## Matrices and Transformations II

REFLECTION

Under reflection, the object and its image have the same size and shape. The object and its image are equidistant from the mirror line. In addition, a straight line that joins a point on the object and its image is perpendicular to the mirror line. This is shown in the following diagram.

REFLECTION

Under reflection, the object and its image have the same size and shape. The object and its image are equidistant from the mirror line. In addition, a straight line that joins a point on the object and its image is perpendicular to the mirror line. This is shown in the following diagram.

M is the mirror line

QQ' is perpendicular to line M

QN = NQ'

Triangle PQR and P'Q'R' have the same shape and size.

ROTATION

Under rotation the object and the image have the same size and shape. The process of rotation involves identification of the centre, angle and direction of rotation.

ENLARGEMENT

Under enlargement the size of an object, increases or decreases in a
given ratio. When an object is enlarged:

- Sides of the object are parallel to their corresponding sides on
the image

- Lengths of the sides of an object and lengths of their images
are in the same ratio.

- Angles do not change

In the illustration below, line AB is parallel to A'B'.

Angle ADC = Angle A'D'C'

Angle ABC = Angle A'B'C'

Angle BAD = Angle B'A'D'

Angle BCD = Angle B'C'D'

LINEAR SCALE FACTOR OF ENLARGEMENT

When the linear scale factor is greater than
positive one (+ve 1), the image is larger and on the same side of the
centre of enlargement as the object. When the linear scale factor is
less than +ve 1, but greater than 0 the image is smaller and on the same
side of the centre of enlargement as the object.

When the scale factor of enlargement is negative, the centre of
enlargement lies between the object and the image. In this case, the
image is inverted as shown below

Triangle ABC is enlarged using linear scale factor of -2 centre of enlargement O.

MATRICES AND TRANSFORMATIONS II

Introduction

Life is full of changes both the living and the
non living change. Transformation involves changing something in some
way. In Mathematics the knowledge of transformation helps us to
understand how points and objects change in terms of positions, shape,
size and direction.

**Transformations**

Transformation on a Cartesian plane

The following graphs show various types of
transformations. Describe fully the transformation that maps:

Solutions

1.Reflection in the line y = 0

2.Enlargement scale factor positive 2 centre (0,0)

3.Rotation positive 90O
about the origin (0,0)

4.Translation using translation vector (-7)

Determining the matrix of transformation.

We can obtain a matrix of transformation using
the coordinates of the object and its image.

Example 1

A square with vertices O(0,0), A (1,0), B(1,1)
and C(0,1) is mapped onto the image O'(0,0), A' (-1,0), B'(-1,1) and
C'(0,1).

Solution

Example 2

Plot the coordinates of the image on the
graph below by clicking on the vertices of the image.

The transformation is a reflection in the line x = 0

Example 3

a) Use the graph below to determine the matrix
that maps:

i) PQR onto P'Q'R'

ii) P'Q'R'onto P''Q''R''

iii) P''Q''R'' onto P'''Q'''R'''

b) Determine a single matrix of transformation that maps PQR onto
P''Q''R''

b) A single matrix of transformation that maps
PQR and P''Q''R''can be obtained in two ways, namely:

i) multiplying the matrix that maps PQR onto P'Q'R' by the matrix that
maps P'Q'R' onto P''Q''R''.

Let the matrix that maps PQR onto P'Q'R' be A and the one that maps
P'Q'R' onto P''Q''R'' be B.

The single matrix of transformation is

Matrix and equate to the final image matrix and work out unknowns a, b, c and d. That is

So that a = 2, b = 0, c = 0 and d = -2

Relationship between area scale factor and
determinant of a matrix

Under the transformation of enlargement, the
ratio of the sides of the image and the corresponding sides of the
object is constant. This is referred to as the linear scale factor. The
square of the linear scale factor gives the area scale factor. The
determinant of the matrix of transformation is equal to the area scale
factor.

NB, the area scale factor can also be obtained by dividing the area of the image by the area of the object.

Example

Use the following graph to determine:

i)The area of triangle KLM

ii)The area of triangle K'L'M'

iii)Area scale factor.

iv)The matrix **P** that maps triangle KLM onto K'L'M'.

V)The determinant of matrix **P**

Vi) State the relationship between area and scale factor and the
determinant of **P**.

STRETCH

Stretch is a transformation in which all the
points of an object move at right angles to an invariant (i.e. fixed)
line. The distance moved by a given point from the fixed line is
proportional to its original distance from the line

Example

Use the following graph to;

I.Identify the fixed (invariant) line

II.Determine the matrix p, that maps ABCD onto A'B'C'D'

III.Describe the transformation.

Solution

i. The invariant line is y-axis (x=0)

Note: the matrix of
the vertices that moves is pre-multiplied by the transformation matrix
and the product equated to the matrix of the corresponding image.

The points on the invariant line do not move and therefore should not be
used when working out the transformation matrix.

SHEAR

A shear is a transformation in which all points
of an object move parallel to a fixed or invariant line. The distance
moved by a given point is proportional to the distance from the
invariant line.

Example

Use the following graph to

i.Identify the fixed (invariant) line

ii.Determine the matrix Q that maps ABCD onto A'B'C'D'

iii.Describe the transformation that maps ABCD onto A'B'C'D'

SOLUTION

I.The invariant line is x axis (y=0)

**Note:** the matrix of the
vertices that moves is pre-multiplied by the transformation matrix and
the product equated to the matrix of the corresponding image.

.

Matrices and Transformations II

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