Vector components & resultants in two dimensions
Choose sine or cosine from the angle reference, not a memorized axis rule.
Describe one diagonal move in two directions.
Need an earlier step? Start with scalars, vectors & direction →
Move 3 m right and 4 m up on a grid. The straight displacement from start to finish is 5 m. Horizontal and vertical components describe this one displacement from two perpendicular viewpoints.
Words and symbols you will use
- Component
- The signed part of a vector along one chosen axis.
- Magnitude A
- The length of the complete vector: here 5 m.
- Angle θ (theta)
- An angle measured from an explicitly named axis.
Read the picture, one step at a time.
- The horizontal leg is +3 m and the vertical leg is +4 m.
- The complete vector joins start to finish. Its magnitude is √(3² + 4²) = 5 m.
- The components are perpendicular. Do not add 3 + 4 to find the diagonal magnitude.
| Compare | What it means | What follows |
|---|---|---|
| θ from +x toward +y | Horizontal is adjacent to θ. | Aₓ = A cos θ; Aᵧ = A sin θ. |
| θ from +y toward +x | Vertical is adjacent to θ. | Aₓ = A sin θ; Aᵧ = A cos θ. |
The idea to keep: For an acute angle in a right triangle, cosine = adjacent ÷ hypotenuse and sine = opposite ÷ hypotenuse. Sketch the angle first, then assign component signs from direction.
When do I use sine or cosine for vector components?
If the angle is measured from the positive x-axis, the x-component uses cosine and the y-component uses sine. Check which side is adjacent to the stated angle.
Components describe one vector along perpendicular axes. They are not two extra vectors acting alongside the original: together they represent it. For a velocity arrow, the components tell how fast the horizontal and vertical coordinates change.
For a sum, add x-components to x-components and y-components to y-components first. Only then find the resultant magnitude. Use the signs to choose a quadrant; an inverse tangent without quadrant reasoning can point in the wrong direction.
When the angle is given from the vertical, redraw the right triangle before selecting a function. A 10 m displacement 30° from +y toward +x has x = 5 m and y ≈ 8.66 m. Select the other angle reference in the model and describe the same vector using 60° from +x.
A reliable approach
- Draw the axes and mark where the angle is measured.
- Resolve components with signs appropriate to the quadrant.
- Recombine with the Pythagorean theorem and a direction that matches both component signs.
Work through one example.
A velocity has components +3 m/s east and +4 m/s north. Find its speed and direction.
Follow the worked solution
- Speed = √(3² + 4²) = 5 m/s.
- Angle = arctan(4/3) ≈ 53.1° north of east.
- Both components are positive, so the arrow lies in the first quadrant.
Adding component magnitudes, 3 + 4 = 7, does not give the magnitude of a perpendicular resultant.
Explain it without notes: Explain why the same vector can be described as 30° from +y or 60° from +x.
Try explaining it now.
A 10 m vector is 30° from +y toward +x. Is its x-component 10 cos 30°?
Compare with an explanation
No. Here x is opposite the angle: x = 10 sin 30° = 5 m, while y = 10 cos 30° ≈ 8.66 m.
Another explanation, if you need oneOptional external lesson · Flipping Physics
Two-dimensional and projectile motion
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 16:57–20:13. 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 the same vector can be described as 30° from +y or 60° from +x.
Make a prediction. Test it.
Rotate the vector from 30° to 150°. Compare the component signs and the unchanged magnitude. Use the 3D view to see that the motion still lies in the x–y plane.
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-checkVector A is 10 m at 30° from +y toward +x. Vector B has components (−2 m, +1 m).
- Draw the angle and resolve A.
- Find R = A + B in components.
- Find its magnitude.
- State the direction quadrant and explain why x does not use cosine of the stated 30°.
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: A_x = 10 sin 30° = 5 m; A_y = 10 cos 30° = 5√3 m.
- 1 point: R_x = 3 m; R_y = 1 + 5√3 m.
- 1 point: |R| = √[3² + (1 + 5√3)²] m ≈ 10.12 m.
- 1 point: Both components are positive, so R is in quadrant I. The stated angle is measured from +y, making x the opposite side.
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 1What changes if the angle is measured from the y-axis?
The adjacent component changes; draw the triangle instead of memorizing x = cosine blindly.
RECALL 2Can one component exceed the vector magnitude?
No, for perpendicular Cartesian components.
RECALL 3How do you add 2D vectors?
Add matching components, then recombine the resultant.
Come back tomorrow: answer these with the cards closed. Try again a week later, especially the ones you missed.
Keep the key ideas handy.
Vector components & resultants in two dimensions
Core idea: If the angle is measured from the positive x-axis, the x-component uses cosine and the y-component uses sine. Check which side is adjacent to the stated angle.
- A_x = A cos θ; A_y = A sin θ, for θ from +x.
- A = √(A_x² + A_y²)
- R_x = A_x + B_x; R_y = A_y + B_y
- If φ is from +y toward +x: A_x = A sin φ; A_y = A cos φ.
Avoid this: Adding component magnitudes, 3 + 4 = 7, does not give the magnitude of a perpendicular resultant.
Remember why: For an acute angle in a right triangle, cosine = adjacent ÷ hypotenuse and sine = opposite ÷ hypotenuse. Sketch the angle first, then assign component signs from direction.
Draw the stated angle. x uses cosine only when that angle is measured from +x.
Explain, don’t just substituteExplain why the same vector can be described as 30° from +y or 60° from +x.
Refresh Kid · AP Physics 1 · Unit 1 · 1.5.A · 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 selection directly connects to this lesson. 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 1.5. 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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