Show that $||\vec x|-|\vec y||≤|\vec x-\vec y|$. When the equality holds?

Question

Show that $||\vec x|-|\vec y||≤|\vec x-\vec y|$. When the equality holds?

I can show the inequality. Please suggest when the equality holds?

Attempt: Equality holds if any one of the following conditions holds:

• if either $\vec x=\vec 0$ or $\vec y=\vec 0$,
• $\vec y=\mu\vec x$, $\mu\neq 0$