Let α = 3î + ĵ and β = 2î - ĵ + 3k̂. If β = β₁ - β₂, where β₁ is parallel to α and β₂ is perpendicular to α, then β₁ × β₂ is equal to?

The correct option is C
12(-3î + 9ĵ + 5k̂)
α = 3î + ĵ
β = 2î - ĵ + 3k̂
β = β₁ - β₂
β₁ = λ(3î + ĵ), β₂ = λ(3î + ĵ) - (2î - ĵ + 3k̂)
β₂ ⋅ α = 0
(3λ - 2)(3) + (λ + 1) = 0
9λ - 6 + λ + 1 = 0
λ = 1/2
⇒ β₁ = (3/2)î + (1/2)ĵ
⇒ β₂ = -(1/2)î + (3/2)ĵ - 3k̂
Now β₁ × β₂ = | | | î ĵ k̂ 3/2 1/2 0 -1/2 3/2 -3 | | |
= î(-3/2 - 0) - ĵ( -9/2 - 0) + k̂(9/4 + 1/4) = -3/2î + 9/2ĵ + 5/2k̂ = 1/2(-3î + 9ĵ + 5k̂) = 12(-3î + 9ĵ + 5k̂)
Aliter:
β = β₁ - β₂ ⇒ β ⋅ α = β₁ ⋅ α = |β₁|
β₁ = (β ⋅ α)α
β₁ = (2î - ĵ + 3k̂) ⋅ (3î + ĵ) / |3î + ĵ|² (3î + ĵ) = 5/10 (3î + ĵ) = (3/2)î + (1/2)ĵ
β₂ = (β ⋅ α)α - β = (5/10)(3î + ĵ) - (2î - ĵ + 3k̂) = -(1/2)î + (3/2)ĵ - 3k̂
β₁ × β₂ = -((β ⋅ α)α × β) = -5/10(3î + ĵ) × (2î - ĵ + 3k̂) = -1/2( -3î + 9ĵ + 5k̂) = 12(-3î + 9ĵ + 5k̂)