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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!

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    $|a - b| \ge |a| - |b|$ – 2017-01-22
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    @OpenBall - Interesting! Then my little calculation must contain a mistake somewhere, right? – 2017-01-22
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    Your conclusion seems right, but the thing that tell us she's wrong is $|a-b|+|b|\not\gt|a|$. – 2017-01-22
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    @Alec No your work is correct. It's just that you have used the triangle inequality, while that comment is about what is called the [reverse triangle inequality](https://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality). – 2017-01-22
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    @Alec The condition you verified is necessary, but not sufficient, so just because it's satisfied it doesn't imply that the triangle is possible. You need to verify the other necessary condition as well $|a-b| \ge \big||a|-|b|\big|$ which, in this case, fails. – 2017-01-22
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    @dxiv - Ah. Then my problem is reduced to proving the reverse triangle inequality for vectors. So far I've only seen it for real numbers. – 2017-01-22

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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)