$$\begin{gathered} \mathrm{Let} \overline{\mathrm{a}} =\hat{\mathrm{i}} +2\hat{\mathrm{j}}+4\hat{\mathrm{k}} ,\overline{\mathrm{b}}=\hat{\mathrm{i}}+\mathrm{\lambda }\hat{\mathrm{j}}+4\hat{\mathrm{k}} \mathrm{and} \\ \overline{\mathrm{c}} =2\hat{\mathrm{i}}+4\hat{\mathrm{j}}+\left({\mathrm{\lambda }}^2-1\right)\hat{\mathrm{k}} \mathrm{be} \mathrm{copl}\overline{\mathrm{b}}\mathrm{anar} \mathrm{vectors}. \\ \mathrm{Then} \mathrm{the} \mathrm{non} -\mathrm{Zero} \mathrm{vector} \overline{\overline{\mathrm{a}}}\times \overline{\mathrm{c}} \mathrm{is} : \end{gathered}$$

$$\begin{gathered} \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right]=0 \Rightarrow \left|\begin{array}{ccc}1& 2& 4\\ 1& \mathrm{\lambda }& 4\\ 2& 4& {\mathrm{\lambda }}^2-1\end{array}\right|=0 \\ 2(8-4\mathrm{\lambda })-4(4-4)+({\mathrm{\lambda }}^2-1)(\mathrm{\lambda }-2)=0 \\ -8(\mathrm{\lambda }-2)+({\mathrm{\lambda }}^2-1)(\mathrm{\lambda }-2)=0 \\ (\mathrm{\lambda }-2)({\mathrm{\lambda }}^2-9)=0 \Rightarrow \mathrm{\lambda }=2,\pm 3 \\ \overline{\mathrm{a}}\times \overline{\mathrm{c}}=\left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 2& 4\\ 2& 4& {\mathrm{\lambda }}^2-1\end{array}\right| \\ =\mathrm{i}(2{\mathrm{\lambda }}^2-18)-\mathrm{j} ({\mathrm{\lambda }}^2-9)+\mathrm{k}(4-4) \\ \mathrm{\lambda }=2 \Rightarrow \overline{\mathrm{a}}\times \overline{\mathrm{c}}=-10\mathrm{i}+5\mathrm{j} \end{gathered}$$