Adding signed vectors in one dimension
Add one-dimensional vectors without losing their directions.
Walk forward, then come back a little.
Need an earlier step? Start with scalars, vectors & direction →
Start at a door. Walk 7 m right, then 3 m left. To tell a friend where you ended up, you need your net change—not the length of your whole walk.
Words and symbols you will use
- Positive (+)
- Our chosen rightward direction.
- Negative (−)
- Leftward, opposite to our positive direction.
- Resultant
- One vector that has the same net effect as the vectors added together.
Read the picture, one step at a time.
- Follow the first arrow from 0 to 7 m.
- Start the second arrow where the first ends: move left from 7 to 4 m.
- Join the original start to the final finish. That arrow is +4 m: 7 + (−3).
| Compare | What it means | What follows |
|---|---|---|
| Signed displacements | Use directions. | +7 + (−3) = +4 m. |
| Lengths of the two legs | Count every meter traveled. | 7 + 3 = 10 m. |
The idea to keep: Only add compatible quantities. Two displacements can be added; a displacement in meters cannot be added to a velocity in meters per second.
How do you add vectors pointing in opposite directions?
Choose a positive direction, attach signs, then add the components. Use the result’s sign for direction and its absolute value for magnitude.
Walking 7 m east followed by 3 m west is not the same as moving 10 m east. The route is 10 m long, but the endpoint is only 4 m east of the start. Algebra keeps track of that distinction when each step has a sign.
Head-to-tail arrows show the same sum geometrically. Place the second arrow at the tip of the first without changing its length or direction. The resultant joins the original tail to the final tip. Only add vectors that describe compatible physical quantities.
A reliable approach
- Label an axis and convert every vector into a signed component.
- Add components with their units.
- Translate the final sign back into words; check whether cancellation makes sense.
Work through one example.
A cart moves +7 m and then −3 m. Find net displacement and total distance.
Follow the worked solution
- Net displacement: +7 m + (−3 m) = +4 m.
- Total distance: 7 m + 3 m = 10 m.
- The cart ends 4 m in the positive direction from its starting point.
Adding magnitudes always gives the distance for consecutive straight legs, not the net displacement when directions differ.
Explain it without notes: Explain why a zero sum does not mean nothing happened.
Try explaining it now.
A robot moves +6 m, −10 m, then +4 m. Has it traveled zero metres?
Compare with an explanation
No. Its displacement is zero, but it has traveled 20 m. The signed sum and the sum of leg lengths answer different questions.
Another explanation, if you need oneOptional external lesson · Flipping Physics
Vector and scalar refresher
This lesson is complete without a video. For another teacher’s explanation, open the original resource below. Pause after a diagram and explain the idea in your own words.
Suggested section 1:44–3:12. Video by Flipping Physics / Jonathan Thomas-Palmer. Refresh Kid is not affiliated with or endorsed by Flipping Physics. These links open another website. Some videos use g = 9.81 m/s²; our examples state g = 10 m/s². Follow the value given in each problem.
After watching: Explain why a zero sum does not mean nothing happened.
Make a prediction. Test it.
Try +8 m and −11 m. Reverse both signs. Does the resultant’s magnitude change? Explain why only its direction changes.
Motion graphs · same clock, different quantities
Try two questions.
Two original Refresh Kid questions. Choose an answer, explain it to yourself, then check the reasoning. These are not released AP exam questions.
Show your reasoning.
Original mini-FRQ · 4 points · Self-checkA robot moves 8 m right, 11 m left, then 2 m right. Right is positive.
- Write its signed legs.
- Find displacement.
- Find distance.
- Describe the final resultant arrow.
Your response stays on this page and is not submitted or automatically graded. Copy it before leaving.
Compare with the worked solution & scoring guide
- 1 point: +8 m, −11 m, +2 m.
- 1 point: Δx = −1 m.
- 1 point: Distance = 21 m.
- 1 point: A 1 m arrow left, from the original position to the endpoint.
Accept an equivalent correct method. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1When do equal 1D vectors cancel?
When their magnitudes match and their directions oppose.
RECALL 2What does a zero resultant imply?
No net change in the quantity represented; the individual vectors may be nonzero.
RECALL 3Can you add 3 m and 2 m/s?
No. These are different physical quantities.
Come back tomorrow: answer these with the cards closed. Try again a week later, especially the ones you missed.
Keep the key ideas handy.
Adding signed vectors in one dimension
Core idea: Choose a positive direction, attach signs, then add the components. Use the result’s sign for direction and its absolute value for magnitude.
- Δx_total = Δx₁ + Δx₂ + …
- For straight legs: distance = |Δx₁| + |Δx₂| + …
Avoid this: Adding magnitudes always gives the distance for consecutive straight legs, not the net displacement when directions differ.
Remember why: Only add compatible quantities. Two displacements can be added; a displacement in meters cannot be added to a velocity in meters per second.
Combine components along the same axis and with compatible units.
Explain, don’t just substituteExplain why a zero sum does not mean nothing happened.
Refresh Kid · AP Physics 1 · Unit 1 · 1.1.B · Check units, direction and model assumptions.
Connect to released AP practice.
2026 · Question 1 · Version J
Use Part A(i) for component velocity graphs and Part A(ii) for a kinematics derivation. The remaining parts use fluid concepts from Unit 8; they are not Unit 1-only practice.
This is a Unit 1 synthesis task to revisit after learning projectile motion. Historical papers may use different timing or course coverage. Official questions remain on College Board’s site; our questions below are original practice.
Browse released years and scoring information ↗Framework alignment: College Board CED, Topic 1.1. Current exam corrections. Checked September 16, 2026. These explanations and practice items are independently authored by Refresh Kid. The simulation is a mathematical model, not experimental data.
About the videos and learning approach
Optional videos are linked to the publisher’s website and YouTube channel with credit. No external video player is loaded on this lesson page. The written lessons, simulations and practice here are independently authored by Refresh Kid.
We combine worked examples, visual models, explanation and recall practice. See the Institute of Education Sciences study guide ↗ for the underlying learning recommendations.
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