Modeling Modeling Motion Motion

with with MatricesMatrices

Section 2-4

Before finishing this section you should be able to:

• Use matrices to determine the coordinates of polygons under a given transformation

Remember: Your textbook is your friend! This presentation is just a supplement to the text. BEFORE you view this, make sure you read this section in your textbook and look at all the great examples that are also worked there for you.

Matrices can be used to describe transformations. Some transformations we will work with in this lesson are translations (slides), reflections (flips), rotations (turns), and dilations (enlargements or reductions).

We can use matrices by representing the coordinates of the vertices of an object in a vertex matrix. This is simply done by

making the first row of the matrix

the x-coordinates and the second row the y-coordinates.

Suppose quadrilateral RSTU with vertices R(3,2), S(7,4), T(9,8), and U(5,6) is translated 2 units right and 3 units down.

Represent the vertices of the quadrilateral as a matrix. (vertex matrix)

Each column of the matrix is one of the vertices of the quadrilateral. The first row is all of the x coordinates and the second row is all of the y coordinates.

3 7 9 5

2 4 8 6

2 2 2 2

3 3 3 3

We want to translate the quadrilateral 2 units to the right so all of the x coordinates in the translation matrix are 2 and we want to translate the quadrilateral 3 unit s down so all of the y coordinates in the translation matrix are –3.

Write the translation matrix.

3 7 9 5 2 2 2 2 5 9 11 7

2 4 8 6 3 3 3 3 1 1 5 3

Add the vertex matrix and the translation matrix.

Use the translation matrix to find the vertices of R’S’T’U’, the translated image of the quadrilateral.

The original quadrilateral before the transformation is the pre-image. The resulting quadrilateral is the image.

A reflection matrix is a 2 x 2 matrix multiplied on the left side of the vertex matrix.

5 3 0

2 8 4

1 0 5 3 0 5 3 0

0 1 2 8 4 2 8 4

Triangle NPQ is represented by the matrix

Find the coordinates of the triangle after a reflection over the y-axis.

N’(5,2), P’(-3,8), Q’(0,-4)

Similarly, a rotation matrix is also a 2 x 2 matrix multiplied on the left side of the vertex matrix.

All of the above transformations have maintained the size and shape of the figure.

However, a dilation changes the size of the figure. The dilated figure is similar to the original figure.

Dilations using the origin as the center of projection can be achieved by multiplying the vertex matrix by the scale factor needed for the dilation.

Trapezoid WXYZ has vertices W(2,1), X(1,-2), Y(-1,-2), Z(-2,1)Find the coordinates of dilated trapezoid W’X’Y’Z’ for a scale factor of 2.5.

2 1 1 2 5 2.5 2.5 52.5

1 2 2 1 2.5 5 5 2.5

More Examples

Suppose triangle ABC with vertices A(-3, 1), B(1, 4), and C(-1, -2) is translated 2 units right and 3 units down.

a. Represent the vertices of the triangle as a matrix.

The matrix representing the coordinates of the vertices of triangle ABC will be a 2 3 matrix.

3 1 1

1 4 2

A B C

x - coordinate

y - coordinate

b. Write the translation matrix.

The translation matrix is 2 2 2

3 3 3

.

c. Use the translation matrix to find the vertices of A’B’C’, the translated image of the triangle.

Add the two matrices.

3 1 1 2 2 2 1 3 1

1 4 2 3 3 3 2 1 5

A B' C

d. Graph triangle ABC and its image.

Graph the points representedby the resulting matrix.

ANIMATION To create an image that appears to be reflected in a mirror, an animator will use a matrix to reflect an image over the y-axis. Use a reflection matrix to find the coordinates of the vertices of a parallelogram reflected in a mirror (the y-axis) if the coordinates of the points connected to create the parallelogram are (-4, 3), (2, 3), (-1, -2),and (-7, -2).

First, write the vertex matrix for the points used to define the parallelogram.

4 2 1 7

3 3 2 2

Multiply by the y-axis reflection matrix.

1 0 4 2 1 7 4 2 1 7

0 1 3 3 2 2 3 3 2 2

The vertices used to define the reflection are (4, 3), (-2, 3), (1, -2), and (7, -2).

ANIMATION Suppose a figure is animated to spin around a certain point. Numerous rotation images would be necessary to make a smooth movement image. If the image has key points at (2, 1), (4, 2), and (3, 5) and the rotation is about the origin, find the location of these points at the 90, 180, and 270 counterclockwise rotations.

First write the vertex matrix. Then multiply it by each rotation matrix.The vertex matrix is

2 4 3

1 2 5

.0 1 2 4 3 1 2 5

1 0 1 2 5 2 4 3

1 0 2 4 3 2 4 3

0 1 1 2 5 1 2 5

0 1 2 4 3 1 2 5

1 0 1 2 5 2 4 3

Rot90

Rot180

Rot270

A trapezoid has vertices at A(2, 3), B(-3, 3), C(-5, -2), and D(3, -2). Find the coordinates of the dilated trapezoid ABCD for a scale factor of 2. Describe the dilation.

First write the coordinates of the vertices as a matrix. Then do a scalar multiplication using the scale factor. 2 3 5 3 4 6 10 6

23 3 2 2 6 6 4 4

The vertices of the image are A(4, 6),B(-6, 6), C(-10, -4), and D(6,-4).

The image has sides that are twicethe length of the original figure.

HELPFUL WEBSITESAnimation with Matrices:

http://www.ltcconline.net/greenl/java/Other/MatrixAnimation/MatrixAnimation.htmlhttp://www.willamette.edu/~gorr/classes/GeneralGraphics/Transforms/transforms2d.htm

2-4 Self-Check Quiz: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-860861-9&chapter=2&lesson=4&quizType=1&headerFile=4&state=