How to visualize Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$? pp 152 and 294,  Kuldeeph Singh's Linear Algebra: Step by Step (2013) illustrates Triangle Inequality. But without referring to Triangle Inequality at all, how to picture  Reverse Triangle Inequality for all $\mathbf{a,b} \in C^n$? I don't want proofs or rigour.

 A: Let do this in terms of vector ... 

sorry for bad drawing. I hope you will understand :D
ADDED: Here is clarification as you requested.
The length $|a| = AD = AE $ because they are radius of the same circle. And $|b| = AC$ so $||a|-|b|| = EC = AC - AE$
A: Draw two circles as follows:

Let $z_1$ be any complex number on the larger red circle and $z_2$ be any complex number on the smaller black circle. Note that $|z_1|$ and $|z_2|$ are the radii of the red circle and black circle respectively.
$\big||z_1| - |z_2|\big|$ (purple on the real axis) is always some positive number representing the absolute value of the difference between these radii.
$|z_1 - z_2|$ (blue) is a positive number representing the distance between $z_1$ and $z_2$ (some different choices for $z_1$ and $z_2$ are shown)
As you can see no matter how you choose the points on the red and black circle, the inequality
$$||z_1| - |z_2||≤|z_1 - z_2|$$ holds
The smallest distance between two points on two concentric circles is the difference between the circles's radii.
A: The intuition is that the inequality is equality whenever there's no cancellation happening. I'll keep this in the reals but the idea generalises. Suppose that $a,b > 0$. Then
\begin{equation}
| |a| - |b| | = |a - b|
\end{equation}
whereas, if $a > 0 > b$, then
\begin{align}
| |a| - |b| | &= | a + b | \leq \max\{|a|, |b|\} \leq \max\{a, -b\} \leq \max\{a, -b\} + \min\{-b, a\} = a - b \\ &= |a - b|
\end{align}
If you want to think more geometrically then $|a - b|$ is the size of the vector from $b$ to $a$ which is minimised when they're both pointing in the same direction and, at that minimum, is the LHS above. When they're not pointing in the same direction then $|a - b|$ is bigger but $||a| - |b||$ stays the same size (as it doesn't care about the angle between $a$ and $b$).
