Vector Algebra
Angle Bisector using Vectors
Grade 12
Question:
<p>The vector \(\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k}\) lies in the plane of the vectors \(\vec{b} = \hat{i} + \hat{j}\) and \(\vec{c} = \hat{j} + \hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then which one of the following gives possible values of \(\alpha\) and \(\beta\)?</p>
<p>\(\alpha = 2, \beta = 2\)</p>
<p>\(\alpha = 1, \beta = 2\)</p>
<p>\(\alpha = 2, \beta = 1\)</p>
<p>\(\alpha = 1, \beta = 1\)</p>
Step-by-Step Solution
Key Concept: A vector lying in a plane spanned by two vectors can be expressed as their linear combination, AND if it bisects the angle between them, it must be proportional to the sum of their unit vectors (angle bisector property).
Step 1: Coplanarity Condition If a lies in the plane of b and c , then a = λ b + μ c for some scalars λ, μ. α î + 2 ĵ + β k̂ = λ( î + ĵ ) + μ( ĵ + k̂ ) Comparing coefficients: α = λ, 2 = λ + μ, β = μ Step 2: Angle Bisector Condition For a to bisect the angle between b and c , it must be proportional to b̂ + ĉ (unit vectors in directions of b and c). | b | = √2, | c | = √2 b̂ = (1/√2)( î + ĵ ), ĉ = (1/√2)( ĵ + k̂ ) b̂ + ĉ = (1/√2)( î + 2 ĵ + k̂ ) Step 3: Apply Proportionality a ∝ ( î + 2 ĵ + k̂ ) Therefore: α/1 = 2/2 = β/1 This gives α = 1 and β = 1 Step 4: Verification From Step 1: α = λ, β = μ, and λ + μ = 2 With α = 1, β = 1: λ = 1, μ = 1, and 1 + 1 = 2 ✓ ∴ Answer: α = 1, β = 1 (Option D)
Correct Answer: D