Preliminaries

Today my particle accelerator in Geneva is not operating, and I have decided to have the day off. I take a bus ride to a nearby village, and get off at the bus-stop in the main square. The village is very pretty, and I wander around enjoying the chalets and the views. I also venture into the surrounding hillside, with its criss-crossing paths. After a couple of hours, I begin to think it is time to return to the village, to catch the bus back to Geneva. If I know that I have walked a total of 6 miles, how far am I from the bus-stop?

The answer of course is that we cannot tell – it could be anything from zero distance to 6 miles. It all depends on how straight or crooked my walk had been up till then. This is because the problem is basically one involving vectors, in which not only a magnitude is involved, but also a direction. The distance that I am from the bus-stop is given by the length of the vector formed by adding all the vectors corresponding to the various straight bits of path along which I had walked.

Vectors are thus very useful in any problem in which directions are implied.