Suppose that vectors $x_1,x_2,…,x_n$ have the following property: for each $i$ the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1,x_2,\dots ,x_n$ are not parallel, what is the value of the sum $x_1+x_2+\dots +x_n=$
I know that answer is $x_1+x_2+\dots +x_n=0$ but I am not really understand why. So I can make these steps: $s = x_1+x_2+\dots +x_n$ and $s-x_1 = x_2 + \dots + x_n$. So $s-x_1 \mid\mid x_1$. But what after?
I am think now (after made question) that:
$x_1 + x_2 \mid\mid s - x_1 + s - x_2$
$\dots$
$x_1 + \dots + x_n \mid\mid s_1 + \dots + s_n - x_1 - x_2 - \cdots - x_n$
$s \mid\mid s_1 + \cdots + s_n - (s) = s_1 + \cdots + s_{n-1}$ where $s_i$ -- just sum of $x_1,\cdots,x_n$. Am I right?
The fact that $s-x_k\Vert x_k$ gives that $\exists \lambda_k\in\mathbb{R}$ such that $s-x_k=\lambda_k x_k\iff s=(1+\lambda_k)x_k$.
Assume that $x_1\not\Vert x_2$. From the previous statements it follows that $s=(1+\lambda_1)x_1=(1+\lambda_2)x_2\Rightarrow s=0.$