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Cambridge IGCSE Maths (0580) · Topic 7: Transformations and vectors · 7.2, 7.3, 7.4

Cambridge IGCSE Maths 0580 Vectors Questions

Vectors questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.

Practise vectors →Vectors notes

What the syllabus asks

Summarised in our own words: check the exact wording in the official 0580 syllabus.

Vectors questions: Extended

Extended content, for Papers 2 and 4.

Question 1 Extended non-calculator · easy

\(\mathbf{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\).
Find (a) \(\mathbf{a} + \mathbf{b}\) (b) \(2\mathbf{a} - \mathbf{b}\).

Show the worked answer
  1. (a) Add the parts: \(3 + (-2) = 1\), \(-1 + 5 = 4\).
  2. (b) \(2\mathbf{a} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}\); subtract \(\mathbf{b}\): \(6 - (-2) = 8\), \(-2 - 5 = -7\).

Answer: (a) \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\) (b) \(\begin{pmatrix} 8 \\ -7 \end{pmatrix}\)

Question 2 Extended non-calculator · easy

Calculate \(|\mathbf{v}|\) for \(\mathbf{v} = \begin{pmatrix} -8 \\ 15 \end{pmatrix}\).

Show the worked answer
  1. \(\sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17\)

Answer: \(17\)

Question 3 Extended calculator · medium

Find the magnitude of the vector \(\begin{pmatrix} 4 \\ 7 \end{pmatrix}\), correct to 3 significant figures.

Show the worked answer
  1. \(\sqrt{4^2 + 7^2} = \sqrt{65} = 8.062\ldots\)

Answer: \(8.06\)

Question 4 Extended non-calculator · medium

\(A\) is the point \((2, 3)\) and \(B\) is the point \((7, -1)\).
Write \(\overrightarrow{AB}\) as a column vector.

Show the worked answer
  1. \(\overrightarrow{AB}\) is the movement from \(A\) to \(B\): \(7 - 2 = 5\) across and \(-1 - 3 = -4\) up.

Answer: \(\begin{pmatrix} 5 \\ -4 \end{pmatrix}\)

Question 5 Extended non-calculator · medium

The vector \(\begin{pmatrix} k \\ 6 \end{pmatrix}\) is parallel to \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\).
Find \(k\).

Show the worked answer
  1. Parallel vectors are multiples of each other: \(6 = 2 \times 3\), so the multiplier is \(2\).
  2. \(k = 2 \times 2 = 4\)

Answer: \(k = 4\)

Question 6 Extended non-calculator · hard

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

Show the worked answer
  1. (a) \(\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = -\mathbf{a} + \mathbf{b}\).
  2. (b) \(\overrightarrow{OM} = \mathbf{a} + \frac{1}{2}(\mathbf{b} - \mathbf{a}) = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}\).

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

Question 7 Extended non-calculator · hard

\(\overrightarrow{OP} = 2\mathbf{a} + \mathbf{b}\), \(\overrightarrow{OQ} = 4\mathbf{a} + 3\mathbf{b}\) and \(\overrightarrow{OR} = 8\mathbf{a} + 7\mathbf{b}\).
Show that \(P\), \(Q\) and \(R\) lie on a straight line.

Show the worked answer
  1. \(\overrightarrow{PQ} = (4\mathbf{a} + 3\mathbf{b}) - (2\mathbf{a} + \mathbf{b}) = 2\mathbf{a} + 2\mathbf{b}\).
  2. \(\overrightarrow{QR} = (8\mathbf{a} + 7\mathbf{b}) - (4\mathbf{a} + 3\mathbf{b}) = 4\mathbf{a} + 4\mathbf{b} = 2\overrightarrow{PQ}\).
  3. \(\overrightarrow{QR}\) is a multiple of \(\overrightarrow{PQ}\) and they share the point \(Q\), so \(P\), \(Q\) and \(R\) are collinear.

Answer: \(\overrightarrow{QR} = 2\overrightarrow{PQ}\), so \(P\), \(Q\) and \(R\) are collinear

Question 8 Extended non-calculator · hard

The position vectors of \(P\) and \(Q\) are \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 1 \end{pmatrix}\).
Find the distance \(PQ\).

Show the worked answer
  1. \(\overrightarrow{PQ} = \begin{pmatrix} -4 \\ -3 \end{pmatrix}\)
  2. \(|\overrightarrow{PQ}| = \sqrt{16 + 9} = 5\)

Answer: \(5\)

Examiner Insights: common mistakes in IGCSE Maths 0580, with fixes (our analysis of public sources) →

Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).

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Questions students ask

Which Cambridge 0580 papers can test vectors?

Vectors is Extended content (syllabus 7.2, 7.3, 7.4), so it is tested only on the Extended papers, Paper 2 (non-calculator) and Paper 4 (calculator). Core candidates do not need it.

Are these Cambridge past-paper questions?

No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.

Where can I practise more vectors questions?

In the 0580 practice for topic 7 (Transformations and vectors), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.

More transformations and vectors questions