= Projective joins and meets by cross products
{title2=$\ell=p\times q,\quad x=\ell\times m$}
In the <real projective plane>, represent a <projective point> by a nonzero vector and a <projective line> by its nonzero coefficient covector, identified with a vector in chosen coordinates. The join of distinct <projective points> represented by $p,q$ has coefficient vector $p\times q$, using the <cross product>. It is nonzero because the representatives are <linearly independent>, and its <dot product> with each representative vanishes. Dually, two distinct <projective lines> with coefficients $\ell,m$ meet in the <projective point> represented by $\ell\times m$. Scaling an input scales the output and leaves the projective object unchanged. This makes joins, intersections, concurrency, and <collinearity> exact calculations with <homogeneous coordinates>.
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