Any two distinct lines in the projective plane meet at a unique point

Question

I am trying to do an exercise from Rick Miranda's book on Algebraic Curves and Riemann surfaces, where he defines

A degree one curve in the projective plane, defined by a homogeneous polynomial in x,y,z of degree one, is called a line.

Then im asked

Prove that any two distinct lines in the projective plane meet at one unique point and give a formula for that point.

So basically he suggests doing this by linear algebra i think, but i cant quite seem to find an invertible matrix or system that gives an invertible matrix so im kinda stuck , so any tips are aprecciated, just something to get me rolling. Thanks in advance!

Answer 1

A line $L$ in the projective plane has the equation $ax+by+cz=0$ where $a,b,c$ are not all 0. Let $L'$be the line $a'x+b'y+c'z=0$. The lines $L$ and $L'$ are distinct iff the vectors $[a,b,c]$ and $[a',b',c']$ are not proportional in which case the cross-product $[a,b,c] \times [a',b',c']$ represents the unique point of intersection, i.e. the point is $$(bc'-cb':-(ac'-ca'):ab'-ba')$$