Question

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 $\vec{a} \times(\vec{b} \times \vec{c})=\vec{b}+\lambda \vec{c}$. If $\vec{b}$ and $\vec{c}$ are non-parallel, then the value of $\lambda$ is:

• A. $-5$
• B. 5
• C. 1
• D. $-1$

Answer 1

Answer: (A)

$\vec{a}=3 \hat{i}+\hat{j}, \vec{b}=\hat{i}+2 \hat{j}+\hat{k}$
As $\vec{a} \times(\vec{b} \times \vec{c})=\vec{b}+\lambda \vec{c}$
$\Rightarrow \vec{a} \cdot \vec{c}(\vec{b})-(\vec{a} \cdot \vec{b}) \vec{c}=\vec{b}+\lambda \vec{c}$
$ \Rightarrow \vec{a} \cdot \vec{c}=1, \vec{a} \cdot \vec{b}=-\lambda $
$\Rightarrow(3 \hat{ i }+\hat{ j }) \cdot(\hat{ i }+2 \hat{ j }+\hat{ k })=-\lambda $
$ \Rightarrow \lambda=-5$