Let a→=3i^+j^ and b→=i^+2j^+k^ . Let c→ be a vector satisfying a→×(b→×c→)=b→+λc→ . If b→ and c→ are non-parallel, then the value of λ is:

Here we are given that a→=3i^+j^,b→=i^+2j^+k^As a→×(b→×c→)=b→+λc→=a→.c→(b→)−(a→.b→)c→=b→+λc→ ⇒ a→.c→=1,a→.b→=−λ ⇒ (3i^+j^).(i^+2j^+k^)=−λ ⇒ λ=−5

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Let $\vec{a} = 3 \hat{i} + \hat{j}$ and $\vec{b} = \hat{i} + 2 \hat{j} + \hat{k}$ . Let $\vec{c}$ be a vector satisfying $\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + λ \vec{c}$ . If $\vec{b}$ and $\vec{c}$ are non-parallel, then the value of $λ$ is:

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answer is -5.

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Detailed Solution

Here we are given that
$\vec{a} = 3 \hat{i} + \hat{j}, \vec{b} = \hat{i} + 2 \hat{j} + \hat{k}$
As $\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + λ \vec{c}$
$= \vec{a} . \vec{c} (\vec{b}) - (\vec{a} . \vec{b}) \vec{c} = \vec{b} + λ \vec{c} \Rightarrow \vec{a} . \vec{c} = 1, \vec{a} . \vec{b} = - λ \Rightarrow (3 \hat{i} + \hat{j}) . (\hat{i} + 2 \hat{j} + \hat{k}) = - λ \Rightarrow λ = - 5$