Let a→=2i^+λ1j^+3k^, b→=4i^+3−λ2j^+6k^and c→=3i^+6j^+λ3−1k^be three vectors such that b→=2a→ and a→is perpendicular to c→.Then a possible value of λ1,λ2,λ3is

∴4i^+3−λ2j^+6k=4i^+2λ1j^+6k^⇒3−λ2=2λ1⇒2λ1+λ2=3a→⋅c→=0⇒6+6λ1+3λ3−1=0⇒2λ1+λ3=−1 Now, λ1,λ2,λ3=λ1,3−2λ1−1−2λ1

Let a→=2i^+λ1j^+3k^, b→=4i^+3−λ2j^+6k^and c→=3i^+6j^+λ3−1k^be three vectors such that b→=2a→ and a→is perpendicular to c→.Then a possible value of λ1,λ2,λ3is

∴4i^+3−λ2j^+6k=4i^+2λ1j^+6k^⇒3−λ2=2λ1⇒2λ1+λ2=3a→⋅c→=0⇒6+6λ1+3λ3−1=0⇒2λ1+λ3=−1 Now, λ1,λ2,λ3=λ1,3−2λ1−1−2λ1

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Let $\vec{a} = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, \vec{b} = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}$ and $\vec{c} = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}$ be three vectors such that $\vec{b} = 2 \vec{a} and \vec{a}$ is perpendicular to $\vec{c} .$ Then a possible value of $(λ_{1}, λ_{2}, λ_{3})$ is

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$(1 / 2, 4, - 2)$

$(- 1 / 2, 4, 0)$

$(1, 3, 1)$

$(1, 5, 1)$

answer is B.

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Detailed Solution

$\begin{aligned} ∴ 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 k = 4 \hat{i} + 2 λ_{1} \hat{j} + 6 \hat{k} \\ \Rightarrow 3 - λ_{2} = 2 λ_{1} \\ \Rightarrow 2 λ_{1} + λ_{2} = 3 \end{aligned}$

$\begin{aligned} \vec{a} \cdot \vec{c} = 0 \\ \Rightarrow 6 + 6 λ_{1} + 3 (λ_{3} - 1) = 0 \\ \Rightarrow 2 λ_{1} + λ_{3} = - 1 \\ Now, (λ_{1}, λ_{2}, λ_{3}) = (λ_{1}, 3 - 2 λ_{1} - 1 - 2 λ_{1}) \end{aligned}$