IQ modulation and phasors: which first?
The fundamental motivation is: &quot;Arbitrary AM modulation and adding two sine waves together is easy, arbitrary phase modulation is hard.&quot;
Here's the details. When we send information on a carrier wave with frequency \$f\$ (radio, optical, etc.), we send this:
$$
S(t) = A(t)\sin(2\pi f t + \phi(t))
$$
Where we modulate \$A(t)\$ for amplitude modulation and modulate \$\phi(t)\$ for FM modulation (or phase modulation).
So right there we already have the phasor notation:
$$
S(t) = A(t)\sin(2\pi f t+\phi(t)) = \Im \left\{A(t)e^{i(2\pi f t+\phi(t))} \right\}\\
= \Im\left\{\hat{R}(t)e^{i 2\pi f t}\right\}
$$
where \$\Im\$ is the imaginary part of the expression, and \$\hat{R}(t)\$ is the complex phasor in magnitude and angle of the signal, as a function of time.
So why do we use IQ? Because it's hard to make circuits that can control things with phase. But it is easy to make circuits that add two amplitude modulated signals together. So rather than thinking &quot;phase + magnitude&quot; we can think &quot;sine + cosine&quot;:
So let's look at that:
$$
S(t)  = A(t)\sin(2\pi f t+\phi(t))\\
 = A(t)\cos(\phi(t))\sin(2\pi f t)+A(t)\sin(2\pi f t +\pi/2)\cos(\phi(t))\\
 = A(t)\cos(\phi(t))\sin(2\pi f t)+A(t)\cos(\phi(t))\cos(2\pi f t)\\
 = I(t)\sin(2\pi f t) + Q(t) \cos(2\pi f t)
$$
Where \$I(t)\$ and \$Q(t)\$ are simply AM modulation on top of a sine and cosine wave (formally, people write I(t) to include the carrier too, but here let's just think of it as the baseband AM modulation of the carrier). That's very easy to make with circuits, and opens up all these arbitrary modulation techniques. So you gave two examples: you can do high efficiency single-sideband communications (which you can do in analog as well) - but you can also do any type of crazy modulation scheme that would be impractical to implement using analog circuits and AM and FM alone. For example, QPSK https://en.wikipedia.org/wiki/Phase-shift_keying#Quadrature_phase-shift_keying_(QPSK) can now be extended easily to any arbitrary number of symbols throughout the IQ plane just as easily as doing 2 - and you can place them at optimal distances away from each other for minimum bit error rate.
AND you still get to do normal AM, FM, and whatever other modulation you want!
EDIT: Demodulation too! You can modulate and demodulate any arbitrary signal if you have I and Q, which is why software defined radios can pretty much do anything on the fly. You just amplitude modulate and add, or you just detect the relatively slow I(t) and Q(t) signals - which is really easy to do in a circuit - then process in software.