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

The Correct Option is A

We are given the vectors \(\vec{a} = 3\hat{i} + \hat{j}\), \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\), and a vector \(\vec{c}\) such that \(\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda \vec{c}\). Our task is to determine the value of \(\lambda\).

First, we need to solve the expression \(\vec{a} \times (\vec{b} \times \vec{c})\). We can use the vector triple product identity:

\(\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\)

We proceed by calculating each dot product:

1. Calculate \(\vec{a} \cdot \vec{c}\): Since \(\vec{c}\) is unknown, we designate its components as \(\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}\). Then:

\(\vec{a} \cdot \vec{c} = (3\hat{i} + \hat{j}) \cdot (x\hat{i} + y\hat{j} + z\hat{k}) = 3x + y\)

2. Calculate \(\vec{a} \cdot \vec{b}\):

\(\vec{a} \cdot \vec{b} = (3\hat{i} + \hat{j}) \cdot (\hat{i} + 2\hat{j} + \hat{k}) = 3 \times 1 + 1 \times 2 = 3 + 2 = 5\)

Substitute these results into the triple product identity:

\(\vec{a} \times (\vec{b} \times \vec{c}) = (3x + y)\vec{b} - 5\vec{c}\)

We are given:

\((3x + y)\vec{b} - 5\vec{c} = \vec{b} + \lambda \vec{c}\)

Equating coefficients gives us two simultaneous equations:

1. 3x + y = 1
2. -5 = \lambda\), from the equation for \(\vec{c}\)

Thus, the value of \(\lambda\) is clearly:

Option: -5