Medium

Position Vectors and the Origin

Simple Explanation

A position vector is a special vector that starts at a fixed origin O and ends at a particular point — it describes exactly where that point is, relative to O. Every point in a plane has exactly one position vector.

Why Do We Need It?

Position vectors let every point be treated as a vector, which is what makes it possible to do vector algebra (like the ratio formula) directly with locations, not just displacements.

See It

Position vectors OA and OB from a common origin O

Worked Example

Decide whether two position vectors are equal

The origin O is fixed. Point A is located 4 units from O in one direction, and point C is located 4 units from O in a different direction. Are OA and OC the same vector?

Why Does This Work?

This follows directly from the definition of vector equality (matching magnitude AND direction), combined with the fact that each point has exactly one position vector from a fixed O — so two position vectors are equal precisely when the two points they describe are actually the same point.

Real-Life Example

GPS coordinates as position vectors

A mapping system measures every location as a distance and direction from one fixed reference point.

Two places at the same distance from that reference point, but in different directions, are clearly different locations — exactly like two position vectors with equal magnitude but different direction.

Practice

Point A has position vector OA with magnitude 7. A new point A' is placed so that OA' = 2·OA (same direction, doubled). Find |OA'|.

Medium

Common mistake

Assuming two points are different just because they're labeled differently — if their position vectors from O genuinely match in both magnitude and direction, they are literally the same point.

Quick Review

A position vector runs from the fixed origin O to a specific point.
Every point has exactly one position vector.
Two position vectors are equal only when magnitude AND direction both match — meaning the same point.