Explanation Given: Let Z , Z ′ , and R be the matrices given as, Z = [ x y ] Z ′ = [ X Y ] R = [ cos ϕ − sin ϕ sin ϕ cos ϕ ] Approach: Suppose the x and the y axes in a coordinate plane are rotated through the acute angle ϕ to produce the X and the Y axes. Then the coordinates ( x , y ) and ( X , Y ) of a point in the x y and the X Y planes are related as follows, x = ( X cos ϕ − Y sin ϕ ) y = ( X sin ϕ + Y cos ϕ ) X = ( x cos ϕ + y sin ϕ ) Y = ( − x sin ϕ + y cos ϕ ) Calculation: Multiply the matrices R and Z ′ as follows, R Z ′ = [ cos ϕ − sin ϕ sin ϕ cos ϕ ] [ X Y ] R Z ′ = [ X cos ϕ − Y sin ϕ X sin ϕ + Y cos ϕ ] … ( 1 ) Use x for ( X cos ϕ − Y sin ϕ ) and y for ( X sin ϕ + Y cos ϕ ) in equation ( 1 ) as, R Z ′ = [ X cos ϕ − Y sin ϕ X sin ϕ + Y cos ϕ ] = [ x y ] = Z Therefore, Z = R Z ′ . The inverse of the matrix R is given as, R − 1 = 1 | R | ( Adjoint of R ) … ( 2 ) The determinant of R is given as, | R | = cos 2 ϕ + sin 2 ϕ = 1 The Adjoint of R is given as, To determine (b) The rotation of the product matrix R 1 R 2 .

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN: 9781305071742

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 12.5, Problem 37E

To determine

(a)

To show:

That the Rotation of Axes Formulas is given as,

Z=RZ′ and Z′=R−1Z.

To determine

(b)

The rotation of the product matrix R1R2.

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Chapter 12.5, Problem 36E

Chapter 12.5, Problem 38E

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(a) To show: That the Rotation of Axes Formulas is given as, Z = R Z ′ and Z ′ = R − 1 Z .

Solution Summary: The author explains that the Rotation of Axes Formulas is given as, Z=RZprime and

Videos

(a)

To show:

That the Rotation of Axes Formulas is given as,

Z=RZ′ and Z′=R−1Z.

(b)

The rotation of the product matrix R1R2.

Want to see the full answer?

Chapter 12 Solutions

(a) To show: That the Rotation of Axes Formulas is given as, Z = R Z ′ and Z ′ = R − 1 Z .

Solution Summary: The author explains that the Rotation of Axes Formulas is given as, Z=RZprime and

Videos

(a) To show: That the Rotation of Axes Formulas is given as, Z=RZ′ and Z′=R−1Z.

(b) The rotation of the product matrix R1R2.

Want to see the full answer?

Chapter 12 Solutions

(a)

To show:

That the Rotation of Axes Formulas is given as,

Z=RZ′ and Z′=R−1Z.

(b)

The rotation of the product matrix R1R2.