To express a displacement, write the journey from one point to another as a sum of known vectors. The key rule is end minus start, or equivalently, walk along known legs and keep every sign.
This skill is the first step of every proof in two-dimensional vector proofs.
How do you build a displacement step by step?
- Name the start and end points of the displacement you want.
- Choose a route using vectors you know, often through the origin O.
- Write each leg. Going against a known vector gives its negative.
- Add the legs and simplify, collecting the coefficients of a and b.
For points A and B with →OA = a and →OB = b, going A to O to B gives −a + b, so →AB = b − a.
Worked example
In triangle OAB, →OA = a and →OB = b. P is the midpoint of OA and Q is the midpoint of OB.
Find →AB, →PQ and →PB. What do your answers show about PQ and AB?
→AB: route A to O to B: →AB = −a + b = b − a.
→PQ: route P to O to Q. Since P is the midpoint of OA, →PO = −½a. Since Q is the midpoint of OB, →OQ = ½b.
→PQ = −½a + ½b = ½(b − a)
→PB: route P to O to B: →PB = −½a + b = b − ½a.
Conclusion: →PQ = ½→AB, so PQ is parallel to AB and half its length.
The mistake to watch for
A common slip is to write →AB = a − b.
Mistaken answer: →AB = a − b
The student subtracted in the order the letters appear in the question, instead of end minus start.
To check, trace the journey with your finger: from A you must first go back to O, which is −a. Then you go out to B, which is +b.
So →AB = −a + b = b − a. A sketch with arrows on each leg catches this every time.
Check yourself
1. With →OA = a and →OB = b, write →BA.
Show answer
→BA = −→AB = −(b − a) = a − b. Check: B to O is −b, then O to A is +a.
a − b
2. OABC is a parallelogram with →OA = a and →OC = c. Write →OB and →AC.
Show answer
In a parallelogram →AB = →OC, so →OB = →OA + →AB = a + c. For →AC, go A to O to C: −a + c.
→OB = a + c, →AC = c − a
3. →OP = 3a − 2b and →OQ = a + 4b. Find →PQ.
Show answer
→PQ = →OQ − →OP = (a + 4b) − (3a − 2b) = a − 3a + 4b + 2b.
−2a + 6b
Where this leads next
Next, use displacements to place a point on a line in find a dividing point on a segment. The non-calculator working trainer is useful for the fraction work that comes with it.
If your diagrams are right but your first line keeps going wrong, our teachers look for exactly that in online one-to-one Additional Mathematics tuition.