Question:

Step 1: Let the 2019 vectors be v ₁, v ₂, ..., v ₂₀₁₉. Given: sum of every 2018 vectors is a scalar multiple of the remaining vector. Step 2: This means: v ₁ + v ₂ + ... + v ₂₀₁₈ = k_1 v ₂₀₁₉ for some scalar k_1, and similarly for other combinations. Step 3: Let S = v ₁ + v ₂ + ... + v ₂₀₁₉. Then: S - v _i = kᵢ v _j (where j ≠ i), meaning S = v _i + kᵢ v _j. Step 4: For this to hold for all pairs, all vectors must be collinear (lie on the same line). If they weren't, the sum of 2018 vectors couldn't be a scalar multiple of the 19th. Step 5: Since all vectors are collinear but not all scalar multiples of each other, they point in possibly different directions on the same line. The constraint that every subset of 2018 has sum proportional to the remaining vector implies the vectors must be arranged such that their total sum is zero . Step 6: With 2019 (odd number) collinear vectors where the condition holds, the magnitude of the sum equals 0 . ∴ Answer: A (0)