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Vectors questions and answers

A column vector (32)\begin{pmatrix} 3 \\ 2 \end{pmatrix} means 3 right and 2 up. Add or subtract vectors by adding or subtracting the top and bottom numbers; multiply by a number by multiplying both. In vector geometry, AB→=−a+b\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}: go back along a\mathbf{a} to O, then out along b\mathbf{b}.

8 questions · Foundation and Higher tier · Every answer is checked as you type it, with a full worked solution. Updated .

Start the questions Vectors guide

Where vectors comes up

GCSE (AQA, Edexcel, OCR)Foundation and Higher: describe translations as 2D vectors; add, subtract and multiply vectors by a scalar, with column and diagram representations. Higher only: vector arguments and proofs.
Cambridge IGCSE (0580)Core: translation by a vector. Extended only: vectors in two dimensions, the magnitude of a vector and vector geometry.

The method

  1. Add/subtract: top with top, bottom with bottom.
  2. Scalar: multiply both numbers.
  3. Magnitude: √(x² + y²).
  4. Paths: write the route as a sum of known vectors; going backwards along a vector makes it negative.

Vectors questions

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Warm-up

Question 1Warm-upFoundation

a=(32)\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} and b=(1−4)\mathbf{b} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}. Find a+b\mathbf{a} + \mathbf{b}.

Question 2Warm-upFoundation

With the same vectors, find a−b\mathbf{a} - \mathbf{b}.

Exam-style

Question 3Exam-styleFoundation

a=(32)\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} and b=(1−4)\mathbf{b} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}. Find 2a−3b2\mathbf{a} - 3\mathbf{b}.

Question 4Exam-styleFoundation

(2k)+(35)=(51)\begin{pmatrix} 2 \\ k \end{pmatrix} + \begin{pmatrix} 3 \\ 5 \end{pmatrix} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}. Find kk.

Question 5Exam-styleHigher

Find the magnitude of the vector (68)\begin{pmatrix} 6 \\ 8 \end{pmatrix}.

Question 6Exam-styleFoundation

OA→=a\overrightarrow{OA} = \mathbf{a} and OB→=b\overrightarrow{OB} = \mathbf{b}. Find AB→\overrightarrow{AB}.

Challenge

Question 7ChallengeHigher

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

Question 8ChallengeHigher

OA→=a\overrightarrow{OA} = \mathbf{a} and OB→=b\overrightarrow{OB} = \mathbf{b}. P is on AB with AP : PB = 1 : 2. Find OP→\overrightarrow{OP}.

Common mistakes

  • Subtracting the wrong way round.
  • Forgetting to multiply both components.
  • Getting AB as a − b (it's b − a).

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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