<p><strong>67.</strong> Let \(\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}\) and \(\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}\) be two given vectors where vectors \(\vec{a}\) and \(\vec{b}\) are non-collinear. The value of \(|\lambda|\) for which vectors \(\vec{\alpha}\) and \(\vec{\beta}\) are collinear, is ________.</p>
Step-by-Step Solution
Key Concept: Two vectors are collinear if and only if one is a scalar multiple of the other, meaning their coefficients with respect to linearly independent vectors must be proportional. Since $\vec{a}$ and $\vec{b}$ are non-collinear (linearly independent), we need $(\lambda - 2) : (4\lambda - 2) = 1 : 3$.
Step 1: For $\vec{\alpha}$ and $\vec{\beta}$ to be collinear, one must be a scalar multiple of the other. That is, $\vec{\beta} = k\vec{\alpha}$ for some scalar $k$. Step 2: Write: $(4\lambda - 2)\vec{a} + 3\vec{b} = k[(\lambda - 2)\vec{a} + \vec{b}]$ Expanding: $(4\lambda - 2)\vec{a} + 3\vec{b} = k(\lambda - 2)\vec{a} + k\vec{b}$ Step 3: Since $\vec{a}$ and $\vec{b}$ are linearly independent, equate coefficients: Coefficient of $\vec{a}$: $4\lambda - 2 = k(\lambda - 2)$ ... (1) Coefficient of $\vec{b}$: $3 = k$ ... (2) Step 4: From equation (2), $k = 3$. Substitute into equation (1): $4\lambda - 2 = 3(\lambda - 2)$ $4\lambda - 2 = 3\lambda - 6$ $\lambda = -4$ Step 5: Verify: When $\lambda = -4$, $\vec{\alpha} = (-6)\vec{a} + \vec{b}$ and $\vec{\beta} = (-18)\vec{a} + 3\vec{b} = 3[(-6)\vec{a} + \vec{b}] = 3\vec{\alpha}$ ✓ Therefore, $|\lambda| = |-4| = \boxed{4}$ Note: If the answer key states 2, verify the problem statement. With the given equations, $|\lambda| = 4$ is correct. If $\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$ should be $(4\lambda - 2)\vec{a} + 3\vec{b}$ with different coefficients, the answer may differ.
Correct Answer: 2