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GCSE Maths / Edexcel

Vectors in geometry

Use column vectors and vector notation to describe movement, combine vectors and identify parallel vector relationships.

Geometry and MeasuresFoundation and HigherGrades 5 to 8Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Vectors

Pearson Edexcel GCSE Maths geometry G25: apply addition and subtraction of vectors, multiplication by a scalar, and vector geometry.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A vector describes a movement. It has size and direction.

A column vector has two numbers. The top number is horizontal movement and the bottom number is vertical movement.

Positive horizontal movement is right and negative horizontal movement is left. Positive vertical movement is up and negative vertical movement is down.

To add vectors, add the horizontal components and add the vertical components.

To subtract vectors, subtract the components in the same positions.

Multiplying a vector by a scalar changes its length. If the scalar is negative, the vector points in the opposite direction.

Two vectors are parallel if one is a multiple of the other, for example [6, -4] = 2[3, -2].

In geometry questions, follow the route carefully. If AB is vector a and BC is vector b, then AC = a + b.

Key ruleColumn vector [x, y] means x across and y up or down. Parallel vectors are scalar multiples.

Diagram guide

3 right3 upcolumn vector [3, 3]top number is across, bottom number is up or down
Column vectorThe top component moves across. The bottom component moves up or down.
aba + badd vectors tip to tail
Adding vectorsPlace vectors tip to tail. The result is the direct movement from start to finish.
v2vsame direction means parallelone vector is a multiple of the otherparallel vectors have proportional components
Parallel vectorsParallel vectors have the same or opposite direction because one is a scalar multiple of the other.

Worked examples

Find a column vector

Point A is (2, 1) and point B is (7, 4). Find vector AB.

3 right3 upcolumn vector [3, 3]top number is across, bottom number is up or down
Example: movement from A to BSubtract starting coordinates from finishing coordinates.
  1. Horizontal movement = 7 - 2 = 5.
  2. Vertical movement = 4 - 1 = 3.
  3. So AB = [5, 3].

Answer: [5, 3]

Add vectors

a = [3, -2] and b = [1, 5]. Find a + b.

aba + badd vectors tip to tail
Example: add componentsAdd across with across and vertical with vertical.
  1. Top components: 3 + 1 = 4.
  2. Bottom components: -2 + 5 = 3.
  3. a + b = [4, 3].

Answer: [4, 3]

Parallel vectors

Show that [6, -4] is parallel to [3, -2].

v2vsame direction means parallelone vector is a multiple of the otherparallel vectors have proportional components
Example: scalar multipleOne vector is exactly twice the other.
  1. [6, -4] = 2[3, -2].
  2. Because one vector is a scalar multiple of the other, the vectors are parallel.

Answer: They are parallel because [6, -4] = 2[3, -2].

Common mistakes

  • Putting the vertical movement on top.
  • Changing the sign when moving left or down.
  • Adding one component but subtracting the other for no reason.
  • Saying vectors are parallel because the numbers look similar rather than proving a scalar multiple.
  • Forgetting that a negative scalar gives the opposite direction but still parallel.

Quick exercise

Try these before moving to the exam-style questions.

  1. A movement is 4 right and 2 down. Write the column vector.
    3 right3 upcolumn vector [3, 3]top number is across, bottom number is up or down
    Quick check: movementDown is negative.
  2. Find [2, 5] + [3, -1].
    aba + badd vectors tip to tail
    Quick check: additionAdd matching components.
  3. Find 3[2, -4].
    v2vsame direction means parallelone vector is a multiple of the otherparallel vectors have proportional components
    Quick check: scalarMultiply both components.
  4. Are [10, 15] and [2, 3] parallel?
    v2vsame direction means parallelone vector is a multiple of the otherparallel vectors have proportional components
    Quick check: parallelCheck whether one vector is a multiple of the other.
  5. A is (1, 6) and B is (4, 2). Find vector AB.
    3 right3 upcolumn vector [3, 3]top number is across, bottom number is up or down
    Quick check: coordinatesFinish minus start.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundation and HigherNon-calculator2 marks

Point A is (3, 2) and point B is (8, 6). Find vector AB.

3 right3 upcolumn vector [3, 3]top number is across, bottom number is up or down
Question diagram: vector from coordinatesFind the horizontal and vertical movement.
column vectorcoordinatesfoundation vectors
Standard exam styleFoundation and HigherNon-calculator3 marks

a = [4, -1] and b = [-2, 5]. Find 2a + b.

aba + badd vectors tip to tail
Question diagram: vector additionMultiply first, then add components.
vector additionscalar multiplemethod marks
ChallengeHigherNon-calculator4 marks

Show that the vectors [9, -6] and [-3, 2] are parallel, and state whether they point in the same or opposite direction.

v2vsame direction means parallelone vector is a multiple of the otherparallel vectors have proportional components
Question diagram: parallel vectorsLook for a scalar multiple.
parallel vectorsscalar multiplehigher geometry