The triangle inequality states:
$|z_1 \pm z_2|\le |z_1|+|z_2|$
Equality holds only when the three points are collinear.
It's intuitively obvious for the equality to hold when all vectors are in same direction but I am unable to understand the case with $-$
For example, lets take two vectors $8 \hat{i} , - 3\hat{i}$. They are collinear. According to the triangle inequality:
$|8\hat{i} + (-3\hat{i})|= |8\hat{i|} + |-3\hat{i}|$
$\implies 5 = 11 ,$ which is clearly false.
Please tell me why this is giving the wrong result and how I should interpret the equality case of triangle inequality.
The necessary condition for this equality to work is that the origin $(0,0),z_1,z_2$ must be collinear and in that order, i.e., the origin must not be in between the line joining $z_1$ and $z_2$.
In your example, the origin is between $z_1$ and $z_2$, hence, this - $|8\hat{i} + (-3\hat{i})|= |8\hat{i|} + |-3\hat{i}|$ - fails. However, $|8\hat{i} - (-3\hat{i})|= |8\hat{i|} + |-3\hat{i}|$ works as intended because now, the origin $,-z_2,z_1$ are collinear without the origin being in between them.
I hope this helps!