Wait a Minute

Wait a Minute!

I smelled a rat. "Hold on there, Roscoe," I exclaimed. "I can draw Mr. Long Vector's path so that the total length is the same as, or less than, Mr. Short Vector's."

"True enough," replied Roscoe. "There is nothing, geometrically speaking, that says Mr. Short Vector will always take a shorter total path. If Mr. Long Vector is very lucky, he can come out better. But that is not what we observe in practice. Statistically, averaged over many Mr. Long Vector projects and Mr. Short Vector projects, we find that Mr. Short Vector projects come out better. The reason for this is that short vectors allow you to do some other things that you can't usually accomplish with long vectors. I'll talk about those in a minute. Although you are right that there is no geometrical 'proof' of this, it is an empirical fact, and the geometrical analogy still has a lot to recommend it."

Just as I was beginning to accept this line of reasoning, Roscoe drove his point home: "Besides, neither of these drawings takes into account the third factor: that you are constantly aiming at a moving target. Because, unlike the horseshoe stake, targets always move in the real world. That means Mr. Long Vector faces much more risk than Mr. Short Vector, because Mr. Short Vector gets to take aim more often. As the target moves, he gets to readjust and therefore spends less time moving in the wrong direction."