In physics we often have a complex vector $\vec E^*$ in 3 dimensions, which is used to keep track of what we really care about, which is the oscillating real vector $\vec E(t) \equiv \text{Re}[\vec E^* e^{i \omega t}]$ where $\omega$ and $t$ are real numbers and Re entails taking the real part of each of the vector components.
A question that came up is: What is the maximum value of $|E(t)|^2$, and what $t$ does that occur at?
For readers who prefer symbols to prose & context: For complex numbers $a,b,c$, is there a way to compute
$${\arg \max}_{\phi\in\mathbb{R}} \left[(\text{Re}[ae^{i\phi}])^2 + (\text{Re}[be^{i\phi}])^2 + (\text{Re}[ce^{i\phi}])^2\right]$$
which is more elegant / efficient than brute-force numerical maximization?
Write $E=E_r+E_ii$. You have the vector valued function $$f(t):={\rm Re}(Ee^{it})={\rm Re}((E_r+E_ii)(\cos(t)+i\sin(t)))=E_r\cos(t)-E_i\sin(t).$$ Note also that $$ f^{\prime}(t)=-(E_r\sin(t)+E_i\cos(t))\;. $$ Setting the derivative of the dot product $\vert f(t) \vert^2=f(t)\cdot f(t)$ equal to zero, you get $$2f^{\prime} (t)\cdot f(t)=0\Leftrightarrow f^{\prime}(t) \perp f(t)\;.$$ Bilinearity of the dot product gives $$0=f^{\prime}(t)\cdot f(t)=E_i\cdot E_r (\sin^2(t)-\cos^2(t))+\sin(t)\cos(t)(\vert E_i \vert^2-\vert E_r \vert^2)$$ so $$ E_i\cdot E_r (\sin^2(t)-\cos^2(t))=\sin(t)\cos(t)(\vert E_i \vert^2-\vert E_r \vert^2) $$ Using the double angle formula, $\sin(t)\cos(t)=\frac{1}{2}\sin(2t)$ and $\sin^2(t)-\cos^2(t)=-\cos(2t)$, you have $$ -2E_i\cdot E_r \cos(2t)=\sin(2t)(\vert E_i \vert^2-\vert E_r \vert^2) $$ Notice that if $E_r$ and $E_i$ are orthonormal, then $f^{\prime}(t)$ and $f(t)$ are perpendicular for all $t$. hence $\vert f(t) \vert$ is constant. Otherwise, we solve $$ \frac{2E_i\cdot E_r }{\vert E_r \vert^2-\vert E_i \vert^2}=\tan(2t)\Rightarrow t=\frac{1}{2}\tan^{-1}\Big(\frac{2E_i\cdot E_r }{\vert E_r \vert^2-\vert E_i \vert^2}\Big)+\frac{n\pi}{2} $$ Substituting this value of $t$ into $$\vert f(t)\vert^2=\vert E_r\vert^2\cos(t)^2-2E_r\cdot E_i\cos(t)\sin(t)+\vert E_i\vert^2\sin^2(t)$$ will give you the result (you have to check which values of $n$ correspond to maxima).