Does $|(a,b)|=1 \implies (a,b) \neq 0$?

Question

So $(a,b) \in \mathbb{R^2}$ and if $\sqrt{a^2+b^2} =1$ does this mean that the vector $(a,b) \neq (0,0)$?

In general is this statement true?

$(a,b) =(0,0) \iff \sqrt{a^2+b^2}=0?$

Going from left to right is trivial but right to left implication is not clear to me.

Thanks in advance.

Answer 1

Suppose $|(a,b)| = 1$. Then, $\sqrt{a^2 + b^2} = 1 \rightarrow a^2 + b^2 = 1$ so clearly $a, b \not= 0$.

You can follow a similar chain of logic for the implication $|(a,b)| = 0 \rightarrow (a,b) = 0$.

In general, it is a property of norms that $||x|| = 0 \leftrightarrow x = 0$.

Answer 2

$$\sqrt{a^2+b^2}=0\implies a^2+b^2=0. $$ Note that the square of a real number is always non-negative. So the only solution to $a^2+b^2=0$ is $a=b=0$.