Compose and resolve velocity vectors
Add velocity components to find a resultant, and resolve a speed and direction into horizontal and vertical components.
Section 13 · Vectors in two dimensions
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
When resolving V at θ above east, which component is V cos θ?
What you will learn
- Compose two or more velocities by vector addition
- Resolve a velocity with known direction into perpendicular components
- Find a resultant speed and direction from its components
Add velocities using one shared coordinate frame
Choose positive coordinate directions and express each velocity in those components. Add horizontal components together and vertical components together to obtain the resultant velocity.
If a velocity has speed V at angle θ above the positive horizontal direction, its components are (V cos θ, V sin θ). To find its speed from components (u, v), use √(u² + v²).
Keep the direction convention attached to the components
A vector (−3, 4) points left and up. The angle from east is measured into the north-west quadrant; state a bearing or a named direction such as north of west to avoid ambiguity.
When resolving a given speed and angle, draw or imagine the right triangle: the adjacent component uses cosine and the opposite component uses sine. Convert degrees consistently with the calculator mode.
Pause and explain
What is the resultant velocity of (3, 2) and (−1, 4)?
Worked example
A moving object has velocity (3, 4) m/s. Find its speed and direction north of east.
Show the worked solution
- The speed is √(3² + 4²) = 5 m/s.
- The direction angle θ above east satisfies tan θ = 4/3.
- Evaluate the angle and state the reference direction.
Answer Speed 5 m/s; direction arctan(4/3) ≈ 53.1° north of east.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A boat's velocity is (3, 2) m/s and a current is (−1, 4) m/s. Find the resultant velocity and speed.
Give me a hint
Add components first, then use Pythagoras for the speed.
Compare my reasoning
- Resultant components are (3 − 1, 2 + 4) = (2, 6).
- Its speed is √(2² + 6²) = √40.
- Simplify the surd.
Resultant (2, 6) m/s; speed 2√10 m/s.
Look for these in your work
- Common directions used
- Components added
- Speed simplified
Resolve a velocity of 10 m/s at 30° north of east into components.
Give me a hint
East is adjacent to the angle; north is opposite.
Compare my reasoning
- East component = 10 cos 30° = 5√3.
- North component = 10 sin 30° = 5.
- Attach the coordinate directions and units.
(5√3, 5) m/s, east and north components respectively.
Look for these in your work
- Correct trig ratios used
- Exact components simplified
- Directions and units stated
A drone has velocity (−3, 4) m/s relative to the air, and the air moves at (5, 0) m/s relative to the ground. Find the drone's ground velocity and speed.
Give me a hint
Compose the two velocities in the same east-north coordinate frame.
Compare my reasoning
- Ground velocity is (−3, 4) + (5, 0) = (2, 4) m/s.
- Speed = √(2² + 4²) = √20.
- Simplify the magnitude.
Ground velocity (2, 4) m/s; speed 2√5 m/s.
Look for these in your work
- Relative velocities composed
- Vector components summed
- Resultant magnitude stated
Common mistakes
- Adding speeds instead of velocity components
- Using sine for the adjacent component
- Giving an angle without its reference direction
- Using different positive directions for vectors in the same sum
What can you explain now?
Resolve a speed of 10 m/s at 30° north of east into components.
Compare with the explanation
(5√3, 5) m/s
East component is 10 cos 30°; north component is 10 sin 30°.
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Make it stick
How do you resolve one velocity into components and compose several velocities into a resultant?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.