## Part 3 - the metric

We already mentioned the notion of the magnitude of a vector, but we said nothing about what it actually is. On a plane it's easy - when we move by in the axis and by in the axis, the distance between the starting and the ending point is (which can be seen by drawing a right triangle and using the Pythagorean theorem - see the picture). It doesn't have to be always like that, though, and here is where the **metric** comes into play.

The metric is a way of generalizing the Pythagorean theorem. The coordinates don't always correspond to distances along perpendicular axes, and it is even not always *possible* to introduce such coordinates (but let's not get ahead of ourselves). We want then to have a way of calculating the distance between points apart, where are some unspecified coordinates.