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