Mini-Lesson

Transformations & Vectors

2 missions. Guess, learn, then check. Your place is saved.

Mission 1A picture was spun round one nail. Can you find where the nail is?
Mission 2A drone flies 3 east, 4 north, then 6 east, 8 north. Straight line?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

A picture was spun round one nail. Can you find where the nail is?

Make a guess: A picture was spun round one nail. Can you find where the nail is?

Geometry · transformations

Describing a transformation fully

A full description names the transformation and gives every fact needed to repeat it exactly.

Transformation	Give	Image is
Reflection	the mirror line as an equation, e.g. $x = 3$ x = 3 or $y = - 1$ y = −1	congruent
Rotation	centre, angle, direction	congruent
Translation	a column vector	congruent
Enlargement	centre, scale factor	similar

Reflections, rotations and translations keep lengths and angles. Enlargements keep angles but change lengths.
Give one transformation only. Two moves for a "single transformation" question score nothing.
Slanted mirror lines such as $y = x$ y = x and $y = - x$ y = −x: Extended.

Worked example Geometry · transformations

Describing a rotation

Question

Describe fully the single transformation that maps triangle A, with vertices

(1, 1)

,

(3, 1)

,

(1, 2)

, onto triangle B, with vertices

(- 1, 1)

,

(- 1, 3)

,

(- 2, 1)

.

1
Same size and shape, but turned round: it is a rotation.
2
Find the angle: side $(1, 1)$ (1, 1) to $(3, 1)$ (3, 1) points right; its image $(- 1, 1)$ (−1, 1) to $(- 1, 3)$ (−1, 3) points up. That is a quarter-turn anticlockwise, 90°.
3
Find the centre: try $(0, 0)$ (0, 0) with tracing paper. Each vertex and its image are the same distance from it.
4
Write all three facts with the word "rotation".

Answer

Rotation, 90° anticlockwise, centre

(0, 0)

.

⚠️ Watch out

leaving out the centre, the angle or the direction each loses a mark, and "turn" is not accepted for "rotation".

Quick check

?Triangle P has vertices

(1, 1)

,

(3, 1)

and

(1, 4)

. Triangle Q has vertices

(7, 1)

,

(5, 1)

and

(7, 4)

. Which is the single transformation that maps P onto Q?

Geometry · transformations

Translations and enlargements

Translation by $(a b)$ ( a ; b ): the top number moves right (+) or left (−); the bottom number moves up (+) or down (−).
Enlargement: every distance from the centre is multiplied by the scale factor.
A scale factor between 0 and 1 makes the image smaller. Scale factor $\frac{1}{2}$ 1/2, centre $(0, 0)$ (0, 0): $(4, 6) \to (2, 3)$ (4, 6) → (2, 3).
Fractional scale factors: Core and Extended.
Extended: a negative scale factor puts the image on the other side of the centre, upside down. Scale factor $- 2$ −2, centre $(0, 0)$ (0, 0): $(1, 3) \to (- 2, - 6)$ (1, 3) → (−2, −6).

Worked example Geometry · transformations

Enlargement from a centre

Question

Enlarge triangle T, with vertices

(2, 3)

,

(3, 3)

,

(2, 5)

, by scale factor 2, centre

(1, 2)

.

1
Count from the centre to each vertex: $(11)$ ( 1 ; 1 ), $(21)$ ( 2 ; 1 ), $(13)$ ( 1 ; 3 )
2
Multiply each by 2: $(22)$ ( 2 ; 2 ), $(42)$ ( 4 ; 2 ), $(26)$ ( 2 ; 6 )
3
Count out from the centre again: $(3, 4)$ (3, 4), $(5, 4)$ (5, 4), $(3, 8)$ (3, 8)
4
Check: the base was 1 and is now 2; the height was 2 and is now 4.

Answer

The image has vertices

(3, 4)

,

(5, 4)

and

(3, 8)

.

⚠️ Watch out

doubling the coordinates themselves, e.g.

(2, 3) \to (4, 6)

, is an enlargement about the origin, not about

(1, 2)

.

Calculate

?Triangle T has a vertex at

P (1, 2)

. T is enlarged by scale factor 3, centre

(- 1, 0)

. Find the

x

-coordinate of the image of P.

Geometry · transformations

Finding the centre of a rotation

The centre is the one point that does not move: an invariant point.
Every other point stays the same distance from the centre: $CP = C P^{'}$ CP = CP'.
So the centre lies on the perpendicular bisector of $P P^{'}$ PP'. Two such bisectors cross at the centre.
On a grid: pin tracing paper at a guess and turn it. If the shape lands on the image, that is the centre.
Core centres: origin, a vertex or an edge midpoint.

Calculate

?The point

P (5, 3)

is rotated 90° clockwise about

(2, 1)

. Find the

y

-coordinate of its image

P^{'}

.

Worked example Geometry · transformations

Two reflections make one move (Extended)

Question

Triangle T has vertices

(2, 1)

,

(3, 1)

,

(2, 3)

. Reflect T in

x = 1

to give

T_{1}

, then reflect

T_{1}

in

x = 4

to give

T_{2}

. Describe fully the single transformation that maps T onto

T_{2}

.

1
Reflect in $x = 1$ x = 1: $T_{1}$ T₁ is $(0, 1)$ (0, 1), $(- 1, 1)$ (−1, 1), $(0, 3)$ (0, 3)
2
Reflect in $x = 4$ x = 4: $T_{2}$ T₂ is $(8, 1)$ (8, 1), $(9, 1)$ (9, 1), $(8, 3)$ (8, 3)
3
Compare T with $T_{2}$ T₂: every vertex has moved 6 right and 0 up, and $T_{2}$ T₂ is the same way round as T.
4
Invariance: each reflection fixes the points on its own mirror, but the combined move fixes no point at all.

Answer

Translation by

(60)

.

⚠️ Watch out

the shift is twice the gap between the mirrors (

2 \times 3 = 6

), and doing the reflections in the other order gives

(- 6 0)

.

Mission check

?The nail problem: a picture turns 90° clockwise about a nail. Corner

A (5, 6)

moves to

A^{'} (7, 0)

and corner

B (1, 6)

moves to

B^{'} (7, 4)

. Where is the nail?

Mission 2

A drone flies 3 east, 4 north, then 6 east, 8 north. Straight line?

Make a guess: A drone flies 3 east, 4 north, then 6 east, 8 north. Straight line?

Geometry · vectors

Column vectors

A vector has a size and a direction. The column vector $(x y)$ ( x ; y ) means $x$ x across (right is +) and $y$ y up (up is +).
Vectors are printed in bold, $a$ a, or written $A B$ AB→ (from A to B). By hand, underline the letter.
A translation is described by a column vector: $(3 - 4)$ ( 3 ; −4 ) moves $(1, 2)$ (1, 2) to $(4, - 2)$ (4, −2).
Equal vectors have the same size and direction, wherever they are drawn.
$B A = - A B$ BA→ = −AB→: same size, opposite direction.

Geometry · vectors

Adding, subtracting and scaling (Extended)

Add or subtract matching components: $(2 - 1) + (- 3 4) = (- 1 3)$ ( 2 ; −1 ) + ( −3 ; 4 ) = ( −1 ; 3 )
On a diagram, $a + b$ a + b means "follow a, then b": draw b starting where a ends. The resultant joins the start to the finish.
$a - b = a + (- b)$ a − b = a + (−b): follow a, then b backwards.
Multiplying by a scalar (a number) multiplies both components: $3 (2 - 1) = (6 - 3)$ 3( 2 ; −1 ) = ( 6 ; −3 )

Worked example Geometry · vectors

Combining column vectors (Extended)

Question

a = (2 - 1)

and

b = (- 3 4)

. Find

3 a - 2 b

as a column vector.

1
Scale a: $3 a = (6 - 3)$ 3a = ( 6 ; −3 )
2
Scale b: $2 b = (- 6 8)$ 2b = ( −6 ; 8 )
3
Subtract the matching components: $(6 - (- 6) - 3 - 8)$ ( 6 − (−6) ; −3 − 8 )

Answer

3 a - 2 b = (12 - 11)

⚠️ Watch out

6 - (- 6) = 12

. Adding

2 b

by mistake gives

(05)

.

Calculate

?

a = (4 - 2)

and

b = (- 1 3)

. Find the bottom component of

2 a - 3 b

.

Geometry · vectors

Parallel vectors (Extended)

$k a$ ka is parallel to $a$ a and $k$ k times as long. If $k > 0$ k > 0 it points the same way; if $k < 0$ k < 0, the opposite way.
Test: is one vector a number times the other? $(- 6 3) = - 3 (2 - 1)$ ( −6 ; 3 ) = −3( 2 ; −1 ), so these are parallel (opposite directions).
$(23)$ ( 2 ; 3 ) and $(32)$ ( 3 ; 2 ) are not parallel: no single number turns 2 into 3 and 3 into 2.
Two journeys line up exactly when the second is a positive multiple of the first.

Sort it

?Which vectors are parallel to

(2 - 3)

?

Tap a statement, then tap the group it belongs in. Then press Check.

Worked example Geometry · vectors

Magnitude of a vector (Extended)

Question

Find the magnitude

(5 - 12)

.

1
The magnitude is the length of the arrow, by Pythagoras: $∣ a ∣ = x^{2} + y^{2}$ |a| = √(x² + y²)
2
Square the components: $5^{2} + (- 12)^{2} = 25 + 144 = 169$ 5² + (−12)² = 25 + 144 = 169
3
Square-root: $169 = 13$ √169 = 13

Answer

13

⚠️ Watch out

(- 12)^{2} = + 144

, and adding the components (

5 + 12 = 17

) is not the length.

Geometry · vectors

Vectors in geometry (Extended)

Follow a route along known vectors: $A B = A O + OB = - a + b$ AB→ = AO→ + OB→ = −a + b
Midpoint M of AB: $OM = a + \frac{1}{2} (b - a) = \frac{1}{2} a + \frac{1}{2} b$ OM→ = a + 1/2(b − a) = 1/2a + 1/2b
Show lines are parallel: one vector is a multiple of the other.
Show X, Y, Z are collinear: $Y Z = k X Y$ YZ→ = kXY→ and they share point Y. Example: $X Y = 2 a + b$ XY→ = 2a + b, $Y Z = 6 a + 3 b = 3 X Y$ YZ→ = 6a + 3b = 3XY→.
$OP$ OP→ is the position vector of P.

Calculate

?Exam-style (Extended). OAB is a triangle with

O A = a

and

OB = b

. P lies on AB with

A P : PB = 1 : 3

. Write

OP = m a + n b

. Find

m

as a decimal.

Mission check

?A drone flies

(34)

and then

(68)

. Did it fly in a straight line?

Recap

Big ideas

Describe fully: reflection (mirror line equation), rotation (centre, angle, direction), translation (column vector), enlargement (centre, scale factor).
The centre of a rotation is its invariant point: it is equally far from every point and that point's image.
Two reflections in parallel mirrors make a translation of twice the gap between them.
Column vector $(x y)$ ( x ; y ): $x$ x right, $y$ y up. Add, subtract and scale component by component.
$k a$ ka is parallel to $a$ a; a positive $k$ k keeps the direction.
Magnitude: $∣ a ∣ = x^{2} + y^{2}$ |a| = √(x² + y²).
Extended: route along known vectors; collinear means parallel and sharing a point.

🏆

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⭐⭐⭐

You've worked through Transformations & Vectors for Cambridge IGCSE Maths (0580). 🎉

Mission 1A picture was spun round one nail. Can you find where the nail is?
Mission 2A drone flies 3 east, 4 north, then 6 east, 8 north. Straight line?

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Specification points and sources for this lesson are listed at the bottom of the page.