\(\vec{A}, \vec{B}, \vec{C}\) are three vectors respectively given by 2\(\hat{i}\) + \(\hat{k}\), \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\) and 4\(\hat{i}\) - 3 \(\hat{j}\) + 7 \(\hat{k}\). 

Then vector \(\vec{R}\), which satisfies the relation \(\vec{R}\) x \(\vec{B}\) = \(\vec{C}\) x \(\vec{B}\) and \(\vec{R}\).\(\vec{A}\) = 0 is : 

- \(\hat{i}\) - 8\(\hat{j}\) + 2\(\hat{k}\)

We have : \(\vec{R}\) x \(\vec{B}\) = \(\vec{C}\) x \(\vec{B}\) and \(\vec{R} . \vec{A} = 0\)

\(\vec{A}\) x [ \(\vec{R}\) x \(\vec{B}\) ] = \(\vec{A}\) x [ \(\vec{C}\) x \(\vec{B}\) ]

[\(\vec{A}\).\(\vec{B}\)] \(\vec{R}\) - [\(\vec{A}\).\(\vec{R}\)] \(\vec{B}\) = [\(\vec{A}\).\(\vec{B}\)] \(\vec{C}\) - [\(\vec{A}\).\(\vec{C}\)] \(\vec{B}\) 

(2 + 1)\(\vec{R}\) = 3 \(\vec{C}\) - (8 + 7)\(\vec{B}\)

\(\vec{R}\) = \(\vec{C}\) - 5\(\vec{B}\) = - \(\hat{i}\) - 8\(\hat{j}\) + 2\(\hat{k}\)