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Edexcel IGCSE Maths (4MA1) · Unit 7: Vectors & Proof

IGCSE Maths Vectors Questions and Answers

Vectors questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 3 worked examples first, then try the 3 exam-style questions yourself and check your working against the mark scheme.

Practise vectors in Unit 7 →Vectors revision notes

What this topic covers

Vectors worked examples

Worked example 1: Column vectors

\(\mathbf{a}=\begin{pmatrix}2\\-5\end{pmatrix}\). Find \(3\mathbf{a}\) and the magnitude of \(\begin{pmatrix}5\\-12\end{pmatrix}\).

Solution

  1. \(3\mathbf{a}=\begin{pmatrix}6\\-15\end{pmatrix}\).
  2. Magnitude \(=\sqrt{5^2+(-12)^2}=13\).

Answer: \(\begin{pmatrix}6\\-15\end{pmatrix}\); 13

Tip: Magnitude uses Pythagoras.

Worked example 2: Vector arithmetic

\(\mathbf{a}=\begin{pmatrix}4\\1\end{pmatrix}\), \(\mathbf{b}=\begin{pmatrix}-2\\3\end{pmatrix}\). Find \(\mathbf{a}-2\mathbf{b}\).

Solution

  1. \(2\mathbf{b}=\begin{pmatrix}-4\\6\end{pmatrix}\).
  2. \(\mathbf{a}-2\mathbf{b}=\begin{pmatrix}8\\-5\end{pmatrix}\).

Answer: \(\begin{pmatrix}8\\-5\end{pmatrix}\)

Tip: Subtracting a negative adds.

Worked example 3: Midpoint vector

\(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and M is the midpoint of AB. Find \(\overrightarrow{OM}\).

Solution

  1. \(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\).
  2. \(\overrightarrow{OM}=\mathbf{a}+\tfrac{1}{2}(\mathbf{b}-\mathbf{a})=\tfrac{1}{2}(\mathbf{a}+\mathbf{b})\).

Answer: \(\tfrac{1}{2}(\mathbf{a}+\mathbf{b})\)

Tip: Travel along known vectors from O to M.

Vectors exam-style questions

Try each question before you open the answer. The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

\( \overrightarrow{AB} = 3\mathbf{a} - 6\mathbf{b} \) and \( \overrightarrow{CD} = -\mathbf{a} + 2\mathbf{b} \). Show that AB is parallel to CD.

Show the answer and mark scheme
  • B1 AB = −3(−a + 2b) = −3 CD
  • B1 one vector is a multiple of the other, so AB is parallel to CD

Answer: \( \overrightarrow{AB} = -3\overrightarrow{CD} \), so AB is parallel to CD (shown)

Question 2 · medium · 3 marks

\( \overrightarrow{OA} = \mathbf{a} \) and \( \overrightarrow{OB} = \mathbf{b} \). P lies on AB with \( AP : PB = 1 : 3 \). Find \( \overrightarrow{OP} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \). Give your answer in its simplest form.

Show the answer and mark scheme
  • M1 AB = b − a
  • M1 OP = a + ¼(b − a)
  • A1 ¾a + ¼b oe

Answer: \( \dfrac{3}{4}\mathbf{a} + \dfrac{1}{4}\mathbf{b} \)

Question 3 · hard · 4 marks

\( \overrightarrow{OA} = 4\mathbf{a} \) and \( \overrightarrow{OB} = 4\mathbf{b} \). P is the midpoint of OA. Q lies on AB with \( AQ : QB = 3 : 1 \). R lies on OB extended so that \( \overrightarrow{OR} = 6\mathbf{b} \). (a) Find \( \overrightarrow{PQ} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \). (b) Prove that P, Q and R lie on a straight line.

Show the answer and mark scheme
  • M1 OQ = 4a + ¾(4b − 4a) = a + 3b
  • A1 PQ = −a + 3b
  • M1 PR = −2a + 6b
  • A1 PR = 2PQ, so PQ and PR are parallel and share the point P: P, Q, R are collinear

Answer: (a) \( -\mathbf{a} + 3\mathbf{b} \) (b) \( \overrightarrow{PR} = -2\mathbf{a} + 6\mathbf{b} = 2\overrightarrow{PQ} \), parallel with the common point P, so collinear (proved)

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and each answer was re-solved independently before publishing.

Revise vectors for the exam

Questions students ask

Which Edexcel IGCSE Maths papers test vectors?

Either Higher tier paper, 1H or 2H, can test vectors: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.

Where can I practise more vectors questions?

In the IGCSE student hub: Unit 7 (Vectors & Proof) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.

More Unit 7 topics: Vectors & Proof