a position vector

A vector usually floats free — only its length and direction matter. But to talk about WHERE a point sits, we need to anchor things. A position vector does this by tethering one end to a fixed origin: it is the arrow from the origin O straight to the point P. With this convention, every point in space gets its own personal vector, and the geometry of locations becomes the algebra of vectors.

If P has coordinates (x, y, z), its position vector is simply r = OP = (x, y, z) — the components and the coordinates coincide, because the tail is pinned at O = (0, 0, 0). This single idea unlocks a beautifully uniform language. The displacement from a point A (position vector a) to a point B (position vector b) is just b - a, because going from A to B is the same as undoing a and then doing b. The midpoint of AB has position vector (a + b)/2. A point dividing AB in the ratio m : n has position vector (n a + m b)/(m + n). Whole theorems about midpoints and centroids fall out as one-line vector calculations.

Position vectors are the workhorse behind the vector equations of lines and planes: a line through A in direction d is the set of points r = a + t d as the parameter t runs over all real numbers. The one thing to remember is that a position vector is meaningful only relative to a chosen origin — move the origin and every position vector changes. By contrast a displacement like b - a is origin-independent, since both endpoints shift together and the difference is unchanged. So use position vectors to name points, and differences of them to describe genuine geometry.