If a,b,c are three non-zero, non-coplanar vectors and \[{{b}_{1}}=b-\frac{b.a}{|a{{|}^{2}}}a,\,{{b}_{2}}=b+\frac{b.a}{|a{{|}^{2}}}a\],\[{{c}_{1}}=c-\frac{c.a}{|a{{|}^{2}}}a-\frac{c.b}{|b{{|}^{2}}}b\], \[{{c}_{2}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{1}}}{|{{b}_{1}}{{|}^{2}}}{{b}_{1}}\], \[{{c}_{3}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{2}}}{|{{b}_{2}}{{|}^{2}}}{{b}_{2}}\], \[{{c}_{4}}=a-\frac{c.a}{|a{{|}^{2}}}a\]. Then which of the following is a set of mutually orthogonal vectors is [IIT Screening 2005]

• A.             \[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{1}}\}\]
• B.             \[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{2}}\}\]
• C.             \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{3}}\}\]
• D.             \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{4}}\}\]