Let î, ĵ and k̂ be the unit vectors along the three positive coordinate axes. Let

\(\begin{array}{l} \vec{a}=3 \hat{i}+\hat{j}-\hat{k}, \\ \vec{b}=\hat{i}+b_{2} \hat{j}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R}, \\ \vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}, c_{1}, c_{2}, c_{3} \in \mathbb{R} \end{array}\)

be three vectors such that \(b_{2} b_{3}>0, \vec{a} \cdot \vec{b}=0\) and 

\(\left(\begin{array}{ccc} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{array}\right)\left(\begin{array}{l} 1 \\ b_{2} \\ b_{3} \end{array}\right)=\left(\begin{array}{c} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{array}\right) .\)

Then, which of the following is/are TRUE?