JEE Main 2005 Mathematics question on Vector Algebra

Question

If

\vec{a}, \vec{b}, \vec{c}

are three non-zero, non-coplanar vectors and

\vec{b}_1 = \vec{b} - \dfrac{\vec{b}\cdot\vec{a}}{|\vec{a}|^2}\vec{a}

,

\vec{b}_2 = \vec{b} + \dfrac{\vec{b}\cdot\vec{a}}{|\vec{a}|^2}\vec{a}

,

\vec{c}_1 = \vec{c} - \dfrac{\vec{c}\cdot\vec{a}}{|\vec{a}|^2}\vec{a} - \dfrac{\vec{c}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}

,

\vec{c}_2 = \vec{c} - \dfrac{\vec{c}\cdot\vec{a}}{|\vec{a}|^2}\vec{a} - \dfrac{\vec{c}\cdot\vec{b}_1}{|\vec{b}_1|^2}\vec{b}_1

,

\vec{c}_3 = \vec{c} - \dfrac{\vec{c}\cdot\vec{a}}{|\vec{a}|^2}\vec{a} - \dfrac{\vec{c}\cdot\vec{b}_2}{|\vec{b}_2|^2}\vec{b}_2

,

\vec{c}_4 = \vec{c} - \dfrac{\vec{c}\cdot\vec{a}}{|\vec{a}|^2}\vec{a}

Then, which of the following is a set of mutually orthogonal vectors?