<p><strong>72.</strong> Let \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\), \(\vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k}\) and \(\vec{c} = \hat{i} + \alpha\hat{j} + \beta\hat{k}\) are linearly dependent and \(|\vec{c}| = \sqrt{3}\), then \(|\alpha| + \beta =\) ________.</p>
Step-by-Step Solution
Key Concept: Linear dependence means vectors are scalar multiples of each other. Use the linear dependence condition along with the magnitude constraint to find α and β, recognizing that one vector must express as a linear combination of others with specific coefficients.
Step 1: Since a⃗, b⃗, c⃗ are linearly dependent, we can write c⃗ as a linear combination: c⃗ = λ a⃗ + μ b⃗ Step 2: Substitute: (1, α, β) = λ(1, 1, 1) + μ(4, 3, 4) This gives: 1 = λ + 4μ, α = λ + 3μ, β = λ + 4μ Step 3: From equations 1 and 3: 1 = λ + 4μ and β = λ + 4μ, so β = 1 Step 4: Use magnitude constraint | c⃗ | = √3: 1 + α^2 + β^2 = 3 1 + α^2 + 1 = 3 ⟹ α^2 = 1 ⟹ α = ±1 Step 5: From α = λ + 3μ and β = λ + 4μ = 1: λ = 1 - 4μ So α = (1 - 4μ) + 3μ = 1 - μ If α = 1: μ = 0, λ = 1 (check: c⃗ = a⃗ ✓) If α = -1: μ = 2, λ = -7 (check: c⃗ = -7 a⃗ + 2 b⃗ ✓) Step 6: For α = 1, β = 1: |α| + β = 1 + 1 = 2 For α = -1, β = 1: |α| + β = 1 + 1 = 2 ∴ Answer: 2
Correct Answer: 2