Is this statement about vectors true?

Question

If vectors $A$ and $B$ are parallel, then, $|A-B| = |A| - |B|$

Is the above statement true?

Answer 1

Let $A$ and $B\neq 0$ be vectors in $\mathbb R^{n}$ and let us consider $B$ as a vector in the positive ray of $A$, i.e. $A=\lambda B$, with $\lambda\in \mathbb R$ ($\lambda> 0$). Then your identity is reduced to

$|\lambda-1|=|\lambda|-1$, which is not true in general if $0<\lambda<1$.