proove involving dot product and zero vector

Show that if $\displaystyle \vec u \cdot \vec v = \vec 0$ for all vectors $\displaystyle \vec v$ then $\displaystyle \vec u=\vec 0$
I know that 0 divided by v is 0, but how do I right that in terms of linear algebra?

It makes sense because any number multiplied with 0 will be 0, then it's just a sum of 0s, but I don't know how to give a formal proof

There is no statement. Your sentence makes as much sense as "If I take the bus."

Moreover, whatever you meant to say, you cannot divide by $\displaystyle \vec v$, it's a vector.

Hint : consider $\displaystyle \vec u \cdot \vec u$.