11.4PureCore
Position Vectors
A position vector pins a point to the origin. Once points have position vectors, the vector joining any two of them follows from one simple rule — the key to all the geometry problems in the next lesson.
What you'll be able to do
- Understand what a position vector is
- Write the position vector of a point
- Find the vector between two points
- Find the distance between two points using vectors
1
Position vectors
The of a point is , the vector from the origin to . It is usually written . So the point has position vector .
2
The vector joining two points
To get from to , go back to the origin then out to : . Always “end minus start”.
End point’s position vector minus the start point’s.
1.
Answer
Tip — AB = b − a, not a − b. Getting the order wrong reverses the vector.
3
Distance between points
The distance from to is the magnitude of — exactly the distance formula in disguise.
Find the joining vector, then its magnitude.
1.
2Magnitude .
Answer
Formula recap
Position vector of A.
Vector joining two points.
Distance between points.
Common mistakes to avoid
Writing AB = a − b.
It is end minus start: AB = b − a.
Confusing a position vector with a displacement between two general points.
A position vector is always measured from the origin O.
Key takeaways
- A position vector OA = a runs from the origin to the point.
- AB = b − a (end minus start).
- Distance from A to B is |b − a|.
Test yourself
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