= Parallel lines as a quotient vector space
{title2=$\mathbb R^3/\operatorname{span}(u)$}
For fixed $u\ne0$, the parallel <affine lines> $l_x=x+\operatorname{span}(u)$ are precisely the cosets of the indicated <quotient vector space>. Changing representatives adds a multiple of $u$, so $l_x+l_y=l_{x+y}$ and $a l_x=l_{ax}$ are well-defined. The zero is the whole line $l_0$, not a single point. If $(u,b_1,b_2)$ is a <basis>, the two line classes $l_{b_1},l_{b_2}$ form a <basis> of the quotient by spanning and independence modulo $\operatorname{span}(u)$.
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