Let $\mathbf u = [3,-1,2,5,0]$, and v = $[1,0,-2,1,4]$.
3u = $[9, -3, 6, 15, 0]$
2v = $[2, 0, -4, 2, 8]$
3u - 2v = $[9, -3, 6, 15, 0]$ - $[2, 0, -4, 2, 8]$ = $[7, -3, 10, -13, 8]$
$$\|[7, -3, 10, -13, 8]\| = \sqrt{(7)^2 + (-3)^2 + (10)^2 + (-13)^2 + (8)^2} = \sqrt{391}$$
This is one answer, and I think it is correct, but there is also a law that says:
$\|xu\| = |x|\cdot\|u\|$, where $x$ is a constant, $u$ is a vector ($\|3u\| = |3|\cdot\|u\|$)
So,
$$\|3u - 2v\| = 3\|u\| - 2\|v\| = 3\sqrt{39} - 2\sqrt{22}$$
These two answers do not agree. I think the second one is false, but then how is the law wrong in this case?
You're incorrect in the second portion, with your mistake being that you are assuming that $$||3u - 2v|| = ||3u|| + ||-2v||$$ This is not true, as it is not true that $||x+y||=||x||+||y||$. Instead, it should be said that $$||3u - 2v|| \le ||3u|| + ||-2v||$$ as shown here.