Transformations and vectors
Maths · GCSE · The School
Four moves, four descriptions
Translation: give the vector. Reflection: give the mirror LINE (an equation, e.g. y = x). Rotation: give angle, direction and CENTRE. Enlargement: give scale factor and CENTRE. Marks are lost for incomplete descriptions more than for wrong ones — "rotation 90°" without a centre is not an answer.
Vectors as instructions
The column vector (3, −2) means 3 right, 2 down. Adding vectors means doing one move then the other: (3, −2) + (1, 5) = (4, 3). Multiplying by a scalar stretches without turning: 2(3, −2) = (6, −4), the same direction, twice as far. Negative flips the direction entirely.
Enlargement with fractions and negatives
A scale factor between 0 and 1 makes the shape SMALLER while still being called an enlargement — the word is technical, not descriptive. A negative scale factor puts the image on the opposite side of the centre, upside down. Both appear in exams precisely because they defeat guessing from the picture.
Finding a centre you have not been given
Exams ask you to DESCRIBE a transformation from two drawn shapes, which means finding the centre yourself. For a rotation: join each point to its image, construct the perpendicular bisector of two such lines, and the centre is where the bisectors cross — every point of the shape has turned about it, so it is the one point equidistant from each pair. For an enlargement it is easier: draw a ray through each point and its image and extend them backwards; they all meet at the centre. Worked: triangle A(1, 1), B(3, 1), C(1, 2) maps to A′(3, 3), B′(7, 3), C′(3, 5). Side AB was 2 units and A′B′ is 4, so the scale factor is 2. The ray through A(1, 1) and A′(3, 3) and the ray through B(3, 1) and B′(7, 3) meet at (−1, −1), and that is the centre. State all of it: enlargement, scale factor 2, centre (−1, −1).
Combining two transformations
Applying a transformation and then a second one to the result can often be described as a single transformation of its own — two successive reflections in parallel mirror lines, for instance, are equivalent to one translation perpendicular to both, of twice the distance between them. Exams ask you to carry out both transformations in order on paper first, and only then look at the start and end shapes to name the single transformation that connects them directly.
Using vectors to show points are collinear
If vector AB is a scalar multiple of vector AC, such as AB = 2AC, then A, B and C all lie on the same straight line, because one is just a stretched copy of the other in the same direction. This is how vector questions prove three points are collinear or that a line is parallel to another, without measuring anything — matching direction is enough, and the scalar tells you the ratio of the distances too.