Vectors & Transformations

### Introduction to Vectors

A vector is a mathematical quantity that has both magnitude (size) and direction. Geometrically, we can represent a vector as a directed line segment. In the context of the Cartesian plane, we use column vectors to describe a movement or displacement.

A column vector is written in the form $\begin{pmatrix} x \\ y \end{pmatrix}$.

* The top number, x, represents the horizontal movement (positive for right, negative for left).

* The bottom number, y, represents the vertical movement (positive for up, negative for down).

For example, the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ represents a movement of 3 units to the right and 2 units down.

### Vector Operations

We can perform arithmetic operations with vectors.

1. Vector Addition: To add two vectors, we add their corresponding components. This is equivalent to performing one movement followed by another.

If a = $\begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and b = $\begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$, then a + b = $\begin{pmatrix} x_1+x_2 \\ y_1+y_2 \end{pmatrix}$.

2. Vector Subtraction: To subtract a vector, we subtract its corresponding components. Subtracting vector **b** is the same as adding the negative of vector **b**.

If a = $\begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and b = $\begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$, then a - b = $\begin{pmatrix} x_1-x_2 \\ y_1-y_2 \end{pmatrix}$.

3. Scalar Multiplication: To multiply a vector by a scalar (a regular number), we multiply each component by that scalar. This scales the magnitude of the vector.

If a = $\begin{pmatrix} x \\ y \end{pmatrix}$ and *k* is a scalar, then *k*a = $\begin{pmatrix} kx \\ ky \end{pmatrix}$.

### Geometric Transformations

A transformation moves a shape, called the object, to a new position. The new shape is called the image. We will study four main types.

#### 1. Translation

A translation is a 'slide' where every point of the object moves by the same amount in the same direction. It is fully described by a single translation vector.

If a point P(x, y) is translated by the vector T = $\begin{pmatrix} a \\ b \end{pmatrix}$ to the image point P'(x', y'), then:

P'(x', y') = (x + a, y + b).

#### 2. Reflection

A reflection is a 'flip' of an object across a fixed line called the line of reflection or mirror line. Every point on the image is the same perpendicular distance from the mirror line as the corresponding point on the object.

Common lines of reflection you must know are:

* The x-axis (equation y = 0)

* The y-axis (equation x = 0)

* The line y = x

* The line y = -x

* Other vertical or horizontal lines like x = c or y = c.

#### 3. Rotation

A rotation is a 'turn' of an object about a fixed point. To fully describe a rotation, you must state three things:

A 90° anticlockwise rotation about the origin (0,0) maps a point (x, y) to (-y, x).

A 180° rotation about the origin maps a point (x, y) to (-x, -y).

#### 4. Enlargement

An enlargement changes the size of an object. To fully describe an enlargement, you must state two things:

The distance of any image point from the centre of enlargement is *k* times the distance of the corresponding object point from the centre.

* If k > 1, the image is larger than the object.

* If 0 < k < 1, the image is smaller than the object.

* If k is negative, the image is formed on the opposite side of the centre of enlargement and is inverted (upside down). For example, a scale factor of -2 means the image is twice the size and inverted.