Vectors Worked Examples for IGCSE Maths
Working through solved examples is one of the most effective ways to master vectors in IGCSE Mathematics. These worked examples, curated by Teacher Rig, cover the most common question types you will e
Working through solved examples is one of the most effective ways to master vectors in IGCSE Mathematics. These worked examples, curated by Teacher Rig, cover the most common question types you will encounter in the Cambridge IGCSE 0580 exam. Each solution shows every step of working with clear explanations of the reasoning behind each step.
Example 1: Finding a vector between two points
Question
OA = a and OB = b. M is the midpoint of AB. Find the vector OM in terms of a and b.
Solution
- 1
Find vector AB
AB = AO + OB = -a + b = b - a
To go from A to B, go backwards along OA (giving -a) then forwards along OB (giving +b).
- 2
Find vector AM
AM = (1/2)AB = (1/2)(b - a)
M is the midpoint of AB, so AM is half of AB.
- 3
Find vector OM
OM = OA + AM = a + (1/2)(b - a) = a + (1/2)b - (1/2)a = (1/2)a + (1/2)b
Go from O to A, then from A to M.
Final Answer: OM = (1/2)a + (1/2)b = (1/2)(a + b)
Exam Tip
The midpoint of AB always has position vector (1/2)(a + b) when OA = a and OB = b. This is worth memorising.
Example 2: Proving three points are collinear
Question
OA = 2a + 3b, OB = 4a + 6b, and OC = 7a + 10.5b. Show that A, B, and C are collinear.
Solution
- 1
Find vector AB
AB = AO + OB = -(2a + 3b) + (4a + 6b) = 2a + 3b
Go from A to O (which is -OA) then from O to B (which is OB).
- 2
Find vector AC
AC = AO + OC = -(2a + 3b) + (7a + 10.5b) = 5a + 7.5b
Go from A to O then from O to C.
- 3
Check if AC is a scalar multiple of AB
AC = 5a + 7.5b = 2.5(2a + 3b) = 2.5 AB
Factor out 2.5 from AC. Since AC = 2.5 AB, the vectors are parallel.
- 4
State the conclusion
AC = 2.5 AB and A is a common point, so A, B, and C are collinear.
Two conditions needed: vectors must be scalar multiples AND share a common point.
Final Answer: AC = 2.5 AB with A as a common point, therefore A, B, and C are collinear.
Exam Tip
For collinearity proofs: (1) find two vectors from a common point, (2) show one is a scalar multiple of the other, (3) state both conditions clearly for full marks.
Example 3: Vector addition with column vectors
Question
a = (3, -1) and b = (-2, 5). Find: (a) a + b, (b) 2a - b, (c) |a|.
Solution
- 1
Find a + b
(3, -1) + (-2, 5) = (3 + (-2), -1 + 5) = (1, 4)
Add corresponding components: top with top, bottom with bottom.
- 2
Find 2a - b
2(3, -1) - (-2, 5) = (6, -2) - (-2, 5) = (6-(-2), -2-5) = (8, -7)
First multiply a by 2, then subtract b component by component.
- 3
Find |a|
|a| = sqrt(3 squared + (-1) squared) = sqrt(9 + 1) = sqrt(10) = 3.16 (3 s.f.)
The magnitude is found using Pythagoras: sqrt(x squared + y squared).
Final Answer: (a) (1, 4), (b) (8, -7), (c) sqrt(10) = 3.16
Exam Tip
When subtracting vectors, be very careful with negative signs. Write out each component calculation separately.
Example 4: Vector proof of parallelism
Question
OABC is a quadrilateral. OA = a, OC = c, and AB = c. Show that CB is parallel to OA.
Solution
- 1
Find OB
OB = OA + AB = a + c
Go from O to A then from A to B.
- 2
Find CB
CB = CO + OB = -c + (a + c) = a
Go from C to O (which is -OC = -c) then from O to B (which is a + c).
- 3
Compare CB with OA
CB = a and OA = a, so CB = 1 times OA.
CB is a scalar multiple of OA (with scale factor 1), so they are parallel.
- 4
State the conclusion
CB = OA, so CB is parallel to OA and equal in length. OABC is a parallelogram.
Since CB = OA, the vectors have the same magnitude and direction, confirming parallelism.
Final Answer: CB = a = OA, therefore CB is parallel to OA.
Exam Tip
To prove two lines are parallel, show their direction vectors are scalar multiples of each other. Always state your conclusion clearly in words.
Explore Vectors Subtopics
Frequently Asked Questions
How many vectors questions appear in the IGCSE exam?
Vectors typically appears in both Paper 2 (non-calculator) and Paper 4 (calculator). You can expect 2-4 questions on vectors across both papers, worth a combined 15-25 marks depending on the session.
What is the best way to practise vectors for IGCSE?
Start by understanding the methods through worked examples like these, then practise past paper questions under timed conditions. Teacher Rig recommends working through at least 20 vectors past paper questions before your exam, checking your method against mark schemes.
Should I memorise vectors formulas for the exam?
Some formulas are given on the formula sheet in the exam, but you should still be very familiar with them. Key formulas that are NOT on the sheet should be memorised. Practice using the formulas so that applying them becomes automatic.
Need Help with Vectors?
Book a free 60-minute trial class with Teacher Rig. Work through Vectors problems together and build your confidence.