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TRANSFORMATIONS + VECTORS LAB
Learn it by playing it
Four short missions on one live grid. Slide a triangle by a column vector, then chain two arrows into their sum and measure the journey. When a mission glows green, read why it worked.
1234
Mission 1 of 4
Slide onto the ghost
Two dials slide the solid triangle across the grid: the top dial moves it right (positive) or left (negative), the bottom dial moves it up (positive) or down (negative). Land it exactly on the dashed ghost.
move = (0 over 0) · ghost needs (6 over 4)
Before you touch the dials: count the squares. Which column vector slides the triangle onto the ghost?
Currently move = (0 over 0) - the ghost wants (6 over 4).
Study notes
Translations: slides by column vector C7.1 / E7.1
A translation slides every point of a shape the same distance in the same direction. Object and image are congruent: same size, same shape, same way round.
The instruction is a column vector (x over y): the top number steps right (left when negative), the bottom number steps up (down when negative).
To describe a translation fully, name the transformation and give the vector: 'translation by the column vector (5 over -2)'. 'Moved right 5 and down 2' is the same idea in words, but the vector is the expected form.
Finding the vector from a diagram: pick ONE matching corner on object and image, and count the squares right/left then up/down. Every corner gives the same answer - use that as a check.
Drawing the image: move each vertex by the vector, then join the new vertices. The shape does not turn, flip or resize - if your image does, it is not a translation.
Reflections C7.1 / E7.1
A reflection flips a shape over a mirror line. Every point and its image are the same perpendicular distance from the line, on opposite sides.
Core mirrors are vertical or horizontal lines, like x = 2 or y = -1. Extended adds any straight line, with y = x and y = -x the favourites.
Describing a reflection fully needs two things: the word 'reflection' and the equation of the mirror line.
Reflecting in the y-axis sends (3, 2) to (-3, 2) - the x-coordinate flips sign. Reflecting in the x-axis sends (3, 2) to (3, -2) - the y-coordinate flips.
Reflecting in y = x swaps the coordinates: (3, 2) goes to (2, 3). Reflecting in y = -x swaps AND negates both: (3, 2) goes to (-2, -3).
The image is congruent but turned over - lettering on the shape reads backwards, which is how you tell a reflection from a rotation on a diagram.
Rotations C7.1 / E7.1
A rotation turns a shape about a fixed point called the centre of rotation. Object and image are congruent, and the centre itself never moves.
Core turns are multiples of 90° about the origin, a vertex, or the midpoint of an edge; Extended allows any centre with the same 90° steps.
A full description needs three facts: 'rotation', the angle with its direction (90° clockwise, 90° anticlockwise), and the centre. 180° needs no direction - both ways land the same.
Handy results about the origin: 90° clockwise sends (2, 5) to (5, -2); 90° anticlockwise sends (2, 5) to (-5, 2); 180° sends (2, 5) to (-2, -5).
Finding the centre of a rotation: join a point to its image, draw the perpendicular bisector, and repeat for a second pair - the bisectors cross at the centre. (This is the Extended method examiners like to ask for.)
Tracing paper turns the shape physically: pin it at the candidate centre, rotate the stated angle, and see whether the drawing lands on the image.
Enlargements C7.1 / E7.1
An enlargement resizes a shape from a centre of enlargement by a scale factor k: every distance from the centre is multiplied by k, and every length in the shape is multiplied by k.
Core scale factors are positive - whole numbers and fractions. With k = 3 the image is three times as far from the centre; with k = 1/2 it sits halfway to the centre (a smaller image is still called an enlargement).
Extended adds negative scale factors: k = -2 doubles the size AND flips the shape to the opposite side of the centre, upside down relative to the object.
With centre the origin, coordinates just multiply: (1, 3) under k = 2 lands at (2, 6); under k = -2 at (-2, -6). With any other centre, count from the centre to the point, multiply that journey by k, and step it off again from the centre.
Describing an enlargement fully needs 'enlargement', the scale factor, and the centre. Finding the centre: join matching points with straight lines (rays) - they all cross at the centre.
Areas scale by k², not k: an enlargement with k = 3 makes the image's area nine times the object's. This catches people in perimeter/area follow-ups.
Vectors in two dimensions (Extended) E7.2
A vector carries a size and a direction. You will see it as a column vector (x over y), as an arrowed segment AB, or as a bold lowercase letter a.
Vectors add nose to tail: (3 over 1) + (2 over 1) = (5 over 2) - add the tops, add the bottoms. The sum is the single journey that replaces the two steps.
Subtraction uses the reversed arrow: -b flips every sign of b, so a - b = a + (-b). In arrow notation, BA = -AB: the same road driven backwards.
Multiplying by a scalar scales both components: 3 x (-2 over 4) = (-6 over 12). The arrow gets three times as long (and reverses if the scalar is negative) but keeps its line's direction.
The route rule links journeys: AB + BC = AC - if you can get there in two steps, the direct vector is their sum. This is the engine of every vector-geometry proof.
This whole subtopic is Extended only on the 2025-2027 syllabus; Core candidates meet column vectors inside C7.1 translations but are not examined on vector arithmetic.
Magnitude of a vector (Extended) E7.3
The magnitude |a| is the arrow's length. A column vector is a right triangle in disguise, so |a| = √(x² + y²) - Pythagoras on the components.
Worked example: |(3 over 4)| = √(9 + 16) = √25 = 5. Watch for the classic slips: adding the legs (3 + 4 = 7) or stopping at the square (25).
Negative components square away: |(-6 over 8)| = √(36 + 64) = 10, the same length as (6 over 8) pointing the other way.
Leave answers exact (like √45 = 3√5) unless the question asks for a rounded value; if it does, round only at the end.
This subtopic is Extended only. It pairs naturally with E7.2: find the journey first, then measure it.
Vector geometry (Extended) E7.4
A position vector starts at the origin: point A(1, 2) has position vector a = (1 over 2) - the coordinates themselves written as a column vector.
The key relation: AB = b - a. The journey from A to B is 'B's position vector minus A's', because going A to O then O to B gives -a + b.
Choosing a route is the whole method: to express an unknown vector in terms of a and b, walk from its tail to its head along arrows you already know, adding each step (and negating any arrow you walk backwards).
The midpoint M of AB has position vector m = (a + b) / 2 - the average of the two position vectors.
Parallel vectors are scalar multiples: if p = 2q, the arrows point along the same line. Proving three points collinear: show AB is a multiple of AC - parallel AND sharing point A means one straight line.
Typical proof that a quadrilateral is a parallelogram: show the two opposite-side vectors are equal, for example AB = DC. Equal vectors mean equal length AND equal direction, which is exactly 'parallel and equal'.
This is Extended-only content and the algebra can run long; write each route step on a new line and simplify at the end.
EduTribe study notes - original material aligned to the Cambridge IGCSE Mathematics 0580 syllabus (2025-2027). Automated checks completed; not yet reviewed by an educator.