A position vector says where a point is, measured from the origin. A displacement vector says how to get from one point to another. They are linked by one rule: AB = OB − OA.
This lesson sits alongside finding a point using a vector ratio and draws on finding a midpoint and a segment length.
What is the difference between the two?
The position vector of A(3, −2) is OA = (3, −2). It always starts at the origin, so its numbers match the coordinates.
A displacement vector such as AB can start anywhere. It records only the change from A to B. The same displacement (4, 1) could be drawn from many different starting points, and each drawing is the same vector.
How do I turn positions into a displacement?
- Write the position vectors OA and OB from the coordinates.
- Subtract: AB = OB − OA. End minus start, upper and lower separately.
- Check the direction by adding AB to OA. You should reach OB.
Worked example
A is the point (3, −2) and B is the point (−1, 5). Find the position vectors of A and B, and then find AB.
Step 1, position vectors: OA = (3, −2) and OB = (−1, 5).
Step 2, subtract: AB = OB − OA = (−1 − 3, 5 − (−2)) = (−4, 7).
Answer: AB = (−4, 7).
Check: OA + AB = (3 + (−4), −2 + 7) = (−1, 5) = OB. Correct.
The reverse vector is BA = (4, −7), which points from B back to A.
The mistake to watch for
A common slip is computing start minus end, which gives the displacement in the wrong direction.
Mistaken answer: AB = OA − OB = (3 − (−1), −2 − 5) = (4, −7).
This is actually BA, not AB.
The fix is to say the words “end minus start” as you write. A simple check is to add your vector to the start point: (3, −2) + (4, −7) = (7, −9), which is not B(−1, 5), so the check fails. A second slip is treating a displacement as if it were a point, for example writing “AB = (−4, 7)” and then plotting the point (−4, 7) as if it were B.
Check yourself
Try these, then open each answer.
1. OP = (2, 6) and OQ = (5, 1). Find PQ.
Show answer
PQ = OQ − OP = (5 − 2, 1 − 6) = (3, −5).
Check: (2, 6) + (3, −5) = (5, 1) = OQ.
2. AB = (4, −3) and OA = (1, 2). Find OB.
Show answer
OB = OA + AB = (1 + 4, 2 + (−3)) = (5, −1).
3. OA = (6, 2) and OB = (−2, 8). M is the midpoint of AB. Find the position vector of M.
Show answer
Add the two position vectors and halve: (6 + (−2), 2 + 8) ÷ 2 = (4, 10) ÷ 2 = (2, 5).
Check: AB = (−8, 6), half is (−4, 3), and OA + (−4, 3) = (2, 5).
Where this leads next
After this, try the vectors and transformations practice set, which mixes all five skills. Keep a record of the slips you make in the mistake log and retest queue, and use the non-calculator working trainer for signed arithmetic.
Some students can follow every line above but still reverse a vector under exam conditions. That is the sort of pattern our teachers look for in online one-to-one Mathematics tuition.