What side of the lane did they jump on?  They'll appear on the same side after the jump.  This raises various strategic issues.  Sometimes you might want to jump in on one side of the lane or the other to avoid jumping right next to an enemy starbase.  Or maybe you want to be able to make a quick, sharp turn and immediately make another jump (turning the corner--for example--say you want to get a ship through the pirate base, it just grazes the edge of the pirate gravity well).

Huh? Is PSIDAR any different from the phase detection research line of Vasari? Cause if you detect a ship that is executing a jump, you can easily tell where it pops out by looking at its trajectory compared to the phase lane. Hell, you can even select your ship, right click somewhere on the edge of your gravwell and use shift-rclick on the other gravwell - the line representing the queued jump is parallel to the enemy's jump line. That way you can nail down where he is going to emerge, just try out different starting points and see if the queued jump line hits the target ship.

All of that is pretty irrelevant though. After some time you will be easily able to tell where the enemy will pop out just by seeing how far the bastard is to left or right of the phase lane.

Phase detection researches supplement scouting very well. When you scout, you never really know if there ain't something moving between gravwells while your scout is elsewhere.
And if you tell me you have scouts everywhere, then it is not cheaper At least not for Vasari, with their expensive scouts. I like this research line, it gives you a nice heads up if something big is going on from an unexpected direction. Altogether pretty handy, in my opinion.