Doubt about an operation involving vectors Given $\vec{u}=(1;2;-1)$ and $\vec{a}=(1;-3;4)$, calculate $\vec{u}-\frac{3}{\vec{u}} . \vec{a}$.

Well, I think that the operation $\frac{3}{\vec{u}}$ is not defined, is it? In case it's possible, what should I do?
You are correct; there's no concept of dividing by a vector.

It may be a typo in the problem you're solving -- perhaps it should be $ \vec u - \frac3{\vec u\cdot \vec a}\cdot \vec a$ or $\vec u - \frac{3}{|\vec u|}\cdot\vec a $?

Or perhaps it is some idiosyncratic notation that you'll have to ask the person who gave you the problem about.
Perhaps
$\frac1{\vec{u}}
$
is a vector such that
$\frac1{\vec{u}}\times\vec{u}
=I 
$.