Rukasha Maths

GCSE maths Beta

Vectors and geometric proof · Higher notes

Higher · Geometry · AQA references G25

A vector records both size and direction. In a proof, define vectors from a common starting point so paths can be compared algebraically. The vector from A to B is the position vector of B minus that of A. A midpoint splits a displacement into equal halves. Two non-zero vectors that are scalar multiples have parallel directions, but you still need the geometric context when claiming points are collinear. Explain what the resulting vector equation proves rather than ending with algebra alone.

Learning objectives

  • Add, subtract and reverse vectors along a path.
  • Prove parallelism using a non-zero scalar multiple.

Key ideas

Subtract position vectors to find a displacement.

OA→=a,OB→=b⇒AB→=b−a\overrightarrow{OA}=\mathbf a,\quad\overrightarrow{OB}=\mathbf b\quad\Rightarrow\quad\overrightarrow{AB}=\mathbf b-\mathbf a

Following a path adds vectors; reversing direction changes the sign.

BA→=−AB→\overrightarrow{BA}=-\overrightarrow{AB}

A midpoint averages the two position vectors.

OM→=a+b2\overrightarrow{OM}=\frac{\mathbf a+\mathbf b}{2}

For non-zero vectors, a non-zero scalar multiple proves parallelism. A positive scalar preserves direction.

Worked examples

Example 1

O and A are distinct. M is the midpoint of AB; N is the midpoint of OB. Prove NM is parallel to OA and half its length.

OA→=a,OB→=b\overrightarrow{OA}=\mathbf a,\quad\overrightarrow{OB}=\mathbf b
OM→=a+b2,ON→=b2\overrightarrow{OM}=\frac{\mathbf a+\mathbf b}{2},\qquad\overrightarrow{ON}=\frac{\mathbf b}{2}
NM→=OM→−ON→\overrightarrow{NM}=\overrightarrow{OM}-\overrightarrow{ON}

Answer

NM→=a2=12OA→\overrightarrow{NM}=\frac{\mathbf a}{2}=\frac12\overrightarrow{OA}

Example 2

M is the midpoint of AB. Find the vector from A to M.

AB→=b−a\overrightarrow{AB}=\mathbf b-\mathbf a

Answer

AM→=b−a2\overrightarrow{AM}=\frac{\mathbf b-\mathbf a}{2}

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Express a route or position using vector notation.

AO2 · Reasoning

Prove that points are collinear or lines are parallel using vector relationships.

AO3 · Problem solving

Combine ratios and vector routes to establish a geometric result.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Keep the order of subtraction correct. Position vectors start at the origin; a displacement vector may start elsewhere.

Try it yourself

Attempt each question before opening its answer.

Check 1: If OA = a and OB = b, write AB.

Answer: b - a

Travel from A back to O using -a, then O to B using b.

Check 2: M is the midpoint of AB. Write AM in terms of a and b.

Answer: (b - a)/2

AM is half of AB in the same direction.

Explain it in your own words

  • Explain why AB is b - a rather than a - b.
  • Write a midpoint position vector and justify it.
  • Explain the geometric meaning of a non-zero vector equal to 3 times another.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Geometry specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.