Vector addition magnitude

Question

Problem

Carrie tells you she has two vectors $\vec a, \vec b$ such that $$|\vec a| = 7, \\ |\vec b| = 2,\\ |\vec a - \vec b| = 4.$$

Why do you not believe her?

My thoughts

I went with the triangle inequality, even though it is defined for addition.

$|\vec a - \vec b| = \overbrace{|\vec a + -\vec b| \leq |\vec a| + |-\vec b|}^{\text{triangle inequality}} = |\vec a| + |\vec b| = 9$

Now, since $4\leq9$, I don't see the problem. Seems like Carrie's statements may well be true.

Question

Have I made some mistake in my calculation?

Any help appreciated!

Answer 1

Let's begin by stating the other side of the triangle inequality wich is:

$$|\ |\vec{a}|\ -|\vec{b}|\ |\leq|\vec{a}-\vec{b}| $$

Thus, $$|7-2|\leq 4 $$ wich is absurd!

In case you look for the demonstration of this inequality for any normed vector space you can take a look at this link Reverse triangle inequality (the demonstration is very simple)