Mini-Lesson

Transformations & Vectors

This mini-lesson covers Transformations & Vectors for Cambridge IGCSE Mathematics (0580): translations, reflections, rotations, enlargements, column vectors, and vector arithmetic & magnitude (Extended content included).

Honest heads-up: these games test recall of rules and results. In the exam you must draw or fully describe each transformation.

Answer as you go and collect ⭐ stars. Press Start.

The four transformations

To fully describe each transformation you must state:

Translation — a column vector.
Reflection — the mirror line (its equation).
Rotation — angle, direction and centre.
Enlargement — scale factor and centre.

Common slip: a rotation needs three things (angle, direction, centre). Miss one and you lose the mark.

Quick check

?To fully describe a rotation, which three things must you state?

Column vectors

Column vectors & translations

A column vector (x over y) means "move x across and y up". A negative top number moves left; a negative bottom number moves down.

translation by (3 over −2)3 right and 2 down

Worked example

Translate the point (1, 5) by the vector (4 over −3): new point = (1 + 4, 5 − 3) = (5, 2).

Calculate

Your turn

1The point (2, 7) is translated by the column vector (5 over −4). Work out the x-coordinate of its image.

Hint: x-coordinate = 2 + 5.

Calculate

Your turn

2Using the same translation (5 over −4), work out the y-coordinate of the image of (2, 7).

Hint: y-coordinate = 7 − 4.

Enlargement

An enlargement multiplies every distance from the centre by the scale factor. A scale factor between 0 and 1 makes the shape smaller; a negative scale factor flips it through the centre.

image length = object length × scale factorarea scales by (scale factor)²

Worked example

A 4 cm side enlarged by scale factor 2.5 becomes 4 × 2.5 = 10 cm.

Calculate

Your turn

3A shape is enlarged by scale factor 3. A side of length 6 cm on the object becomes how long on the image?

cm

Hint: image length = 6 × 3.

Quick check

?A shape is enlarged by scale factor ½. What happens to it?

Vector arithmetic

Vector arithmetic & magnitude

Add or subtract vectors component by component; multiply by a scalar to scale each component. The magnitude (length) uses Pythagoras.

|(x over y)| = √(x² + y²)the length of the vector

Worked example

(3 over 4) + (1 over 2) = (4 over 6). Magnitude of (3 over 4) = √(9 + 16) = 5.

Calculate

Your turn

4Work out the magnitude of the vector (6 over 8).

Hint: √(6² + 8²) = √(36 + 64) = √100.

Calculate

Your turn

5Vector a = (2 over 5) and vector b = (3 over 1). Work out the top component of a + b.

Hint: add the top components: 2 + 3.

Calculate

Your turn

6Work out the top component of the vector 3 × (4 over 2) (scalar multiplication).

Hint: multiply each component by 3, so top = 3 × 4.

Match game

Transformation ⇄ description

Tap a transformation, then what you must state to describe it.

Sort it

Changes size?

Sort each transformation: does it change a shape's size, or keep it congruent (same size)? Tap it, then its box.

🔍 Changes size

= Same size

Recap

Key points

Describe fully: translation → vector; reflection → mirror line; rotation → angle, direction, centre; enlargement → scale factor, centre.

Column vectors: add across & up; translate by adding to coordinates.

Enlargement: ×SF; SF between 0 and 1 shrinks; areas ×SF².

Vectors: add component-wise; magnitude = √(x² + y²).

You've covered the Transformations & Vectors essentials for Cambridge IGCSE Maths (0580). Press Finish to see your score.

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