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I have a question to the following example:

Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be the set of all lines through the origin of $\mathbb{A}_2$. Take $\mathbb{K}^2$ as vector space and $\pi$ to map the couple $(a,b)$ to the line with equation $ax+by=0$. Then $(\mathbb{P},\mathbb{K}^2,\pi)$ is a one-dimensional projective space over $\mathbb{K}$.

In this one-dimensional projective space $(\mathbb{P},\mathbb{K}^2,\pi)$, take as reference the lines $x=0$, $y=0$ and the line $x-y=0$ as unit point . Then the line $ax+by=0$ has projective coordinates $(a,-b)$.

Now my question is: Why is the coordinate vector $(a,-b)$? I thought it would be $(a,b)$.

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Because you take $x-y=0$ as the unit point $(1,1)$, you have to reverse signs. If you were to take $x+y=0$ as the unit, then $(a,b)$ would be the coordinates of $ax+by=0$.

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