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Question

# If the vectors α^i+^j+^k,^i+β^j+^k,^i+^j+λ^k(α,β,γ≠1) are coplanar, then the value of 11−α+11+β+11−γ is.

A
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C
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D
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Solution

## The correct option is C 1Vectors α^i+^j+^k,^i+β^j+^k,^i+^j+γ^k are co-planarsThus ∣∣ ∣∣α111β111γ∣∣ ∣∣=0α[βγ−1]−1[γ−1]+1[1−β]=0αβγ−α−γ+1+1−β=0αβγ+2=α+β+γαβγ=α+β+γ−2⟶111−α+11−β+11−γ=(1−β)(1−γ)+(1−α)(1−γ)+(1−α)(1−β)(1−α)(1−β)(1−γ)=1−β−γ+βγ+1−α−γ+αγ+1−α−β+αβ[1−α−β+αβ][1−γ]=3−2α−2β−2γ+αβ+βγ+αγ1−α−β+αβ−γ+αγ+βγ−αβγ=3−2α−2β−2γ+αβ+βγ+αγ1−α−β−γ+αβ+βγ+αγ−(α+β+γ−2) [from equation 1]=3−2α−2β−2γ+αβ+βγ+αγ3−2α−2β−2γ+αβ+βγ+αγ=1

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