<p>The vector <strong>a</strong> = α<strong>i</strong> + 2<strong>j</strong> + β<strong>k</strong> lies in the plane of the vectors <strong>b</strong> = <strong>i</strong> + <strong>j</strong> and <strong>c</strong> = <strong>j</strong> + <strong>k</strong> and bisects the angle between <strong>b</strong> and <strong>c</strong>. Then, which one of the following gives possible values of α and β?</p>
Step-by-Step Solution
Key Concept: A vector lies in a plane if it can be expressed as a linear combination of two vectors in that plane. A vector bisects the angle between two vectors if it is proportional to the sum of their unit vectors.
Step 1: Use the angle bisector condition. For vector a to bisect the angle between b and c , it must be proportional to the sum of unit vectors along b and c : | b | = √(1^2 + 1^2) = √2 | c | = √(1^2 + 1^2) = √2 Unit vector along b : b̂ = (1/√2) i + (1/√2) j Unit vector along c : ĉ = (1/√2) j + (1/√2) k The angle bisector direction: b̂ + ĉ = (1/√2) i + (2/√2) j + (1/√2) k ∴ a = λ[(1/√2) i + (2/√2) j + (1/√2) k ] = (λ/√2)[ i + 2 j + k ] Step 2: Match with a = αi + 2j + βk. From the expression above, we need α = β (the coefficients of i and k must be equal). The coefficient of j is 2 in both expressions, which is consistent. Step 3: Verify coplanarity. For a to lie in the plane of b and c , the scalar triple product must be zero: [ a b c ] = 0 Using the determinant: |α 2 β| |1 1 0| = α(1·1 - 0·1) - 2(1·1 - 0·1) + β(1·1 - 1·0) |0 1 1| = α(1) - 2(1) + β(1) = α - 2 + β For coplanarity: α + β = 2 Step 4: Combine both conditions. From angle bisector: α = β From coplanarity: α + β = 2 Substituting α = β into α + β = 2: 2α = 2 ⟹ α = 1, β = 1 Step 5: Verification. With α = 1, β = 1: a = i + 2 j + k This is indeed proportional to i + 2 j + k , matching our angle bisector direction. ∴ Answer: A
Correct Answer: A