Equate vector components to solve unknowns
Use equality of like vectors to form and solve scalar equations for unknown components or coefficients.
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
If (2x, y − 1) = (8, 5), what equation comes from the first component?
What you will learn
- Equate corresponding components of equal vectors
- Solve simultaneous equations arising from vector equality
- Substitute unknowns back into the original vector equation
Equal vectors must match in each component
If (p, q) = (r, s), then p = r and q = s. These are two scalar equations from one vector equation.
When unknowns multiply known vectors, expand each component first. For x(2, 1) + y(1, −1) = (5, 1), the horizontal equation is 2x + y = 5 and the vertical equation is x − y = 1.
Solve the scalar system, then verify the vector
Use substitution or elimination to find the unknown coefficients. Check both components in the original equation; satisfying only one coordinate is not enough.
Like-vector equality is also useful in geometric arguments: if two paths describe the same displacement, equate their horizontal and vertical components separately.
Pause and explain
If (p + 2, 3q) = (7, −6), what is q?
Worked example
Find x and y if x(2, 1) + y(1, −1) = (5, 1).
Show the worked solution
- Compare components: 2x + y = 5 and x − y = 1.
- Add the equations to get 3x = 6, so x = 2.
- Substitute into x − y = 1 to get y = 1; check both components.
Answer x = 2 and y = 1.
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.
Find p and q if p(1, 2) + q(2, −1) = (5, 0).
Give me a hint
Write one scalar equation for each coordinate.
Compare my reasoning
- The horizontal equation is p + 2q = 5.
- The vertical equation is 2p − q = 0, so q = 2p.
- Substitute: p + 4p = 5, so p = 1 and q = 2.
p = 1 and q = 2.
Look for these in your work
- Both component equations formed
- System solved consistently
- Solution checks in each component
Find k if 3(2, k) − (1, 4) = (5, 2).
Give me a hint
Match the first components first, then use the second components to find k.
Compare my reasoning
- The first component is 6 − 1 = 5, which agrees.
- The second component gives 3k − 4 = 2.
- Thus 3k = 6 and k = 2.
k = 2.
Look for these in your work
- Vector operations expanded
- Both components checked
- Unknown solved
Two routes have the same displacement: 2a + b = (8, 3), where a = (2, 1) and b = (x, y). Find b.
Give me a hint
Subtract the known displacement 2a from both sides.
Compare my reasoning
- 2a = (4, 2).
- b = (8, 3) − (4, 2).
- Subtract corresponding components.
b = (4, 1), so x = 4 and y = 1.
Look for these in your work
- Known vector scaled
- Subtraction direction correct
- Both unknown components found
Common mistakes
- Equating only the first components
- Adding vectors before multiplying their scalar coefficients
- Changing a negative component sign when setting up an equation
- Not checking the solution in both coordinates
What can you explain now?
If (a, 4) = (3, b), what are a and b?
Compare with the explanation
a = 3 and b = 4
Corresponding coordinates of equal vectors are equal.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does one vector equation become two scalar equations, and how should you verify the answer?
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.