<p>Given \(\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}\) and \(\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}\). Vectors \(\vec{\alpha}\) and \(\vec{\beta}\) are collinear. Find \(\lambda\).</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 basis vectors must be proportional.
Step 1: For vectors α and β to be collinear, β = k α for some scalar k. Step 2: Write: (4λ - 2) a + 3 b = k[(λ - 2) a + b ] Step 3: Expand: (4λ - 2) a + 3 b = k(λ - 2) a + k b Step 4: Since a and b are linearly independent, equate coefficients:
• Coefficient of a : 4λ - 2 = k(λ - 2)
• Coefficient of b : 3 = k Step 5: Substitute k = 3 into the first equation: 4λ - 2 = 3(λ - 2) Step 6: Simplify: 4λ - 2 = 3λ - 6 Step 7: Solve: 4λ - 3λ = -6 + 2 → λ = -4 Verification: When λ = -4: α = -6 a + b and β = -18 a + 3 b = 3(-6 a + b ) ✓ ∴ Answer: -4
Correct Answer: -4