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

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

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

$2\mathbf{b}=\begin{pmatrix}-4\\6\end{pmatrix}$. $\mathbf{a}-2\mathbf{b}=\begin{pmatrix}8\\-5\end{pmatrix}$. Answer: $\begin{pmatrix}8\\-5\end{pmatrix}$

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

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

$ \overrightarrow{OP} = 2\mathbf{a} + \mathbf{b} $, $ \overrightarrow{OQ} = 5\mathbf{a} + 3\mathbf{b} $ and $ \overrightarrow{OR} = 11\mathbf{a} + 7\mathbf{b} $. Prove that P, Q and R are collinear.

M1 PQ = 3a + 2b or QR = 6a + 4b M1 both PQ and QR (or PR = 9a + 6b) A1 QR = 2PQ, parallel with the common point Q, so collinear Answer: $ \overrightarrow{QR} = 6\mathbf{a} + 4\mathbf{b} = 2\overrightarrow{PQ} $, with the common point Q (proved)

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

Edexcel IGCSE 4MA1 Vectors: Exam 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 8 exam-style questions yourself and check your working against the mark scheme.

3 worked examples
8 questions with mark schemes
Higher tier · calculator allowed
Free, no sign-in

Practise vectors in Unit 7 →Vectors revision notes

What this topic covers

Vectors 1: Column Vectors: Adding, subtracting, scalar multiplying column vectors.
Vectors 2: Vector Arithmetic: Vector magnitude and vector routes on grids.
Vectors 3: Geometric Proofs: Using vector methods to prove points collinear/midpoints.

Vectors worked examples

Worked example 1: Column vectors

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

Solution

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

3 steps · 1 to fill in

Tip: Magnitude uses Pythagoras.

Worked example 2: Vector arithmetic

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

Solution

$2 b = (\begin{matrix} - 4 \\ 6 \end{matrix})$ .
$a - 2 b = (\begin{matrix} 8 \\ - 5 \end{matrix})$ .

3 steps

Tip: Subtracting a negative adds.

Worked example 3: Midpoint vector

$\vec{O A} = a$ , $\vec{O B} = b$ and M is the midpoint of AB. Find $\vec{O M}$ .

Triangle OAB with OA = a and OB = b; M is the midpoint of AB and OM is dashed

Solution

$\vec{A B} = b - a$ .
$\vec{O M} = a + \frac{1}{2} (b - a) = \frac{1}{2} (a + b)$ .

3 steps

Tip: Travel along known vectors from O to M.

Vectors exam-style questions

Try each question before you open the answer (one has its full worked solution open). 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

$a = (\begin{matrix} 3 \\ - 2 \end{matrix})$ and $b = (\begin{matrix} - 1 \\ 5 \end{matrix})$ . Write $2 a - 3 b$ as a column vector.

Show the answer and mark scheme

M1 (6, −4) or (−3, 15) seen as column vectors
A1 (9, −19) as a column vector

Answer: $(\begin{matrix} 9 \\ - 19 \end{matrix})$

Question 2 · easy · 2 marks

$\vec{A B} = 3 a - 6 b$ and $\vec{C D} = - a + 2 b$ . 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: $\vec{A B} = - 3 \vec{C D}$ , so AB is parallel to CD (shown)

Question 3 · medium · 3 marks

(a) Find the magnitude of the vector $p = (\begin{matrix} 5 \\ - 12 \end{matrix})$ . (b) The vector $q = (\begin{matrix} k \\ 6 \end{matrix})$ has magnitude 10, and $k$ is negative. Find the value of $k$ .

Show the answer and mark scheme

B1 13
M1 k² + 6² = 10²
A1 k = −8

Answer: (a) 13 (b) $k = - 8$

Question 4 · medium · 3 marks

$\vec{O A} = a$ and $\vec{O B} = b$ . P lies on AB with $A P : P B = 1 : 3$ . Find $\vec{O P}$ in terms of $a$ and $b$ . 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: $\frac{3}{4} a + \frac{1}{4} b$

Question 5 · medium · 3 marks

$\vec{O P} = 2 a + b$ , $\vec{O Q} = 5 a + 3 b$ and $\vec{O R} = 11 a + 7 b$ . Prove that P, Q and R are collinear.

Full worked solution Free, no sign-in

M1 PQ = 3a + 2b or QR = 6a + 4b
M1 both PQ and QR (or PR = 9a + 6b)
A1 QR = 2PQ, parallel with the common point Q, so collinear

Answer: $\vec{Q R} = 6 a + 4 b = 2 \vec{P Q}$ , with the common point Q (proved)

Every question here has a full worked solution. See them all →

Question 6 · hard · 4 marks

OABC is a parallelogram with $\vec{O A} = a$ and $\vec{O C} = c$ . M is the midpoint of AB, and N is the point on OC with $O N : N C = 1 : 3$ . Find $\vec{N M}$ in terms of $a$ and $c$ . Give your answer in its simplest form.

Show the answer and mark scheme

B1 OM = a + ½c
B1 ON = ¼c
M1 NM = NO + OM = −¼c + a + ½c
A1 a + ¼c

Answer: $a + \frac{1}{4} c$

Question 7 · hard · 4 marks

$\vec{O A} = 4 a$ and $\vec{O B} = 4 b$ . P is the midpoint of OA. Q lies on AB with $A Q : Q B = 3 : 1$ . R lies on OB extended so that $\vec{O R} = 6 b$ . (a) Find $\vec{P Q}$ in terms of $a$ and $b$ . (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) $- a + 3 b$ (b) $\vec{P R} = - 2 a + 6 b = 2 \vec{P Q}$ , parallel with the common point P, so collinear (proved)

Question 8 · very hard · 6 marks

OABC is a parallelogram with $\vec{O A} = a$ and $\vec{O C} = c$ . N is the midpoint of AB. Y is the point on CN with $C Y : Y N = 2 : 1$ . (a) Find $\vec{C N}$ in terms of $a$ and $c$ . (b) Prove that Y lies on the diagonal OB, and find the ratio $O Y : Y B$ .

Show the answer and mark scheme

B1 ON = a + ½c (or AN = ½c)
M1 CN = −c + a + ½c
A1 CN = a − ½c
M1 OY = c + ⅔(a − ½c)
A1 OY = ⅔a + ⅔c
A1 OB = a + c, so OY = ⅔ OB: a multiple of OB with O common, so Y lies on OB; OY : YB = 2 : 1

Answer: (a) $a - \frac{1}{2} c$ (b) $\vec{O Y} = \frac{2}{3} a + \frac{2}{3} c = \frac{2}{3} \vec{O B}$ , so Y lies on OB (proved); $O Y : Y B = 2 : 1$

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

Unit 7 practice: Vectors & Proof in the student hub (more questions with mark schemes)
IGCSE Maths revision notes for this topic and the IGCSE Maths formula sheet
Edexcel IGCSE Maths 4MA1 past papers to practise under exam conditions
IGCSE Maths 4MA1 examiner report summary: where students lose marks

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.

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Edexcel IGCSE 4MA1 Vectors: Exam Questions and Answers

Try it first

What this topic covers

Vectors worked examples

Worked example 1: Column vectors

Solution

Worked example 2: Vector arithmetic

Solution

Worked example 3: Midpoint vector

Solution

Vectors exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · medium · 3 marks

Question 4 · medium · 3 marks

Question 5 · medium · 3 marks

Full worked solution Free, no sign-in

Question 6 · hard · 4 marks

Question 7 · hard · 4 marks

Question 8 · very hard · 6 marks

Revise vectors for the exam

Questions students ask

Practise vectors with your progress saved

More Unit 7 topics: Vectors & Proof