Consider three vectors A=i^+j^-2k^, B=i^-j^+k^ and C=2i^-3j^+4k^. A vector X of the form αA + βB (α and β are numbers) is perpendicular to C. The ratio of α and β is

Vector X of the form αA + βB.X=αA+βB=α(i^+j^-2k^)+β(i^-j^+k^)X=i^(α+β)+j^(α-β)+k^(-2α+β)A vector X is perpendicular to C, i.e. X · C = 0[i^(α+β)+j^(α-β)+k^(-2α+β)]·[2i^-3j^+4k^]=0⇒ 2(α+β)-3(α-β)+4(-2α+β)=0⇒ 2α+2β-3α+3β-8α+4β=0⇒ -9α+9β=0or  α=β⇒αβ=11or  α:β=1:1