Question:

Step 1: Define the vectors and their sum. Let the 2019 vectors be $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_{2019}}$ and denote their sum as: $$S = \sum_{i=1}^{2019} \vec{v_i}$$ Step 2: Translate the given condition into mathematical form. We are told that the sum of every 2018 vectors (i.e., all vectors except one) is a scalar multiple of the remaining vector. This means for each $i$: $$S - \vec{v_i} = \lambda_i \vec{v_i}$$ where $\lambda_i$ is some scalar that may depend on $i$. Step 3: Rearrange to express $S$ in terms of individual vectors. From the equation in Step 2, we can rearrange: $$S = \vec{v_i} + \lambda_i \vec{v_i} = (1 + \lambda_i)\vec{v_i}$$ This equation holds for every value of $i$ from 1 to 2019. Step 4: Deduce that all vectors must be parallel to $S$. Since $S = (1 + \lambda_i)\vec{v_i}$ for each $i$, every vector $\vec{v_i}$ is a scalar multiple of $S$. This means all 2019 vectors are parallel to $S$. Step 5: Apply the constraint that not all vectors are scalar multiples of each other. We are given that not all vectors are scalar multiples of each other. However, from Step 4, we concluded that all vectors must be parallel to $S$. If $S \neq \vec{0}$, then all vectors would be scalar multiples of the single vector $S$, which would mean all vectors are scalar multiples of each other—contradicting the given condition. Therefore, we must have: $$S = \vec{0}$$ Step 6: State the final answer. The magnitude of the sum of all 2019 vectors is: $$|S| = |\vec{0}| = 0$$ The answer is **Option 1: 0**