To add displacement vectors, link them end to start along a path and add the upper numbers together, then the lower numbers together. The single vector from the very start to the very end is the answer.
It builds on describing a translation with a column vector and leads into finding a point using a vector ratio.
Why does the path not matter?
A displacement vector records only the change from one point to another, not the route taken. So AC is the same vector whether you write it as AC, or as AB + BC, or as AD + DC.
This gives you a choice. Pick the route along arrows you already know, and follow the arrows in their direction.
How do I combine vectors along a path?
- Mark the start and the end of the vector you need.
- Trace a route along known vectors.
- If you travel against an arrow, use its reverse: change both signs.
- Add the upper numbers, then the lower numbers.
- Check by testing that your total makes sense when you picture the moves on a grid.
Worked example
On a diagram, AB = (4, 1), BC = (−1, 3) and CD = (2, −5). Find AD.
Step 1, route: A to B to C to D, all following the arrows.
Step 2, write the sum: AD = AB + BC + CD.
Step 3, upper numbers: 4 + (−1) + 2 = 5.
Step 4, lower numbers: 1 + 3 + (−5) = −1.
Answer: AD = (5, −1).
Check: across, the moves are 4 right, 1 left and 2 right, which is 5 right overall. Up and down, the moves are 1 up, 3 up and 5 down, which is 1 down overall. That matches (5, −1).
The mistake to watch for
A common slip is adding two arrows that point into the same place, so they do not link end to start.
Mistaken answer: for AC, the student added AB and CB, getting (4, 1) + (1, −3) = (5, −2).
Here CB = (1, −3) is the reverse of BC = (−1, 3).
AB and CB both end at B, so they cannot be chained.
To get from A to C, go A to B, then B to C. Use BC, not CB: AC = AB + BC = (4, 1) + (−1, 3) = (3, 4).
Equivalently, AC = AB − CB = (4, 1) − (1, −3) = (3, 4). The sign change on CB is exactly the reversal.
Check yourself
Try these, then open each answer.
1. AB = (2, 5) and BC = (−6, 1). Find AC.
Show answer
AC = AB + BC. Upper: 2 + (−6) = −4. Lower: 5 + 1 = 6.
AC = (−4, 6)
2. PQ = (3, −2), QR = (−1, −4) and RS = (5, 0). Find PS.
Show answer
Upper: 3 + (−1) + 5 = 7. Lower: −2 + (−4) + 0 = −6.
PS = (7, −6)
3. AB = (1, 4) and AC = (5, −2). Find BC.
Show answer
Route B to A to C: BC = BA + AC = −AB + AC = (−1, −4) + (5, −2) = (4, −6).
Check: AB + BC = (1, 4) + (4, −6) = (5, −2) = AC.
Where this leads next
Once paths feel natural, use them with fractions of a vector in finding a point using a vector ratio. Then test yourself on the vectors and transformations practice set. The non-calculator working trainer is handy for the signed sums.
If you can do the sums but the diagram route still feels unclear, our teachers can work through it with you in online one-to-one Mathematics tuition.