Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
1st Edition
ISBN: 9780078682278
Author: McGraw-Hill, Berchie Holliday
Publisher: Glencoe/McGraw-Hill
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Chapter 2.4, Problem 25E

a.

To determine

Find the method to derive matrix for the reflection or rotation.

a.

Expert Solution
Check Mark

Answer to Problem 25E

  Rx−xis=[100−1]

Explanation of Solution

Given information:

  A(1,3), B(−2,−1) and C(−1,−3) are the vertices of ΔABC .

  [1−2−1−313] under Rx−axis

Calculation:

Here, we will suppose that the matrix [12−13−1−3] is transformed to [1−2−1−313] under Rx−axis .

Now, let Rx−axis=[abcd]

Thus,

  [abcd][1−2−13−1−3]=[1−2−1−313][a+3b−2a−b−a−3bc+3d−2c−d−c−3d]=[1−2−1−313]

Now, when two matrices are equal then their corresponding elements are equal.

Now, we will equate the first row of both matrices:

  a+3b=1−2a−b=−2−a−3b=−1∴a=1, b=0

Now, we will equate the second row of both matrices:

  c+3d=−3−2c−d=1−c−3d=3∴c=0, d=−1

Now we get:

  Rx−xis=[100−1]

Hence, the required answer is

  Rx−xis=[100−1] .

b.

To determine

Find the method to derive matrix for the reflection or rotation.

b.

Expert Solution
Check Mark

Answer to Problem 25E

  Rx−xis=[−1001]

Explanation of Solution

Given information:

  A(1,3), B(−2,−1) and C(−1,−3) are the vertices of ΔABC .

  [−1213−1−3] under Ry−axis

Calculation:

Here, we will suppose that the matrix [1−2−13−1−3] is transformed to [−1213−1−3] under Rx−axis .

Now, let Rx−axis=[abcd]

Thus,

  [abcd][1−2−13−1−3]=[−1213−1−3][a+3b−2a−b−a−3bc+3d−2c−d−c−3d]=[−1213−1−3]

Now, when two matrices are equal then their corresponding elements are equal.

Now, we will equate the first row of both matrices:

  a+3b=−1−2a−b=2−a−3b=1∴a=−1, b=0

Now, we will equate the second row of both matrices:

  c+3d=3−2c−d=−1−c−3d=−3∴c=0, d=1

Now we get:

  Rx−xis=[−1001]

Hence, the required answer is Rx−xis=[−1001] .

c.

To determine

Find the method to derive matrix for the reflection or rotation.

c.

Expert Solution
Check Mark

Answer to Problem 25E

  Rx−xis=[0110]

Explanation of Solution

Given information:

  A(1,3), B(−2,−1) and C(−1,−3) are the vertices of ΔABC .

  [3−1−31−2−1] under Ry−r

Calculation:

Here, we will suppose that the matrix [1−2−13−1−3] is transformed to [3−1−31−2−1] under Rx−axis .

Now, let Rx−axis=[abcd]

Thus,

  [abcd][1−2−13−1−3]=[3−1−31−2−1][a+3b−2a−b−a−3bc+3d−2c−d−c−3d]=[3−1−31−2−1]

Now, when two matrices are equal then their corresponding elements are equal.

Now, we will equate the first row of both matrices:

  a+3b=3−2a−b=−1−a−3b=−3∴a=0, b=1

Now, we will equate the second row of both matrices:

  c+3d=1−2c−d=−2−c−3d=−1∴c=1, d=0

Now we get:

  Rx−xis=[0110]

Hence, the required answer is Rx−xis=[0110] .

d.

To determine

Find the method to derive matrix for the reflection or rotation.

d.

Expert Solution
Check Mark

Answer to Problem 25E

  R90°=[0−110]

Explanation of Solution

Given information:

  A(1,3), B(−2,−1) and C(−1,−3) are the vertices of ΔABC .

  [−3131−2−1] under Rot90

Calculation:

Here, we will suppose that the matrix [1−2−13−1−3] is transformed to [−3131−2−1] under R90° .

Now, let R90°=[abcd]

Thus,

  [abcd][1−2−13−1−3]=[−3131−2−1][a+3b−2a−b−a−3bc+3d−2c−d−c−3d]=[−3131−2−1]

Now, when two matrices are equal then their corresponding elements are equal.

Now, we will equate the first row of both matrices:

  a+3b=−3−2a−b=1−a−3b=3∴a=0, b=−1

Now, we will equate the second row of both matrices:

  c+3d=−1−2c−d=−2−c−3d=−1∴c=1, d=0

Now we get:

  R90°=[0−110]

Hence, the required answer is R90°=[0−110] .

e.

To determine

Find the method to derive matrix for the reflection or rotation.

e.

Expert Solution
Check Mark

Answer to Problem 25E

  R180°=[−100−1]

Explanation of Solution

Given information:

  A(1,3), B(−2,−1) and C(−1,−3) are the vertices of ΔABC .

  [−121−313] under Rot180

Calculation:

Here, we will suppose that the matrix [1−2−13−1−3] is transformed to [−121−313] under R180° .

Now, let R180°=[abcd]

Thus,

  [abcd][1−2−13−1−3]=[−121−313][a+3b−2a−b−a−3bc+3d−2c−d−c−3d]=[−121−313]

Now, when two matrices are equal then their corresponding elements are equal.

Now, we will equate the first row of both matrices:

  a+3b=−1−2a−b=2−a−3b=1∴a=−1, b=0

Now, we will equate the second row of both matrices:

  c+3d=−3−2c−d=1−c−3d=3∴c=0, d=−1

Now we get:

  R180°=[−100−1]

Hence, the required answer is R180°=[−100−1] .

f.

To determine

Find the method to derive matrix for the reflection or rotation.

f.

Expert Solution
Check Mark

Answer to Problem 25E

  R270°=[01−10]

Explanation of Solution

Given information:

  A(1,3), B(−2,−1) and C(−1,−3) are the vertices of ΔABC .

  [3−1−3−121] under Rot270

Calculation:

Here, we will suppose that the matrix [1−2−13−1−3] is transformed to [1−2−1−313] under R270° .

Now, let R270°=[abcd]

Thus,

  [abcd][1−2−13−1−3]=[3−1−3−121][a+3b−2a−b−a−3bc+3d−2c−d−c−3d]=[3−1−3−121]

Now, when two matrices are equal then their corresponding elements are equal.

Now, we will equate the first row of both matrices:

  a+3b=3−2a−b=−1−a−3b=−3∴a=0, b=1

Now, we will equate the second row of both matrices:

  c+3d=−1−2c−d=2−c−3d=1∴c=−1, d=0

Now we get:

  R270°=[01−10]

Hence, the required answer is R270°=[01−10] .

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Chapter 2.4, Problem 24E
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Chapter 2.4, Problem 26E

Chapter 2 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

Ch. 2.1 - Prob. 1CFUCh. 2.1 - Prob. 2CFUCh. 2.1 - Prob. 3CFUCh. 2.1 - Prob. 4CFUCh. 2.1 - Prob. 5CFUCh. 2.1 - Prob. 6CFUCh. 2.1 - Prob. 7CFUCh. 2.1 - Prob. 8CFUCh. 2.1 - Prob. 9CFUCh. 2.1 - Prob. 10CFU
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