Let $\vec A, \vec B, \vec C$ are three vectors respectively given by $2\hat i + \hat k$, $\hat i + \hat j + \hat k$ and $4\hat i - 3\hat j + 4\hat k$. Then vector $\vec R$, which satisfies the relation $\vec R × \vec B = \vec C × \vec B$ and $\vec R.\vec A = 0$ is

$- \hat i - 8\hat j + 2\hat k$

We have

$\vec R × \vec B = \vec C × \vec B$ and $\vec R.\vec A = 0$

$⇒\vec A×(\vec R×\vec B)=\vec A×(\vec C×\vec B)$

$⇒(\vec A.\vec B)\vec R-(\vec A.\vec R)\vec B=(\vec A.\vec B)\vec C-(\vec A.\vec C)\vec B$

$⇒(2+1)\vec R=3\vec C-(8+7)\vec B⇒\vec R=\vec C-5\vec B=- \hat i - 8\hat j + 2\hat k$

Hence (C) is the correct answer.