Transformations and vectors

📌 Key points

1. In these notes a column vector is written across, so (3, −2) means 3 to the right and 2 down.
2. A reflection is described by its mirror line, e.g. the x-axis, y = x or x = 2.
3. A rotation is described by its centre, angle and direction, e.g. 90° clockwise about (0, 0).
4. A translation is described by a column vector. Shapes keep their size and orientation.
5. An enlargement is described by its centre and scale factor. Lengths are multiplied by the scale factor; a scale factor between 0 and 1 makes the shape smaller.
6. A negative scale factor puts the image on the other side of the centre, upside down.
7. About the origin: rotating 90° clockwise maps (x, y) to (y, −x), rotating 180° maps (x, y) to (−x, −y), and reflecting in y = x maps (x, y) to (y, x).
8. Add vectors by adding their components; 2a is twice as long as a in the same direction. The magnitude of (x, y) is √(x² + y²).
9. If OA = a and OB = b, then AB = b − a. Parallel vectors are multiples of each other.

📖 Key terms

Translation

A slide of a shape, described by a column vector.

Reflection

A flip of a shape in a mirror line.

Rotation

A turn of a shape about a centre by an angle in a given direction.

Enlargement

A change of size from a centre by a scale factor.

Scale factor

The number every length is multiplied by in an enlargement.

Vector

A quantity with both size (magnitude) and direction.

Magnitude

The length of a vector.

Position vector

The vector from the origin O to a point.