Characteristic of a field to make projective points the same In the resolution of a linear algebra exercise, they tell me that given a field $\Bbb K$ and the projective space $\Bbb P^n(\Bbb K)$, the projective points $(1:-1:2)$ and $(2:1:1)$ are the same if and only if $\text{char}(\Bbb K)=3$.
I understand the right to left implication, since if $\text{char}(\Bbb K)=3$ then $3=0$, hence $-1=-1+0=-1+3=2$ and $4=3+1=1$. Therefore, $(1:-1:2)=(1:2:2)=(2:4:4)=(2:1:1)$.
However, I have trouble understanding the left to right implication, I am not really sure of why any other value of the characteristic of the field would result in those two points being different.
Thanks in advance!

The simplest way is to rewrite $(1: -1: 2)$ as $(-1, 1, -2)$.
If we have $(-1, 1, -2) = \lambda(2, 1, 1)$, then by comparing the second coordinate, it's clear that $\lambda = 1$ and hence $-1 = 2$ and $-2 = 1$. This of course implies $3 = 0$.
