Vectors and transformations: translations, rotations, reflections, enlargements, combinations

Vectors – The Building Blocks

Think of a vector as a directed arrow that tells you how to move from one point to another.

In coordinates, a vector is written as $\mathbf{v} = \langle a, b \rangle$ where $a$ is the horizontal change and $b$ the vertical change.

Example: To go from point $A(2,3)$ to point $B(5,7)$, the vector is $\mathbf{AB} = \langle 3, 4 \rangle$.

Vectors can be added or subtracted by adding or subtracting their components:

• $\langle a, b \rangle + \langle c, d \rangle = \langle a+c,\; b+d \rangle$
• $\langle a, b \rangle - \langle c, d \rangle = \langle a-c,\; b-d \rangle$

Scaling a vector by a number $k$ multiplies both components: $k\mathbf{v} = \langle ka,\; kb \rangle$.

Translations – Moving Things Around

Imagine sliding a pizza across the table. The pizza keeps its shape and orientation, just its position changes.

A translation is defined by a translation vector $\mathbf{t} = \langle p, q \rangle$. Every point $P(x, y)$ moves to $P'(x+p,\; y+q)$.

1. Choose the translation vector.
2. Add the vector components to each point’s coordinates.

Example: Translate point $C(4,2)$ by $\mathbf{t} = \langle -1, 3 \rangle$ → $C'(3,5)$.

Exam tip: Remember that a translation does not change distances or angles.

Rotations – Spinning Around

Think of a merry‑go‑round. Every point moves in a circle around a fixed centre.

To rotate a point $P(x, y)$ about the origin by an angle $\theta$ (counter‑clockwise), use the rotation matrix:

Apply it: $\begin{pmatrix}x'\\ y'\end{pmatrix} = \begin{pmatrix}\cos \theta & -\sin \theta\\ \sin \theta & \cos \theta\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}$.

1. Compute $\cos \theta$ and $\sin \theta$ (use a calculator if needed).
2. Multiply the matrix by the coordinate column.

Example: Rotate point $D(1,0)$ by $90^\circ$ → $D'(0,1)$.

Exam tip: Rotations preserve distances and angles. If you’re given a rotated shape, you can often find the original by rotating the opposite way.

Reflections – Mirror Images

Picture a funhouse mirror. Points are flipped across a line.

Reflection across the line $y = x$ swaps the coordinates: $P(x, y) \rightarrow P'(y, x)$.

Reflection across the x‑axis: $P(x, y) \rightarrow P'(x, -y)$.

Reflection across the y‑axis: $P(x, y) \rightarrow P'(-x, y)$.

1. Identify the axis of reflection.
2. Apply the appropriate coordinate change.

Exam tip: Reflections reverse orientation (clockwise ↔ counter‑clockwise) but keep side lengths unchanged.

Enlargements – Scaling Up or Down

Think of zooming in or out on a photo. The shape stays the same but its size changes.

An enlargement with centre at the origin and scale factor $k$ maps $P(x, y)$ to $P'(kx, ky)$.

1. Choose the scale factor $k$ (e.g., $k=2$ doubles size).
2. Multiply both coordinates by $k$.

Example: Enlarge point $E(3, -1)$ by a factor of $3$ → $E'(9, -3)$.

Exam tip: If $k>1$ the figure gets larger; if $0 it shrinks.