Magnitude AND direction. Add, subtract, scale, find magnitude, and reason geometrically with position vectors. The skill that unlocks Paper 4 "prove these points are collinear" or "find AB" questions.
What you’ll learn
Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).
E5.1 — Describe a translation using a vector represented by (xy).
E5.2 — Add and subtract vectors; multiply a vector by a scalar.
E5.3 — Calculate the magnitude of a vector.
E5.4 — Use position vectors to describe paths in geometric problems.
Vector arithmetic
Add and subtract component-wise. Scalar multiplication scales each component.
Addition.(ab)+(cd)=(a+cb+d).
Subtraction.(ab)−(cd)=(a−cb−d).
Scalar multiplication.k(ab)=(kakb).
The scalar k multiplies BOTH components.
Worked.a=(3−2), b=(−14). Find 2a−3b.
2a=(6−4).
3b=(−312).
2a−3b=(6−(−3)−4−12)=(9−16).
Add and subtract component by component.
Scalar multiplication: scale BOTH components.
Don't divide by a vector — undefined.
Always show the column vectors clearly.
Magnitude (length) of a vector
Pythagoras on the components. ∣v∣=x2+y2.
Formula.∣v∣ for v=(xy):
∣v∣=x2+y2.
This is just Pythagoras — the magnitude is the length of the vector if you draw it as an arrow.
Worked.v=(5−12).
∣v∣=25+144=169=13.
Magnitude is always positive. And ∣0∣=0 — only the zero vector has zero magnitude.
∣v∣=x2+y2.
Always non-negative.
Pythagoras applied to the components.
Position vectors and AB
AB is end minus start. Position vectors point from origin.
A position vectorOA points from the origin O to the point A.
Path between two points.AB=OB−OA.
End minus start. The notation tells you direction matters.
Worked.A(2,1), B(7,5).
AB=(7−25−1)=(54).
BA=(−5−4) (just negate).
Midpoint. If M is the midpoint of AB,
OM=21(OA+OB).
Triangle path navigation.AB+BC=AC. (Chain the trips.)
Cambridge geometry style. Often Cambridge gives you OA=a and OB=b and asks you to express AB, AM, etc. in terms of a and b.
Worked.OA=a, OB=b. M is the midpoint of AB. Express OM in terms of a and b.
OM=21(a+b).
AB=OB−OA.
Midpoint: OM=21(OA+OB).
Chain: AB+BC=AC.
Reverse: BA=−AB.
Parallel and collinear vectors
Two vectors are parallel iff one is a scalar multiple of the other. Three points are collinear iff a vector chain factorises this way.
Parallel test.u is parallel to v if and only if there exists a scalar k=0 such that
u=kv.
Worked.u=(4−2), v=(−63). Are they parallel?
Try k: 4−6=−1.5 and −23=−1.5. Same k.
Yes, v=−1.5u. Parallel.
Collinear points. Three points A,B,C are collinear iff AB is a scalar multiple of AC (they share the same line).
Worked example structure. "Show A,B,C are collinear" → compute AB and AC, factor out a common scalar.
Parallel: u=kv.
Collinear: AB=k⋅AC.
k can be negative or fractional.
Both components must give the SAME k.
How it’s examined
Vectors appear every Paper 4 as a 5-7 mark question — typically position-vector geometry with a and b given. Paper 2 has 2-3 mark single-step items. Examiner reports flag sign errors in AB=b−a (writing a−b instead).
Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.
Take this whole topic with you
Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Vectors, ready to print or save as PDF.
Step-by-step worked examples — Vectors
Step-by-step solutions to past-paper-style questions on vectors, written exactly the way a tutor would explain them at the board.
1Add and subtract column vectors
Core• addition
▼
Question
Given a=(3−2) and b=(−14), find a+2b.
Step-by-step solution
Step 1
Compute 2b.
2b=(−28)
Step 2
Add component-wise.
a+2b=(3+(−2)−2+8)=(16)
Answer
(16)
2Find the magnitude of a vector
Core• magnitude
▼
Question
Find ∣v∣ where v=(5−12).
Step-by-step solution
Step 1
∣v∣=x2+y2.
=25+144=169=13
Answer
∣v∣=13
3Use position vectors to find a path
Extended• Adapted from 0580/42 May/Jun 2024 Q16• position vector
▼
Question
OA=a and OB=b. Express AB in terms of a and b.
Step-by-step solution
Step 1
Use AB=OB−OA.
AB=b−a
Answer
AB=b−a
4Position vector of a midpoint
Extended• midpoint
▼
Question
M is the midpoint of AB. Express OM in terms of a and b.
Step-by-step solution
Step 1
Midpoint position vector is the average.
OM=21(a+b)
Answer
OM=21(a+b)
5Show two vectors are parallel
Extended• parallel
▼
Question
u=(4−2) and v=(−63). Show that u is parallel to v.
Step-by-step solution
Step 1
v=−23u since −23×4=−6 and −23×−2=3.
Step 2
One is a scalar multiple of the other → parallel.
Answer
v=−23u, hence parallel.
Key Formulae — Vectors
The formulae you need to memorise for vectors on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.
Magnitude of a 2D vector
(xy)=x2+y2
When to use
Length of a vector or distance between two points (via vector form).
Vector addition / subtraction
(ab)±(cd)=(a±cb±d)
When to use
Combine vectors component-wise.
Path between two points
AB=OB−OA
When to use
When given position vectors and asked for the vector between two points.
Parallel vectors
u∥v⟺v=ku
k
non-zero scalar
When to use
To prove two vectors are parallel.
Key Definitions and Keywords — Vectors
Definitions to memorise and the exact keywords mark schemes credit for vectors answers — sharpened from recent examiner reports for the 2026 0580 sitting.
Vector
Examiner keyword
A quantity with both magnitude and direction. Often written as a column vector (xy).
Scalar
A pure number (no direction).
Magnitude
Examiner keyword
The length of a vector, written ∣v∣.
Position vector
Examiner keyword
A vector from a fixed origin O to a point A, written OA or a.
Parallel vectors
Two vectors are parallel iff one is a scalar multiple of the other.
Common Mistakes and Misconceptions — Vectors
The traps other students keep falling into on vectors questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.
✕Computing AB as a−b
0580/42 — recurring
▼
Why it happens
Confusing the order of subtraction.
How to avoid it
AB=OB−OA=b−a. End point minus start point.
✕Forgetting the square root in magnitude
▼
Why it happens
Speed.
How to avoid it
Magnitude formula has — not just x2+y2.
✕Multiplying only one component by the scalar
▼
Why it happens
Forgetting the scalar applies to BOTH components.
How to avoid it
k(ab)=(kakb) — multiply BOTH.
✕Treating a and b as scalars in proofs
▼
Why it happens
They look like ordinary letters in algebra.
How to avoid it
Vectors are bolded or have arrows. Don't divide by a vector or take square roots of vectors.
Practice questions
Exam-style questions with step-by-step worked solutions. Try one before checking the method.
Past paper style quiz
Get a report showing which sub-topics you've nailed and which ones still need work.
4. Exam Quiz
Assess your understanding
Attempt a past paper style quiz for this sub-topic and get instant feedback to identify your strengths and weaknesses.
Instant AI marking SchemeExaminer's feedbackAI Detailed report
Video lesson
Short walkthrough of the concepts students most often get stuck on.
Vectors — frequently asked questions
The things students keep getting wrong in this sub-topic, answered.