Performing the dot product: ‌=(1+λ)⋅1+(2−λ)⋅1+(1+4λ)⋅1 ‌=(1+λ)+(2−λ)+(1+4λ) ‌=1+λ+2−λ+1+4λ ‌=4+4λ Set the expression to zero for perpendicularity: 4+4λ=0 Solving for λ : ‌4λ=−4 ‌λ=−1 Thus, the correct value of λ is -1 .