IGCSE Mathematics: Vectors Practice Questions
A vector has magnitude and direction, written as a column with the horizontal movement on top. Vectors are added by adding corresponding components, and two vectors are parallel if one is a scalar multiple of the other.
Topic 7.2 of Cambridge IGCSE Mathematics 0580 splits into straightforward column arithmetic and harder geometric proof. The proof questions all reduce to expressing an unknown route in terms of the given vectors. The questions below cover both halves.
What you need to know for Vectors
- Column vectorThe top number is the movement in the x direction and the bottom number the movement in the y direction. Negatives mean left and down.
- MagnitudeThe length of a vector, found using Pythagoras on its two components.
- Adding and scalingAdd corresponding components. Multiplying by a scalar multiplies both components.
- Route notationThe vector AB equals AO plus OB, which is minus a plus b. Travelling backwards along a vector reverses its sign.
- MidpointIf M is the midpoint of AB, then OM = a + half of (b minus a), which simplifies to half of (a + b).
- Proving parallelShow that one vector is a scalar multiple of the other. State the conclusion explicitly, since the reasoning carries a mark.
IGCSE Mathematics Vectors questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Vector a has components 3 and 4. Calculate its magnitude.
Show the worked answer
- The magnitude is the length of the vector, found using Pythagoras on the two components.
- 32 + 42 = 9 + 16 = 25.
- The magnitude is the square root of 25.
- The magnitude of a is 5. Magnitude is never negative, whatever the signs of the components.
Vector a has components 3 and minus 1. Vector b has components 2 and 4. Calculate 2a minus 3b as a column vector.
Show the worked answer
- Multiply a by 2: each component doubles, giving components 6 and minus 2.
- Multiply b by 3: each component triples, giving components 6 and 12.
- Subtract the second from the first, component by component.
- Top: 6 minus 6 = 0. Bottom: minus 2 minus 12 = minus 14. So 2a minus 3b has components 0 and minus 14.
In a diagram, OA = a and OB = b. M is the midpoint of AB. Express OM in terms of a and b, showing your reasoning.
Show the worked answer
- First find AB by travelling from A to O and then O to B: AB = minus a + b, which is b minus a.
- M is the midpoint of AB, so AM is half of AB, giving AM = half of (b minus a).
- Travel from O to M via A: OM = OA + AM = a + half of (b minus a).
- Simplify: a + half b minus half a = half a + half b = half of (a + b).
Vector PQ has components 4 and 6, and vector RS has components 6 and 9. Show that PQ and RS are parallel.
Show the worked answer
- Two vectors are parallel if one is a scalar multiple of the other.
- Compare the components: 6 divided by 4 = 1.5, and 9 divided by 6 = 1.5.
- Both components scale by the same factor, so RS = 1.5 x PQ.
- Since RS is a scalar multiple of PQ, the two vectors are parallel. The scalar tells you RS is 1.5 times as long.
Common mistakes in this topic
- Reversing the sign when travelling backwards along a vector.
- Adding components instead of using Pythagoras for magnitude.
- Sign errors when subtracting negative components.
- Writing a vector answer without keeping it in column or bold form.
- Omitting the concluding statement in a parallel or collinear proof.
Exam tips
- Draw the route on the diagram before writing any algebra. The route dictates the signs.
- Travelling against the arrow means a negative. Travelling with it means a positive.
- For midpoints, go from the origin to a known point first, then half way along the connecting vector.
- In proofs, always finish with a sentence stating what you have shown.
- Check a magnitude answer is positive and larger than either individual component.
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Vectors FAQs
How do I find the magnitude of a vector?
Apply Pythagoras to the two components: square each, add them, then take the square root. A vector with components 3 and 4 has magnitude 5. Magnitude is a length, so it is always positive regardless of the signs of the components.
How do I express a route as a vector?
Break the journey into steps along the vectors you are given. Travelling in the direction of a vector adds it, and travelling against it subtracts it. So the vector from A to B, where OA is a and OB is b, equals minus a plus b, which is b minus a.
How do I prove two vectors are parallel?
Show that one vector is a scalar multiple of the other, meaning both components scale by the same factor. Then state explicitly that because one is a multiple of the other, the vectors are parallel. The concluding statement carries its own mark.
How do I find the vector to a midpoint?
Travel from the origin to one endpoint, then half way along the vector joining the two endpoints. If OA is a and OB is b, then OM equals a plus half of (b minus a), which simplifies neatly to half of (a plus b).
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.