Let i, j ext and k be the unit vectors along the three positive coordinate axes. Let a =3…

Let \(\hat{i, j} \ ext { and } \hat{k}\) be the unit vectors along the three positive coordinate axes. Let
\(\vec{a}=3 \hat{i}+\hat{j}-\hat{k}\)\(\vec{b}=\hat{i}+b_{2} \hat{j}+b_{3} \hat{k},\) b₂, b₃ ∈ R,
\(\vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k},\) c₁, c₂, c₃ ∈ R
be three vectors such that b₂b₃ > 0,
\[\left(\begin{array}{ccc} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{array}\right)\left(\begin{array}{l} 1 \\ b_{2} \\ b_{3} \end{array}\right)=\left(\begin{array}{l} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{array}\right)\]
Then, which of the following is/are true?