   Chapter 11.FOM, Problem 1P ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# The gray square in Table 1 has the following vertices: [ 0 0 ] , [ 1 0 ] , [ 1 1 ] , [ 0 1 ] Apply each of the three transformations given in Table 1 to these vertices and sketch the result to verify that each transformation has the indicated effect. Use c = 2 in the expansion matrix and c = 1 in the shear matrix.

To determine

To verify:

The three transformations of the gray square have indicated effect.

Solution:

It is verified that each transformation has the indicated effect.

Explanation

Given:

The vertices of the gray square are:

, , , and .

c=2 in expansion matrix and c=1 in the shear matrix.

Approach:

Multiplication of Matrices:

If A is an m×n matrix and B is a n×p matrix, then their product AB is the m×p matrix whose ijelement is the inner product of the ith row of A and jth column of B.

Transformation matrix:

Reflection in xaxis

T=

Expansion in xdirection

T=[c001]

Shear in xdirection

T=[1c01]

Calculation:

The data matrix is given by

D=

Plot the points on the graph and connect them by a line segment. Figure 1

Multiply the D matrix by appropriate transformation matrix.

(1) Reflection in xaxis: Apply reflection transformation

T=

The following matrix is obtained

TD==

Obtain the resulting graph by plotting the points and joining them by a line. Figure 2

(2) Expansion in xdirection

T=[c001]

Substitute 2 for c in above matrix.

T=

The following matrix is obtained

TD==

Obtain the resulting graph by plotting the points and joining them by a line. Figure 3

(3) Shear in xdirection

T=[1c01]

Substitute 1 for c in above matrix.

T=

The following matrix is obtained

TD==

Obtain the resulting graph by plotting the points and joining them by a line. Figure 4

Conclusion:

Hence, it is verified that each transformation has the indicated effect.

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