$\begin{aligned} & \vec{a}=\hat{\imath}+\hat{\jmath}+k \\ & \overrightarrow{\mathrm{~b}}=\hat{\imath}+8 \hat{\jmath}+2 \mathrm{k} \\ & \overrightarrow{\mathrm{c}}=4 \hat{\imath}+\mathrm{c}_2 \hat{\jmath}+\mathrm{c}_3 \mathrm{k} \\ & \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}} \\ & (\vec{b}-\vec{c}) \times \vec{a}=0 \\ & \overrightarrow{\mathrm{~b}}-\overrightarrow{\mathrm{c}}=\lambda \vec{\alpha} \\ & \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}+\lambda \vec{\alpha} \\ & -\hat{\imath}-8 \hat{\jmath}+2 k=\left(4 \hat{\imath}+c_2 \hat{\jmath}+c_3 k\right)+\lambda(\hat{\imath}+\hat{\jmath}+k) \\ & \lambda+4=-1 \Rightarrow \lambda=-5 \\ & \lambda+\mathrm{c}_2=-8 \Rightarrow \mathrm{c}_2=-3 \\ & \lambda+\mathrm{c}_3=2 \Rightarrow \mathrm{c}_3=7 \\ & \overrightarrow{\mathrm{c}}=4 \hat{\imath}-3 \hat{\mathrm{j}}+7 \mathrm{k} \\ & \cos \theta=\frac{12-12+7}{\sqrt{26} \cdot \sqrt{74}}=\frac{7}{\sqrt{26} \cdot \sqrt{74}}=\frac{7}{2 \sqrt{481}} \\ & \tan ^2 \theta=\frac{625 \times 3}{49} \\ & {\left[\tan ^2 \theta\right]=38}\end{aligned}$