Vectors and transformations: translations, rotations, reflections, enlargements, combinations

Geometry – Vectors and Transformations (IGCSE 0580)

Learning Objectives

• Write vectors using the standard notation ⟨a, b⟩ and i + j.
• Find the magnitude, direction (bearing) and components of a vector.
• Perform the four core rigid motions – translation, rotation, reflection – and the similarity motion enlargement.
• Combine two or more transformations and describe the overall motion.
• Use vector notation to solve transformation problems efficiently (e.g. find the translation vector that maps one figure onto another).
• Connect vectors and transformations to the related syllabus strands on coordinate geometry (C2) and functions (C3).

1. Vectors

1.1 Definition and notation

A vector has both magnitude (length) and direction. In the plane it is written as

v = ⟨a, b⟩ or v = a i + b j

• Displacement vector: joins an initial point (tail) to a terminal point (head).
• Position vector: vector from the origin O(0, 0) to a point P(x, y); it is ⟨x, y⟩.

1.2 Components, magnitude and direction

1.3 Vector operations (core tier)

• Addition: u** + **v** = ⟨uₓ + vₓ, uᵧ + vᵧ⟩  (commutative)
• Subtraction: u** − **v** = ⟨uₓ − vₓ, uᵧ − vᵧ⟩
• Scalar multiplication: k**v** = ⟨k a, k b⟩ (k may be positive, negative or zero)
• Opposite vector: −**v** = ⟨‑a, ‑b⟩ (same magnitude, opposite direction)

Example – Translate ΔABC by **t** = ⟨3, ‑2⟩:

• A′(x_A + 3, y_A ‑ 2), B′(x_B + 3, y_B ‑ 2), C′(x_C + 3, y_C ‑ 2).

2.2 Rotation

A rotation turns a figure about a fixed centre O(Oₓ, Oᵧ) through an angle θ (anticlockwise positive).

• General formula (useful for any centre):

\[ \begin{aligned} x' &= Oₓ + (x-Oₓ)\cos\theta - (y-Oᵧ)\sin\theta\\ y' &= Oᵧ + (x-Oₓ)\sin\theta + (y-Oᵧ)\cos\theta \end{aligned} \]

• Special‑case shortcuts (centre at the origin):

Example – Rotate P(4, 1) 90° anticlockwise about the origin:

P′ = (‑1, 4)

2.3 Reflection

A reflection produces a mirror image in a line (the “mirror line”). The most common lines and their coordinate rules are:

For a general line y = mx + c the syllabus expects a verbal description (e.g. “reflect in the line y = 2x + 1”) rather than the full matrix formula.

Example – Reflect Q(5, ‑3) in the line y = x:

Q′ = (‑3, 5)

2.4 Enlargement (similarity)

• Scale factor k > 0 and centre of enlargement C(Cₓ, Cᵧ).
• Coordinate rule:

x' = Cₓ + k (x ‑ Cₓ)  y' = Cᵧ + k (y ‑ Cᵧ)

• If C is the origin, the rule simplifies to (x, y) → (k x, k y).

Example – Enlarge ΔDEF by k = 2 with centre C(1, ‑2):

• D′ = (1 + 2(x_D‑1), ‑2 + 2(y_D+2))
• Similarly for E′ and F′.

2.5 Combining transformations (core level)

When two transformations are performed one after the other, the overall effect is a composition. At IGCSE level only the following basic combinations are required:

• Two translations: the result is a single translation whose vector is the sum of the two vectors.
  Example: **t₁** = ⟨2, ‑1⟩ followed by **t₂** = ⟨‑3, 4⟩ ⇒ overall **t** = ⟨‑1, 3⟩.
• Translation followed by a rotation (or vice‑versa): treat the translation as a separate step; the order matters, so write the steps in the order given in the question.
• Reflection followed by a translation: perform the reflection first, then add the translation vector to the reflected coordinates.

For IGCSE exams you are not required to express the composition as a single matrix; a clear step‑by‑step description is sufficient.

IGCSE‑style problem – A point P(2, 3) is first reflected in the x‑axis, then translated by **t** = ⟨4, ‑1⟩. Find the final coordinates.

1. Reflection in x‑axis: P₁ = (2, ‑3).
2. Translation: P₂ = (2 + 4, ‑3 ‑ 1) = (6, ‑4).

3. Summary of Transformation Formulas (core tier)

4. Solving Transformation Problems with Vectors

1. Read the question carefully – note the type(s) of transformation and the order.
2. Write the appropriate vector or coordinate rule for each step.
3. Apply the rule to the given points (use algebra or a calculator for trigonometric values).
4. If the problem asks for the unknown transformation (e.g. “find the translation that maps ΔABC onto ΔA′B′C′”), compute the vector between a pair of corresponding points; verify with a second pair.
5. For a combination, carry out the steps sequentially; remember that the order cannot be changed.
6. Check your answer:
   • For rigid motions, distances and angles should be unchanged.
   • For enlargements, the ratio of any length in the image to the corresponding length in the original should equal the scale factor k.

5. Common Pitfalls & Tips (core tier)

• Direction of a translation vector: the vector points from the original point to its image (tail → head). Do not reverse it.
• Rotation about a point other than the origin: always subtract the centre coordinates, perform the rotation, then add the centre back.
• Direction angle: use atan2(b, a) or consider the signs of a and b to place θ in the correct quadrant.
• Reflection line: write the line exactly as given (e.g. y = x, not x = y) before applying the rule.
• Enlargement scale factor: k < 1 shrinks, k > 1 enlarges, k = 1 leaves the figure unchanged.
• Order of composition: perform transformations in the order stated; “AB” means do A first, then B.
• Checking work: after a rigid motion, verify that a side length (e.g. AB) is the same before and after; after an enlargement, verify that the ratio of corresponding sides equals k.

6. Links to Other Syllabus Areas

• C2 – Coordinate Geometry: the formulas for translation, rotation and reflection are applications of the coordinate plane; knowing how to manipulate (x, y) pairs is essential.
• C3 – Functions and Graphs: the rotation and reflection rules can be viewed as functions mapping (x, y) to (x′, y′). Recognising these as transformations helps when interpreting graphs of linear and trigonometric functions.
• C5 – Vectors (core): the vector notation introduced here is the same as that used for representing forces, velocities, etc., in other parts of the syllabus.