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 $λ$ is

• A. -5
• B. 5
• C. 1
• D. -1

Answer 1

Answer: (A)

-5

Answer 2

$\vec{a} = 3\hat{i} + \hat{j} \quad \text{and} \quad \vec{b} = \hat{i} + 2\hat{j} + \hat{k}$
$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} = \vec{b} + \lambda \vec{c}$
If $\vec{b} \quad \text{and} \quad \vec{c}$ are non-parallel, then
$\vec{a} \cdot \vec{c} = 1 \quad \text{and} \quad \vec{a} \cdot \vec{b} = -\lambda$
but $\vec{a} \cdot \vec{b} = 5$
$⇒λ=−5$
So, the correct option is (A): -5