Two lines intersect in exactly one point in projective plane, $\mathbb P^2$
Is this a consequence of the Weak Bezout theorem ?
because both lines are described by linear homogeneous polynomials, number of intersection points is at most the product of the degree, namely $1\cdot 1=1$ but, they also cannot share a irreducible factor, othwerwise they would be identical so the claim follows.
Is my reasoning correct ?
Best Answer
This reasoning works, but is kind of overkill.
It is more elementary to say that a line in the projective plane corresponds to a 2-dimensional linear subspace of $\mathbb R^3$, and the intersection between two such different subspaces has to have dimension $1$ -- it can't be $2$ because then the subspaces would be equal, contrary to assumtions, and it can't be $0$ because then the subspaces would have a direct sum of dimension $4$ which is too large to fit into $\mathbb R^3$.
So the intersection is a subspace of dimension $1$ in $\mathbb R^3$, which is exactly what a point in the projective plane is, by definition.