Let $\overrightarrow n=x\hat i+y\hat j+z\hat k$
Given: $ |\overrightarrow n|=1$ $\Rightarrow\: x^2+y^2+z^2=1$........(i)
Given $\overrightarrow u=\hat i+\hat j,\overrightarrow v=\hat i-\hat j\:and\:\overrightarrow w=\hat i+2\hat j+3\hat k$
Also given that $\overrightarrow n.\overrightarrow u=\overrightarrow n.\overrightarrow v=0$
$\Rightarrow\:x+y=0\:\:and\:\:x-y=0$............(ii)
$\Rightarrow\:x=y=0$
Substituting the values of $x\:and\:y$ in (i) we get $z=\pm1$
$\therefore |\overrightarrow n.\overrightarrow w|=3$