Vectors, Magnitude, and Direction

A single number can tell you how far something moved, but not which way. An arrow fixes that: it points somewhere and it has a length. That's a vector, and that extra piece of information, direction, is what makes vectors useful for describing velocity, force, or any quantity that cares about where it's headed, not just how much.

To build a vector from two points, subtract. If the tail is at (x0, y0) and the head is at (x1, y1), the horizontal component is x1 - x0 and the vertical component is y1 - y0. Those two numbers, written as <x1 - x0, y1 - y0>, tell you exactly how far to move right/left and up/down to get from tail to head. The magnitude, the length of the arrow, comes straight from the Pythagorean theorem: √((x1-x0)² + (y1-y0)²), since the components form the legs of a right triangle and the vector itself is the hypotenuse.

This sets up everything else in the unit. Once a vector is just a pair of components, you can add vectors by adding components, scale them by multiplying components, and eventually pack them into matrices. One fact makes all of that work cleanly: a vector's identity depends only on its components, not on where you drew it. Slide the same arrow anywhere in the plane and the magnitude and direction never change, only the tail and head coordinates do.