Let $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ and $\hat{\mathrm{k}}$ be the unit vectors along the three positive coordinate axes. Let $$ \begin{aligned} & a=3 i+j-k, \\ & b=i+b_2 j+b_3 k, \\ & \rightarrow \\ & c=c_1 i+c_2 j+c_3 k, \\ & b_2, b_3 \in \mathbb{R}, \\ & c_1, c_2, c_3 \in \mathbb{R} \end{aligned} $$ be three vectors such that $b_2 b_3>0, \vec{a} \cdot \vec{b}=0$ and Then, which of the following is/are TRUE ?