Why $|j\omega|=\omega$

1. Length of $j$

$j$ is an arrow pointing up with length 1. We square the numbers on the axes, not $j$ itself:

$$j=0+1\cdot j\;\Rightarrow\;|j|=\sqrt{0^2+1^2}=1$$

Two turns of $90^\circ$, $180^\circ$ in total, so the arrow points left:

$$j\cdot j=j^2=-1$$

$$j^2=-1\;(\text{rotation})\qquad\qquad |j|^2=1\;(\text{length})$$

$$|j\omega|=|j|\cdot\omega=1\cdot\omega=\omega$$

$$\left|\frac{1}{j\omega}\right|=\frac{1}{\omega},\qquad \varphi=-90^\circ$$