Finding a Vector Between Two Points
Simple Explanation
Once you know the position vectors of two points A and B (from a fixed origin O), the vector that runs directly from A to B is found by subtracting: AB = OB − OA.
Why Do We Need It?
This is the crucial bridge that lets you compute the vector connecting any two points as soon as you know both their position vectors — the basis of nearly every later vector calculation.
See It
Origin O with dashed arrows to A and B (their position vectors), and a solid arrow directly from A to B
Formula
The Vector Between Two Points
AB = OB − OA
The vector from point A to point B equals B's position vector minus A's position vector.
- O
- — the fixed origin
- OA, OB
- — the position vectors of A and B
- AB
- — the vector from A to B
When to use it: Whenever you know the position vectors of two points and need the vector that goes from one to the other.
Worked Example
Derive the formula for the vector between two points
Points A and B have position vectors OA and OB from origin O. Given that OA + AB = OB (by the Triangle Law), rearrange to express AB in terms of OA and OB.
Why Does This Work?
This is the same Triangle Law logic used for vector addition: travelling from O to A, then A to B, is the same overall journey as travelling directly from O to B, so OA + AB = OB. Rearranging that equation to isolate AB gives the formula directly.
Real-Life Example
A robot arm computing the path to a target
A robot arm knows its base position (O), its current gripper position (A), and a target location (B).
Computing AB = OB − OA gives exactly the vector the arm needs to move along to reach the target from its current position.
Practice
OA has magnitude 10, and OB points in exactly the opposite direction to OA with the same magnitude (so OB = −OA). Find the magnitude of AB.
MediumCommon mistake
Computing AB as OA − OB instead of OB − OA — remember the order is "destination minus start."
Quick Review
- AB = OB − OA — destination's position vector minus the start's.
- Derived directly from the Triangle Law: OA + AB = OB.
- Always subtract in the order (destination − start).