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So I get that the norm of a vector is the square root of each of its components squared. I can do that just fine, but I think my book may be confusing me with a wrong answer and there is only one example so I don't know if I am doing it the right way.

Let $u=(-1,3,4)$, $v=(2,1,-1)$, $W=(-2,-1,3)$. Find the norm/length of $3u-v+2w$. The textbook says the answer is $\sqrt{478}$. I just have no idea where this is from.

My Solution: $3u=(-3,9,12)$, $-v=(-2,-1,1)$, $2w=(4,2,-2)$ So, $3u-v+2w =(-1,10,11)$. I should just take the norm of this right? That gives me $\sqrt{222}$.

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    When you tried to compute $2\vec w$ you computed $2\vec v$ instead. – 2017-02-04
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    I spent an hour on this, I feel bitter. But thank You everyone! – 2017-02-04

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Check your value of $3u-v+2w$. It should equal $\langle -9, 6, 19 \rangle$.