I am trying to show if the following statements are true or false.

1. Is it true that $|a + b| = |a| + |b|$ for general vectors $a$ and $b$?
2. If $a \cdot b = a \cdot c$ for equally-sized non-zero vectors $a$, $b$, $c$, does it follow that $b = c$?

For the first one I found a counterexample that shows that the statement is false.

If vector $a=\langle 1,4,5\rangle$ and $b=\langle 2,2,2\rangle$ then $|a|+|b|=\sqrt{42}$+$2\cdot\sqrt{3}=9.945$, and then, $|a+b|=\sqrt{1^2+4^2+5^2+2^2+2^2+2^2}=7.348$,

then we can conclude that $9.945 \ne7.348$ and the statement is false. Also by the triangle identity $|a + b| \le |a| + |b|$

For the second statement I also found an counterexample that proves that is false.
If vector $a=\langle 1,0,0\rangle$, $b=\langle 0,1,0\rangle$ and $c=\langle 0,0,1\rangle$, we will obtain the following dot product:

$a \cdot b = 0$

$a \cdot c = 0$ 

then $a \cdot b = a \cdot c = 0$ and $b \ne c$ 

My question is: 

To prove the two statements is it enough to find a counterexample and say if it is true or false. Or should I try to provide a more mathematical proof like induction or contradiction?