Fewpal
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If any line form an angle α, β, γ, δ with the diagonal of cube then cos2α + cos2β + cos2γ + cos2δ = ____

(A) (8/3)

(B) – (8/3)

(C) (4/3)

(D) – (4/3)   

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The correct option (C) (4/3)

Explanation:

Let OA = OB = OC = a be sides and vector(AL, BM, CN) and vector OP are diagonals.

vector OP = (a, a, a), vector AL = (– a, a, a), vector BM = (a, – a, a), vector CN = (a, a, – a),

ℓ, m, n → direction cosine of line.

Let Diagonal vector(OP, AL, BM, CN) form angle α, β, γ, δ with line.

cos α  = [vector(OP ∙ ℓ)/vector(|OP| |ℓ|)]

= [{(a, a, a) ∙ (ℓ, m, n)}/{√(3a2) ∙ √(ℓ2 + m2 + n2)}]

= [{a(ℓ + m + n)}/{√(3) ∙ a ∙ √(ℓ2 + m2 + n2)}]

= [(ℓ + m + n)/√3] ........ (ℓ2 + m2 + n2 = 1)

cos β = [(– ℓ + m +n)/√3], cos γ = [(ℓ – m + n)/√3],

cos δ = [(ℓ + m – n)/√3]

∴ cos2α + cos2β + cos2γ + cos2δ = (4/3) (ℓ2 + m2 + n2) = (4/3) (1) = (4/3).

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