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05802025–2027 syllabus

MATHEMATICS · CHAPTER 7

Transformations and vectors

Describe movement and scale precisely, then use directed quantities to prove geometrical relationships.

Core + Extended4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • recognise and describe four transformations
  • combine transformations
  • use positive, fractional and negative enlargement factors
  • add, subtract and scale vectors
  • calculate vector magnitudes
  • use position vectors in geometrical proofs

AT A GLANCE

Syllabus0580Coverage2025–2027Sections4LevelCore + Extended

INTRODUCTION · THE BIG IDEA

Describe movement and scale precisely, then use directed quantities to prove geometrical relationships.

A transformation maps every point of a shape to a new position. A vector records a movement with both magnitude and direction.

Precise descriptions matter: naming only rotation or enlargement is incomplete without its centre and other defining information.

01

SECTION 01

The four transformations

Core concept

A reflection needs a mirror line. Each point and its image are the same perpendicular distance from that line. A rotation needs a centre, angle and direction. An enlargement needs a centre and scale factor. A translation needs a column vector.

A negative enlargement places the image on the opposite side of the centre; a fractional factor reduces its size. To locate an enlargement centre, join corresponding vertices and extend the lines until they meet.

A combination applies transformations in the stated order, using each new image as the starting shape for the next operation.

DETAILED EXPLANATION

  • Enlargement preserves angles and multiplies lengths by |scale factor| and area by its square.
  • Column-vector top entry is horizontal movement; bottom entry is vertical movement.
Coordinate-grid examples of reflection, rotation, translation and enlargementtranslation vectormirror line x = 0
A complete description names the mirror line, rotation centre/angle/direction, enlargement centre/scale factor, or translation vector.
Information required for a complete transformation description
TransformationRequired details
Reflectionequation or description of mirror line
Rotationcentre, angle and clockwise/anticlockwise direction
Enlargementcentre and scale factor
Translationtwo-component column vector
Original worked example

Describing a rotation

  1. Join a point and its image to a possible centre.
  2. Repeat with another pair; perpendicular bisectors meet at the centre.
  3. Measure the angle and decide clockwise or anticlockwise.

Answer: A complete answer has the form rotation 90° clockwise about (2, −1).

02

SECTION 02

Vector arithmetic

Core concept

Vectors add head-to-tail. Subtracting b means adding −b, which has the same magnitude as b but opposite direction.

A scalar changes a vector's magnitude and may reverse its direction. Equal vectors have equal magnitude and direction even if drawn in different places.

RULE 1
(a, b) + (c, d) = (a + c, b + d)
RULE 2
k(a, b) = (ka, kb)
RULE 3
AB = b − a for position vectors a and b
Original worked example

Combining column vectors

  1. a = (4, −1) and b = (−2, 5).
  2. 2a = (8, −2).
  3. 2a − b = (8, −2) − (−2, 5).

Answer: 2a − b = (10, −7).

03

SECTION 03

Magnitude and direction

Core concept

The magnitude of a vector is its length. Its horizontal and vertical components form a right-angled triangle, so Pythagoras gives the formula.

A zero vector has magnitude zero and no defined direction.

RULE 1
|(x, y)| = √(x² + y²)
Original worked example

Magnitude of vector (−7, 24)

  1. Square both components: (−7)² = 49 and 24² = 576.
  2. Add: 49 + 576 = 625.
  3. Take the square root.

Answer: The magnitude is 25.

04

SECTION 04

Vector geometry and proof

Core concept

Choose two independent vectors and express every required path in terms of them. Different routes between the same points give equal vectors.

If one vector is a scalar multiple of another, they are parallel. If two directed segments meeting at a point are parallel, the three points are collinear.

Section ratios can be handled by taking the appropriate fraction of a whole vector.

Triangle with midpoint vector constructionOABMabAM = 1/2(b − a)
If M is the midpoint of OB, then OM = 1/2b and AM = 1/2b − a; scalar multiples establish parallel directions.
Original worked example

A point dividing a line in a ratio

  1. OA = a and OB = b. Point P divides AB in the ratio AP : PB = 2 : 1.
  2. AB = b − a.
  3. AP = 2/3 AB = 2/3(b − a).
  4. OP = OA + AP = a + 2/3(b − a).
  5. Collect the coefficients.

Answer: OP = 1/3a + 2/3b.

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • Every transformation needs its defining details.
  • Negative enlargement reverses direction through the centre.
  • Vectors add by components or head-to-tail.
  • Magnitude uses Pythagoras.
  • Position-vector differences describe directed segments.
  • Scalar multiples provide evidence for parallel or collinear lines.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • recognise and describe four transformations
  • combine transformations
  • use positive, fractional and negative enlargement factors
  • add, subtract and scale vectors
  • calculate vector magnitudes
  • use position vectors in geometrical proofs
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