Question

The value of λ for which the vectors \(2i-3j+4k\) and \(-4i+λj-8k\) are collinear is 

• A. \(0\)
• B. \(1\)
• C. \(3\)
• D. \(6\)
• E. \(4\)

Answer 1

The Correct Option is D

Step 1: Recall that two vectors are collinear if and only if one is a scalar multiple of the other. Given vectors:
\[ \vec{a} = 2\vec{i} - 3\vec{j} + 4\vec{k} \]
\[ \vec{b} = -4\vec{i} + \lambda\vec{j} - 8\vec{k} \]

Step2: Set up the proportionality condition for collinear vectors:
\[ \frac{-4}{2} = \frac{\lambda}{-3} = \frac{-8}{4} \]

Step3: Simplify each ratio:
\[ -2 = \frac{\lambda}{-3} = -2 \]

Step4: Solve for λ using the middle equality:
\[ -2 = \frac{\lambda}{-3} \]
\[ \lambda = (-2)(-3) \]
\[ \lambda = 6 \]

Conclusion: The value of λ that makes the vectors collinear is \(\boxed{D}\) (6).

Answer 2

Two vectors are collinear if one is a scalar multiple of the other. Let the vectors be:

\[ \mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \] \[ \mathbf{b} = -4\mathbf{i} + \lambda\mathbf{j} - 8\mathbf{k} \]

For \( \mathbf{a} \) and \( \mathbf{b} \) to be collinear, there must exist a scalar \( k \) such that \( \mathbf{b} = k\mathbf{a} \). Let's compare the components:

\[ -4 = 2k \quad \Rightarrow \quad k = -2 \] \[ \lambda = -3k \quad \Rightarrow \quad \lambda = -3(-2) = 6 \] \[ -8 = 4k \quad \Rightarrow \quad k = -2 \]

Since the value of \( k \) is consistent across all components (\( k = -2 \)), the vectors are collinear when \( \lambda = 6 \).

Therefore, the value of \( \lambda \) for which the vectors are collinear is 6.