Let A→=2i^+k^,B→=i^+j^+k^ and C→=4i^−3j^+7k^.then a vector R→ satisfying R→⋅A→=0 and R→×B→=C→×B→.

R→×B→=C→×B→⇒(R→−C→)×B→=0⇒R→=C→+λB→R→⋅A→=0⇒C→⋅A→+λB→⋅A→=0⇒15+3λ=0 or λ=−5∴ R→=C→−5B→=(4i^−3j^+7k^)−5(i^+j^+k^)or R=−i^−8i^+2k^

Let A→=2i^+k^,B→=i^+j^+k^ and C→=4i^−3j^+7k^.then a vector R→ satisfying R→⋅A→=0 and R→×B→=C→×B→.

R→×B→=C→×B→⇒(R→−C→)×B→=0⇒R→=C→+λB→R→⋅A→=0⇒C→⋅A→+λB→⋅A→=0⇒15+3λ=0 or λ=−5∴ R→=C→−5B→=(4i^−3j^+7k^)−5(i^+j^+k^)or R=−i^−8i^+2k^

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Let $\vec{A} = 2 \hat{i} + \hat{k}, \vec{B} = \hat{i} + \hat{j} + \hat{k} and \vec{C} = 4 \hat{i} - 3 \hat{j} + 7 \hat{k}$ .
then a vector $\vec{R}$ satisfying $\vec{R} \cdot \vec{A} = 0$ and
$\vec{R} \times \vec{B} = \vec{C} \times \vec{B}$ .

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$- \hat{i} + 8 \hat{j} + 2 \hat{k}$

$- \hat{i} + 7 \hat{j} + 5 \hat{k}$

$- \hat{i} - 7 \hat{j} + 5 \hat{k}$

$- \hat{i} - 8 \hat{j} + 2 \hat{k}$

answer is D.

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Detailed Solution

$\vec{R} \times \vec{B} = \vec{C} \times \vec{B} \Rightarrow (\vec{R} - \vec{C}) \times \vec{B} = 0 \Rightarrow \vec{R} = \vec{C} + λ \vec{B}$

$\begin{array}{l} \vec{R} \cdot \vec{A} = 0 \Rightarrow \vec{C} \cdot \vec{A} + λ \vec{B} \cdot \vec{A} = 0 \Rightarrow 15 + 3 λ = 0 or λ = - 5 \\ ∴ \vec{R} = \vec{C} - 5 \vec{B} = (4 \hat{i} - 3 \hat{j} + 7 \hat{k}) - 5 (\hat{i} + \hat{j} + \hat{k}) \end{array}$