Let $\mathrm{\lambda }\in \mathrm{Z},\overrightarrow{\mathrm{a}}=\mathrm{\lambda }\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{j}}-\stackrel{\wedge }{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3\stackrel{\wedge }{\mathrm{i}}-\stackrel{\wedge }{\mathrm{j}}+2\stackrel{\wedge }{\mathrm{k}}.$ Let $\overrightarrow{\mathrm{c}}$ be a vector such that $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})\times \overrightarrow{\mathrm{c}}=\overrightarrow{0},\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{c}}=-17\mathrm{and}\overrightarrow{\mathrm{b}}.\overrightarrow{\mathrm{c}}=-20.$Then ${|\overrightarrow{\mathrm{c}}\times (\mathrm{\lambda }\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{j}}+\stackrel{\wedge }{\mathrm{k}})|}^2$ is equal to

$$\begin{gathered} (\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})\times \overline{\mathrm{c}}=\overline{0} \\ \overline{\mathrm{c}}=\mathrm{t}(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \\ \overline{\mathrm{c}}=\mathrm{t}(\mathrm{\lambda }\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{j}}-\stackrel{\wedge }{\mathrm{k}}+3\stackrel{\wedge }{\mathrm{i}}-\stackrel{\wedge }{\mathrm{j}}+2\stackrel{\wedge }{\mathrm{k}}) \\ \Rightarrow \overline{\mathrm{c}}=\mathrm{t}((\mathrm{\lambda }+3)\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{k}}) \\ \overline{\mathrm{a}}.\overline{\mathrm{c}}=-17\Rightarrow (\mathrm{\lambda }\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{j}}-\stackrel{\wedge }{\mathrm{k}}).\mathrm{t}((\mathrm{\lambda }+3)\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{k}})=-17 \\ \Rightarrow \mathrm{t}({\mathrm{\lambda }}^2+3\mathrm{\lambda }-1)=-17 .....\left(1\right) \\ \overline{\mathrm{b}}.\overline{\mathrm{c}}=-20\Rightarrow (3\stackrel{\wedge }{\mathrm{i}}-\stackrel{\wedge }{\mathrm{j}}+2\stackrel{\wedge }{\mathrm{k}=-20}).\mathrm{t}((\mathrm{\lambda }+3)\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{k}}) \end{gathered}$$
$\Rightarrow \mathrm{t}(3\mathrm{\lambda }+9+2)=-20 ....\left(2\right)$
$\frac{\mathrm{Eq}\left(1\right)}{\mathrm{Eq}\left(2\right)} \mathrm{gives} \frac{{\mathrm{\lambda }}^2+3\mathrm{\lambda }-1}{3\mathrm{\lambda }+11}=\frac{17}{20}$
$$\begin{gathered} \Rightarrow 20{\mathrm{\lambda }}^2+60\mathrm{\lambda }-20=51\mathrm{\lambda }+187 \\ \Rightarrow 20{\mathrm{\lambda }}^2+9\mathrm{\lambda }-207=0 \\ \Rightarrow \mathrm{\lambda }=3,\frac{-69}{20}\Rightarrow \mathrm{\lambda }=3 \\ \Rightarrow \mathrm{t}(9+9-1)=-17 \mathrm{t}=\frac{-17}{17}=-1 \\ \therefore {\left|\overline{\mathrm{c}}\times \left(\mathrm{\lambda }\stackrel{\wedge }{\mathrm{i}}+\stackrel{\wedge }{\mathrm{j}}+\stackrel{\wedge }{\mathrm{k}}\right)\right|}^2={\left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -6& 0& -1\\ 3& 1& 1\end{array}\right|}^2 \\ ={|\stackrel{\wedge }{\mathrm{i}}-3\stackrel{\wedge }{\mathrm{j}}-6\stackrel{\wedge }{\mathrm{k}}|}^2 \\ =1+9+36=46 \end{gathered}$$