\beta)/2}.$$

This simplification does not extend to $n=3$ because the symmetry is broken. In

$$e^{i\alpha}+e^{i\beta}+e^{i\gamma}=2\cos(\alpha-\beta)e^{i(\alpha+\beta)/2}+e^{i\gamma},$$ the first term has a non-unit coefficient.

I doubt that there is anything better than $$\arg(e^{i\alpha}+e^{i\beta}+e^{i\gamma}).$$

Answer 2

Let ${\bf u}_i=(\cos\theta_i,\sin\theta_i)$ $(1\leq i\leq n)$ be the given unit vectors. Then $${\rm s}:=\sum_{i=1}^n{\bf u}_i=(p,q),\qquad p:=\sum_{i=1}^n\cos\theta_i,\quad q:=\sum_{i=1}^n\sin\theta_i\ .$$ If ${\bf s}\ne{\bf 0}$ we simply have $${\rm arg}({\bf s})={\rm arg}(p,q)\ .$$ There is not more to it. The essential point is having the function $${\rm arg}:\quad\dot{\mathbb R}^2\to{\mathbb R}/(2\pi{\mathbb Z})$$ in your mathematical (and computational) toolkit. This function is one of the most important functions of multivariate analysis. It gives for each point $(x,y)\ne(0,0)$ the polar angle of $(x,y)$ modulo $2\pi$. When $x>0$ (and only then) one has $${\rm arg}(x,y)=\arctan{y\over x}+2k\pi\ .$$ On some pocket calculators this function is accessible as second output of the command ${\tt pol}$ (or similar), and certain outdated programming languages refer to it under ${\tt atan2}$, but with reversed order of the variables. In both cases a single real number is outputted. The waving away of the multivaluedness of ${\rm arg}$ is paid for by a discontinuity along the negative $x$-axis.