Question

If $\vec{a} = 4\hat{i} + \lambda \hat{j} - 6\hat{k}$ and $\vec{b} = -6\hat{i} + 12\hat{j} + 9\hat{k}$ are collinear, then the value of $\lambda$ is equal to

• A. -4
• B. 4
• C. -6
• D. -8
• E. 8

Hint:

Collinear vectors are just scalar multiples of each other. Identify the scale factor from the fully numerical components to find the unknown part immediately.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Two vectors \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) are collinear if their components are proportional: \( \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} = k \).

Step 2: Detailed Explanation:
From the given vectors:
\( \frac{4}{-6} = \frac{\lambda}{12} = \frac{-6}{9} \)
Simplify the known ratios:
\( \frac{4}{-6} = -\frac{2}{3} \)
\( \frac{-6}{9} = -\frac{2}{3} \)
Since the ratios are equal, the condition for collinearity is satisfied.
Equate the second ratio to the common value:
\[ \frac{\lambda}{12} = -\frac{2}{3} \]
\[ 3\lambda = -24 \]
\[ \lambda = -8 \]

Step 3: Final Answer:
The value of \( \lambda \) is -8.