Let a = i + j + k, b =- i -8 j +2 k and c =4 i +c_2 j +c_3 k be three vectors such that b × a = c × a. If the angle between the vector c and the…

Let $\displaystyle \vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=-\hat{i}-8 \hat{j}+2 \hat{k}$ and $\displaystyle \vec{c}=4 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ be three vectors such that $\displaystyle \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$. If the angle between the vector $\displaystyle \overrightarrow{\mathrm{c}}$ and the vector $\displaystyle 3 \hat{i}+4 \hat{j}+\hat{k}$ is $\displaystyle \theta$, then the greatest integer less than or equal to $\displaystyle \tan ^2 \theta$ is $\displaystyle \_\_\_\_$.