If the volume of a parallelopiped, whose coterminous edges are given by the vectors a → = i ^ + j ^ +…

The volume of the parallelepiped is, $v=\left|\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]\right|$

$\Rightarrow \left|\begin{array}{ccc}1& 1& n\\ 2& 4& -n\\ 1& n& 3\end{array}\right|=\pm 158$

$\Rightarrow 1\left(12+n^2\right)-1(6+n)+n(2n-4)=\pm 158$

$\Rightarrow 3n^2-5n-152=0$ or $3n^2-5n+164=0$ 

($\because D<0$, no real roots of  $3n^2-5n+164=0$)

$n=8, -\frac{19}{3} \Rightarrow n=8$, as $n\ge 0$

then, $\overrightarrow{\mathrm{b}}·\overrightarrow{\mathrm{c}}=2+4\mathrm{n}-3\mathrm{n}=10$

$\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{c}}=1+n+3n=33$