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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

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.

Answer

Solution:

It is verified that each transformation has the indicated effect.

Explanation

Given:

The vertices of the gray square are:

[00], [10], [11], and [01].

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=[1001]

Expansion in xdirection

T=[c001]

Shear in xdirection

T=[1c01]

Calculation:

The data matrix is given by

D=[01100011]

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

Algebra and Trigonometry (MindTap Course List), Chapter 11.FOM, Problem 1P , additional homework tip  1

Figure 1

Multiply the D matrix by appropriate transformation matrix.

(1) Reflection in xaxis: Apply reflection transformation

T=[1001]

The following matrix is obtained

TD=[1001][01100011]=[01100011]

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

Algebra and Trigonometry (MindTap Course List), Chapter 11.FOM, Problem 1P , additional homework tip  2

Figure 2

(2) Expansion in xdirection

T=[c001]

Substitute 2 for c in above matrix.

T=[2001]

The following matrix is obtained

TD=[2001][01100011]=[02200011]

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

Algebra and Trigonometry (MindTap Course List), Chapter 11.FOM, Problem 1P , additional homework tip  3

Figure 3

(3) Shear in xdirection

T=[1c01]

Substitute 1 for c in above matrix.

T=[1101]

The following matrix is obtained

TD=[1101][01100011]=[01210011]

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

Algebra and Trigonometry (MindTap Course List), Chapter 11.FOM, Problem 1P , additional homework tip  4

Figure 4

Conclusion:

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

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