Geometry - Vectors and Transformations

1. Translations

A translation is a transformation that moves every point of a shape by the same distance in the same direction. This is often described as sliding the shape. The translation vector represents the displacement.

Translation Vector

A translation can be represented by a vector $\mathbf{v} = (x, y)$, where $x$ is the horizontal displacement and $y$ is the vertical displacement.

To translate a point $P(a, b)$ by the vector $\mathbf{v} = (x, y)$, we add the components of the vectors:

New position of $P' = (a + x, b + y)$

Example

Translate the point $P(2, 3)$ by the vector $\mathbf{v} = (4, -1)$.

New position of $P' = (2 + 4, 3 + (-1)) = (6, 2)$

Table Summary

Concept	Description
Definition	Sliding a shape by a constant vector.
Representation	Translation vector $\mathbf{v} = (x, y)$
Transformation Rule	$P(a, b) \rightarrow P'(a + x, b + y)$

2. Rotations

A rotation is a transformation that turns a shape about a fixed point called the centre of rotation. A positive angle indicates a counter-clockwise rotation, and a negative angle indicates a clockwise rotation.

Rotation Centre

The centre of rotation is the fixed point about which the shape is rotated.

Angle of Rotation

The angle of rotation is the amount the shape is turned.

Example

Rotate the point $P(1, 1)$ by $90^\circ$ about the origin.

Using the rotation formula: $x' = x \cos(\theta) - y \sin(\theta)$, $y' = x \sin(\theta) + y \cos(\theta)$

$x' = 1 \times \cos(90^\circ) - 1 \times \sin(90^\circ) = 0 - 1 = -1$

$y' = 1 \times \sin(90^\circ) + 1 \times \cos(90^\circ) = 1 + 0 = 1$

New position of $P' = (-1, 1)$

Table Summary

Concept	Description
Definition	Turning a shape about a fixed point.
Centre of Rotation	Fixed point about which the shape is rotated.
Angle of Rotation	Amount the shape is turned (positive for counter-clockwise, negative for clockwise).
Rotation Rule (about origin)	$P(x, y) \rightarrow P'(x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta))$

3. Reflections

A reflection is a transformation that flips a shape across a line called the line of reflection. Each point on the shape is reflected to a corresponding point on the other side of the line.

Line of Reflection

The line of reflection is the line across which the shape is flipped.

Example

Reflect the point $P(2, 3)$ across the line $y = x$

The reflection of a point $(a, b)$ across the line $y = x$ is $(b, a)$

New position of $P' = (3, 2)$

Table Summary

Concept	Description
Definition	Flipping a shape across a line.
Line of Reflection	The line across which the shape is flipped.
Reflection Rule (across $y = x$)	$P(a, b) \rightarrow P'(b, a)$

4. Enlargements

An enlargement is a transformation that changes the size of a shape. It is determined by a centre of enlargement and a scale factor.

Centre of Enlargement

The centre of enlargement is the fixed point from which the shape is enlarged or reduced.

Scale Factor

The scale factor determines how much the shape is enlarged or reduced. A scale factor greater than 1 enlarges the shape, and a scale factor between 0 and 1 reduces the shape.

Example

Enlarge the point $P(1, 1)$ by a scale factor of 2 about the origin.

Using the enlargement formula: $x' = x \times scale \ factor$, $y' = y \times scale \ factor$

$x' = 1 \times 2 = 2$

$y' = 1 \times 2 = 2$

New position of $P' = (2, 2)$

Table Summary

Concept	Description
Definition	Changing the size of a shape.
Centre of Enlargement	Fixed point from which the shape is enlarged or reduced.
Scale Factor	Determines the extent of the size change.
Enlargement Rule (about origin)	$P(x, y) \rightarrow P'(x \times scale \ factor, y \times scale \ factor)$

5. Combinations of Transformations

Transformations can be combined to create more complex transformations. The order in which transformations are applied is important.

For example, to move a shape from point A to point B, you might need to combine a translation, a rotation, and a reflection.

The order of transformations matters. Applying transformations in a different order will result in a different final position.

Example

Consider a shape at point $A(1, 1)$. We want to move it to point $B(4, 5)$ using a combination of translation and rotation.

First, translate $A(1, 1)$ by the vector $(3, 4)$ to a new position $A'(4, 5)$.

Then, rotate $A'(4, 5)$ by $90^\circ$ about the origin to get the final position $B(4, 5)$.

Table Summary

Concept	Description
Definition	Applying multiple transformations sequentially.
Order of Transformations	The sequence in which transformations are applied is crucial.
Example	Moving a point from one location to another using a combination of translation and rotation.