As the given vectors are coplanar, so $\left|\begin{array}{ccc}2& 1& 3\\ 3& 3& 1\\ c_1& c_2& c_3\end{array}\right|=0$

$\Rightarrow 8c_1-7c_2-3c_3=0 ...\left(i\right)$

Also, $\overrightarrow{a}·\overrightarrow{c}=5\Rightarrow 2c_1+c_2+3c_3=5 ...\left(ii\right)$

and $\overrightarrow{b}·\overrightarrow{c}=0\Rightarrow 3c_1+3c_2+c_3=0 ...\left(iii\right)$

Solving the above three equations using Cramer's rule, we get,

$∆=\left|\begin{array}{ccc}8& -7& -3\\ 2& 1& 3\\ 3& 3& 1\end{array}\right|=-122$

${∆}_1=\left|\begin{array}{ccc}0& -7& -3\\ 5& 1& 3\\ 0& 3& 1\end{array}\right|=-10$

${∆}_2=\left|\begin{array}{ccc}8& 0& -3\\ 2& 5& 3\\ 3& 0& 1\end{array}\right|=85$

${∆}_3=\left|\begin{array}{ccc}8& -7& 0\\ 2& 1& 5\\ 3& 3& 0\end{array}\right|=-225$

Hence, $122\left(c_1+c_2+c_3\right)=122\left(\frac{10}{122}-\frac{85}{122}+\frac{225}{122}\right)=150$