Andy, I salute your determination actually to UNDERSTAND all of this! 'Maths' and all things related being difficult for me, I do tend to take the easy way out, accept that I'm unlikely to understand why, and simply accept that it is so (whatever 'it' may be). To quote Hilaire Belloc (who, incidentally and irrelevantly, was an enthusiastic if somewhat gung-ho sailor) - 'But scientists, who ought to know, Assure us that it must be so'.
Peter, thank you for your really valiant attempt to make this subject comprehensible to the likes of Andy and me. The SailNet forum article suggests four practical points which we can take from the maths, parallelograms of forces etc, and can bear in mind when relevant while we are sailing. I wondered whether the final version of your article for the library might incorporate them (I'm sure they are there somewhere in what you have written!), perhaps as a 'Practical Summary' or some such. They are -
1) The direction of the Apparent Wind is always ahead (in terms of boat heading) of the True Wind direction.
2) On a beam reach or close hauled the AW strength is greater than that of the TW, and increases the closer to the wind the boat is sailed.
3) As the TW direction moves aft from being abeam, the strength of the AW progressively lessens compared with that of the TW.
4) When the TW direction is well aft, small changes in TW direction produce much larger changes in AW direction.
If I've got those right, I'll accept that It Is So, and try to ruminate on the practical implications for my sailing.
Michael