Intersection of lines in projective space

Question

I'm given the lines tu+sv and rw+kz where t,s,r,k are constants and u=(-5,0,1/4,0) v=(0,1,1/3,0), w=(4/3,-1,1/2,0), z=(7,1/2,-1/3,0). How can I find their point of intersection?

Thanks for all the help.

Answer 1

Two lines in $\mathbb{RP}^3$ don't neccessarily intersect. But your points all have zero as their last coordinate, so you can consider the projective plane of all such points, and compute the intersection there. In a real projective plane I'd compute join and meet using the cross product. Avoiding fractions I'd write this as

$$\left(\begin{pmatrix}-20\\0\\1\end{pmatrix}\times\begin{pmatrix}0\\3\\1\end{pmatrix}\right)\times\left(\begin{pmatrix}8\\-6\\3\end{pmatrix}\times\begin{pmatrix}42\\3\\-2\end{pmatrix}\right)$$

Compute that, add a zero as fourth coordinate and you are done. If you want to, you can cancel the common factor of $54$ you'll find in the result, in order to simplify the numbers a bit.