Question

If $\overrightarrow{a}$ is a non-zero vector of modulus $a$ and $m$ is a non-zero scalar then $m\overrightarrow{a}$ is a unit vector if

Answer 1

$m\overrightarrow{a}$ is a unit vector so its modulus is $1$.
Therefore
$|m\overrightarrow{a}|=1$ or
$|m||\overrightarrow{a}|=1$
$|\overrightarrow{a}|=\frac{1}{|m|}$
Hence the necessary condition is
$a=\frac{1}{|m|}$